AFastCompactFiniteDifferenceMethodforFractionalCattaneo...
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Research Article A Fast Compact Finite Difference Method for Fractional Cattaneo Equation Based on Caputo–Fabrizio Derivative Haili Qiao, 1 Zhengguang Liu, 2 and Aijie Cheng 1 1 School of Mathematics, Shandong University, Jinan, Shandong, China 2 School of Mathematics and Statistics, Shandong Normal University, Jinan, Shandong, China Correspondence should be addressed to Aijie Cheng; [email protected] Received 15 October 2019; Accepted 16 January 2020; Published 19 March 2020 Academic Editor: Eric Feulvarch Copyright © 2020 Haili Qiao et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e Cattaneo equations with Caputo–Fabrizio fractional derivative are investigated. A compact finite difference scheme of Crank–Nicolson type is presented and analyzed, which is proved to have temporal accuracy of second order and spatial accuracy of fourth order. Since this derivative is defined with an integral over the whole passed time, conventional direct solvers generally take computational complexity of O(MN 2 ) and require memory of O(MN), with M and N the number of space steps and time steps, respectively. We develop a fast evaluation procedure for the Caputo–Fabrizio fractional de- rivative, by which the computational cost is reduced to O(MN) operations; meanwhile, only O(M) memory is required. In the end, several numerical experiments are carried out to verify the theoretical results and show the applicability of the fast compact difference procedure. 1. Introduction Fractional diffusion equations have become a strong and forceful tool to describe the phenomenon of anomalous diffusion, and more research works have been obtained in the last decades [1–6]. However, since the fractional de- rivative is nonlocal and has weak singularity, it is impossible to solve fractional diffusion equations analytically in most cases. Instead, seeking numerical solutions is becoming an indispensable tool for research work about fractional equations. Different from the traditional derivative of the integer order, the fractional derivative depends on the total in- formation in the correlative region, and this is the so- called nonlocal properties. Just because of this, it con- sumes computational time extremely to solve fractional equations. We hope to develop effective numerical schemes, which not only have better stability and higher accuracy but also require less storage memory and save computational cost. About stability and convergence analysis of the nu- merical schemes for fractional equations, the readers can refers to [7, 8] for spatial fractional order equation, [9–18] for temporal fractional diffusion equations, and [19–22] for space-time-fractional equations. About the complexity, i.e., storage requirement and computation cost of an algorithm, researchers devote themselves to reduce storage requirement and computational time by analyzing the particular struc- ture of coefficient matrices arising from the discretization system or reutilizing the intermediate data skillfully. We call these algorithms fast methods, including fast finite difference methods [23–28], fast finite element methods [29], and fast collocation methods [30, 31]. A fast method for Caputo fractional derivatives is proposed [32, 33]. Lu et al. [34] presented a fast method of approximate inversion for tri- angular Toeplitz tridiagonal block matrix, which is suc- cessfully applied to the fractional diffusion equations. Comparatively, there is less research work about the fast method for temporal fractional derivative than that for spatial fractional operators. Hindawi Mathematical Problems in Engineering Volume 2020, Article ID 3842946, 17 pages https://doi.org/10.1155/2020/3842946
Transcript of AFastCompactFiniteDifferenceMethodforFractionalCattaneo...
Research ArticleA Fast Compact Finite Difference Method for Fractional CattaneoEquation Based on CaputondashFabrizio Derivative
Haili Qiao1 Zhengguang Liu2 and Aijie Cheng 1
1School of Mathematics Shandong University Jinan Shandong China2School of Mathematics and Statistics Shandong Normal University Jinan Shandong China
Correspondence should be addressed to Aijie Cheng aijiesdueducn
Received 15 October 2019 Accepted 16 January 2020 Published 19 March 2020
Academic Editor Eric Feulvarch
Copyright copy 2020 Haili Qiao et al )is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
)e Cattaneo equations with CaputondashFabrizio fractional derivative are investigated A compact finite difference scheme ofCrankndashNicolson type is presented and analyzed which is proved to have temporal accuracy of second order and spatialaccuracy of fourth order Since this derivative is defined with an integral over the whole passed time conventional directsolvers generally take computational complexity of O(MN2) and require memory of O(MN) with M and N the number ofspace steps and time steps respectively We develop a fast evaluation procedure for the CaputondashFabrizio fractional de-rivative by which the computational cost is reduced to O(MN) operations meanwhile only O(M) memory is required Inthe end several numerical experiments are carried out to verify the theoretical results and show the applicability of the fastcompact difference procedure
1 Introduction
Fractional diffusion equations have become a strong andforceful tool to describe the phenomenon of anomalousdiffusion and more research works have been obtained inthe last decades [1ndash6] However since the fractional de-rivative is nonlocal and has weak singularity it is impossibleto solve fractional diffusion equations analytically in mostcases Instead seeking numerical solutions is becoming anindispensable tool for research work about fractionalequations
Different from the traditional derivative of the integerorder the fractional derivative depends on the total in-formation in the correlative region and this is the so-called nonlocal properties Just because of this it con-sumes computational time extremely to solve fractionalequations We hope to develop effective numericalschemes which not only have better stability and higheraccuracy but also require less storage memory and savecomputational cost
About stability and convergence analysis of the nu-merical schemes for fractional equations the readers canrefers to [7 8] for spatial fractional order equation [9ndash18]for temporal fractional diffusion equations and [19ndash22] forspace-time-fractional equations About the complexity iestorage requirement and computation cost of an algorithmresearchers devote themselves to reduce storage requirementand computational time by analyzing the particular struc-ture of coefficient matrices arising from the discretizationsystem or reutilizing the intermediate data skillfully We callthese algorithms fast methods including fast finite differencemethods [23ndash28] fast finite element methods [29] and fastcollocation methods [30 31] A fast method for Caputofractional derivatives is proposed [32 33] Lu et al [34]presented a fast method of approximate inversion for tri-angular Toeplitz tridiagonal block matrix which is suc-cessfully applied to the fractional diffusion equationsComparatively there is less research work about the fastmethod for temporal fractional derivative than that forspatial fractional operators
HindawiMathematical Problems in EngineeringVolume 2020 Article ID 3842946 17 pageshttpsdoiorg10115520203842946
A time-fractional Cattaneo equation is considered withthe following form
zu(x t)
zt+
zαu(x t)
ztα
z2u(x t)
zx2+ f(x t) (x t) isin Ω times(0 T]
u(x 0) ϕ(x) zuzt
1113868111386811138681113868t0 ψ(x) x isin Ω
u(x t) 0 x isin zΩ tgt 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(1)
where 1lt αlt 2 Ω (a b) for one-dimensional case andΩ (a b) times (c d) for two-dimensional case f(x t) is thesource term ϕ(x) andψ(x) are the prescribed functions forinitial conditions and zαuztα is a new Caputo fractionalderivative without singular kernel which is defined in thenext section
Our purpose is to establish a fast finite difference scheme ofhigh order for this equation We will extract the recursive re-lation between the (k + 1) time step and the k time step of thefinite difference solution )e computational work is signifi-cantly reduced from O(MN2) to O(MN) and the memoryrequirement from O(MN) to O(M) where M and N are thetotal numbers of points for space steps and time steps re-spectively For improving the accuracy a compact finite dif-ference scheme is established)eoretical analysis shows that thefast compact difference scheme has spatial accuracy of fourthorder and temporal accuracy of second order Several numericalexperiments are implemented which verify the effectivenessapplicability and convergence rate of the proposed scheme
)is paper is organized as follows some definitions andnotations are prepared in Section 2 )e compact finitedifference scheme is described and then the stability andconvergence rates are rigorously analyzed for the scheme inSection 3 )e compact finite difference scheme is extendedto the case of two space dimensions in Section 4 Fastevaluation and efficient storage are established skillfully inSection 5 Some numerical experiments are carried out inSection 6 In the end we summarize the major contributionof this paper in Section 7
2 Some Notations and Definitions
We provide some definitions which will be used in thefollowing analysis
First let us recall the usual Caputo fractional derivativeof order α with respect to time variable t which is given by
Ca D
αt u(x t)
1Γ(n minus α)
1113944
t
a
(t minus s)nminus αminus 1
u(n)
(x s)ds
n minus 1lt αlt n
(2)
By replacing the kernel function (t minus s)minus α with the ex-ponential function exp(minus α(t minus s1 minus α)) and 1(Γ(1 minus α))
with M(α)1 minus α Caputo and Fabrizio [35] proposed thefollowing definition of fractional time derivative
Definition 1 (see [35]) Let u(middot t) isin H1(a b) bgt aα isin (0 1) then the new Caputo derivative of the fractionalorder is defined as
Dαt u(x t)
M(α)
1 minus α1113946
t
auprime(x s)exp minus α
t minus s
1 minus α1113876 1113877ds (3)
where M(α) is a normalization function satisfyingM(0) M(1) 1 When the function u does not belong toH1(a b) then this derivative can be reformulated as
Dαt u(x t)
αM(α)
1 minus α1113946
t
a(u(x t) minus u(x s))exp minus α
t minus s
1 minus α1113876 1113877ds
(4)
Definition 2 (see [36]) )e above new Caputo derivative oforder 0lt αlt 1 can also be reformulated as
CF0 D
αt u(x t)
11 minus α
1113944
t
0uprime(x s)exp minus α
t minus s
1 minus α1113876 1113877ds (5)
Definition 3 (see [35 36]) Let u(middot t) isin H1(a b) if nge 1 andα isin [0 1] the fractional time derivative CF
0 Dαt u(x t) of order
(n + α) is defined byCF0 D
n+αt u(x t)
CF0 D
αt
CF0 D
nt u(x t)1113872 1113873 (6)
Particularly for 1lt αlt 2 we have
CF0 D
αt u(x t)
M(α)
2 minus α1113944
t
0uPrime(x s)exp (1 minus α)
t minus s
2 minus α1113876 1113877ds
(7)
Remark 1 )e CaputondashFabrizio (CF) operator was pro-posed with a nonsingular kernel for describing materialheterogeneities that do not exhibit power-law behavior [35]
Remark 2 An open discussion is ongoing about themathematical construction of the CF operator Ortigueiraand Tenreiro Machado [37] indicated that the CF fractionalderivative is neither a fractional operator nor a derivativeoperator the authors of [38 39] showed that this operatorcannot describe dynamic memory and Giusti [40] indicatedthat this operator can be expressed as an infinite linearcombination of RiemannndashLiouville integrals with integer
2 Mathematical Problems in Engineering
powers As responses to these criticisms Atangana andGomez-Aguilar [41] pointed out the need to account for afractional calculus approach without an imposed index lawand with nonsingular kernels Furthermore Hristov [42]indicated that the CF operator is not applicable forexplaining the physical examples in [37 40] instead hesuggested that the CF operator can be used for the analysis ofmaterials that do not follow a power-law behavior )eauthors of [43] believe that models with CF operatorsproduce a better representation of physical behaviors thando integer-order models providing a way to model theintermediate (between elliptic and parabolic or betweenparabolic and hyperbolic) behaviors
To obtain the accuracy of the fourth order in spatialdirections the following lemma is necessary
Lemma 1 (see [44]) Denote θ(s) (1 minus s)3[5 minus 3(1 minus s)2]If f(x) isin C6[a b] h (b minus a)M xi a + ih(0le ileM)then it holds that
112
fPrime ximinus 1( 1113857 + 10fPrime xi( 1113857 + fPrime xi+1( 11138571113858 1113859
1h2 f ximinus 1( 11138571113890 minus 2f xi( 1113857 + f xi+1( 11138571113891
+h4
3601113946
1
0 f(6)
xi minus sh( 1113857 + f(6)
1113960
middot xi + sh( 1113857]θ(s)ds 1le ileM minus 1
(8)
3 Compact Finite Difference Scheme for One-Dimensional Fractional Cattaneo Equation
In order to construct the finite difference schemes the in-terval [a b] is divided into subintervals withxi a + ih (0le ileM) and [0 T] is discretized withtk kΔt (0le kleN) where h (b minus a)M and Δt TNare the spatial grid size and temporal step size respectivelyDenote Ωh xi 0le ileM1113864 1113865 ΩΔt tk 0le kleN1113864 1113865 thenΩh timesΩΔt becomes a discretization of the practical compu-tational domain [a b] times [0 T] )e values of the function uat the grid points are denoted as uk
j u(xj tk) and theapproximate solution at the point (xj tk) is denoted as uk
j Denote Vh v|v (v0 v1 vM)1113864 1113865 We also introduce
the following notations for any mesh function v isin Vh
δxvjminus (12) vj minus vjminus 1
h
δ2xvj vj+1 minus 2vj + vjminus 1
h2 1le jleM minus 1
(9)
and define the average operator
Avj
112
vj+1 + 10vj + vjminus 11113872 1113873 1le jleM minus 1
vj j 0 M
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(10)
It is easy to see that
Avj I +h2
12δ2x1113888 1113889vj (11)
where I is the identical operator We also denoteAv (Av1Av2 AvM) for vector v (v1 v2 vM)and A(u v) (Au v)
For any two gird functions u v isin V0h
v|v isin Vh v0 vM 01113864 1113865 the discrete inner products andnorms are defined as
(u v) h 1113944Mminus 1
j1ujvj
langu vrang h 1113944Mminus 1
j1δxujminus (12)δxvjminus (12)
u22 (u u)
|u|21 langu urang
uinfin max1lejleMminus 1
uj
11138681113868111386811138681113868
11138681113868111386811138681113868
(12)
By summation by parts it is easy to see that
δ2xu v1113872 1113873 minus δxu δxv( 1113857 minus langu vrang u δ2xv1113872 1113873 (13)
For the average operator A define
A(v v)≜ (Av v) v2A (14)
Additionally let VΔt v|v (v0 v1 middot middot middot vN)1113864 1113865 be thespace of grid function defined on ΩΔt For any functionv isin VΔt a difference operator is introduced as follows
δtvk
vk minus vkminus 1
Δt (15)
31 Compact Finite Difference Scheme We will consider thetime-fractional Cattaneo equation equipped with theCaputondashFabrizio derivative Vivas-Cruz et al [43] gave thetheoretical analysis of a model of fluid flow in a reservoirwith the CaputondashFabrizio operator )ey proved that thismodel cannot be used to describe nonlocal processes since itcan be represented as an equivalent differential equationwith a finite number of integer-order derivatives
)e finite difference methods usually lead to stencilsthrough the whole history passed by the solution whichconsume too much computational work In this paper wewill establish a high-order finite difference scheme andpropose a procedure to reduce the computational cost In[43] the authors proposed a recurrence formula of dis-cretized CF operator and obtained an algorithm which canbe considered a stencil with a one-step expression withoutthe need of integrals over the whole history It seems that theprocedure in our paper and the algorithm in [43] are dif-ferent in approach but equally satisfactory in result
For obtaining effective approximation with high orderwe introduce the numerical discretization for the fractional
Mathematical Problems in Engineering 3
Cattaneo equation by means of compact finite differencemethods
At the node (xi tk+(12)) the differential equation takesthe following form
zu
ztxi tk+(12)1113872 1113873 +
zαu
ztαxi tk+(12)1113872 1113873
z2u
zx2 xi tk+(12)1113872 1113873 + f xi tk+(12)1113872 1113873
1le ileM minus 1 1le kleN minus 1
(16)
)e approximation of the fractional derivative is givenby [45]
CF0 D
αt u xi tk+(12)1113872 1113873
1(α minus 1)Δt
M0δtuk+1i minus 1113944
k
n1Mkminus n(⎡⎣
minus Mkminus n+11113857δtuji minus Mkψi
⎤⎦ + Rk+(12)i
(17)
with truncation error Rk+(12)i O(Δt2) and
Mn exp1 minus α2 minus αΔtn1113876 1113877 minus exp
1 minus α2 minus αΔt(n + 1)1113876 1113877 (18)
Furthermore by Lemma 1 and equation (15) the spaceand time derivative are approximated by
z2u
zx2 xi tk+ (12)1113872 1113873 12
δ2xuk+1i + δ2xuk
i1113872 1113873
A+ O Δt2 + h
41113872 1113873
(19)
zu
ztxi tk+(12)1113872 1113873 δtu
k+1i + O Δt21113872 1113873 (20)
Substituting (17) and (19)sim(20) into (16) we get
Aδtuk+1i +
1(α minus 1)Δt
A M0δtuk+1i minus 1113944
k
n1Mkminus n(⎡⎣
minus Mkminus n+11113857δtuni minus Mkψi1113891
12
δ2xuk+1i + δ2xu
ki1113872 1113873 + Af
k+(12)i + R
k+(12)i
1le ileM minus 1 0le kleN minus 1
(21)
and there exists a constant C depending on the function uand its derivatives such that
Rk+(12)i leC Δt2 + h
41113872 1113873 (22)
By the initial and boundary value conditions we have
u0i ϕi 1le ileM minus 1
uk0 u
kM 0 0le kleN
(23)
A compact finite difference scheme can be established byomitting the truncation term R
k+(12)i and replacing the exact
solution uki in equation (21) with numerical solution uk
i
(α minus 1)ΔtAδtuk+1i + M0Aδtu
k+1i minus
(α minus 1)Δt2
δ2xUk+1
(α minus 1)Δt
2δ2xU
k+ 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857Aδtu
ni + MkAψi
+(α minus 1)ΔtAfk+ (12)i
1le ileM minus 1 1le kleN minus 1
u0i ϕi 1le ileM minus 1
uk0 u
kM 0 0le kleN
(24)
32 Stability Analysis and Optimal Error Estimates
321 Stability Analysis )e following Lemma about Mn isuseful for the analysis of stability
Lemma 2 (see [45]) For the definition of Mn Mn gt 0 andMn+1 ltMn forallnle k are held
Multiplying hδtuk+1i on both sides of equation (24) and
summing up with respect to i from 1 to M minus 1 the followingequation is obtained
(α minus 1)Δt δtUk+1
2
A+ M0 δtU
k+1
2
Aminusα minus 12
δ2xUk+1
Uk+1
1113872 1113873
+α minus 12
δ2xUk+1
Uk
1113872 1113873
α minus 12
δ2xUk U
k+11113872 1113873 minus
α minus 12
δ2xUk U
k1113872 1113873
+ 1113944k
n1Mkminus n minus Mkminus n+1( 1113857 AδtU
n δtU
k+11113872 1113873
+ Mk Aψ δtUk+1
1113872 1113873 +(α minus 1)Δt Afk+(12)
δtUk+1
1113872 1113873
(25)
Observing equation (13) we have
δ2xUk+1 Uk+11113872 1113873 minus δxUk+1 δxUk+1( 1113857 minus δxUk+11113868111386811138681113868
111386811138681113868111386821 le 0
δ2xUk Uk1113872 1113873 minus δxUk δxUk( 1113857 minus δxUk1113868111386811138681113868
111386811138681113868111386821 le 0
δ2xUk+1 Uk1113872 1113873 minus δxUk+1 δxUk( 1113857 minus langUk+1 Ukrang δ2xUk Uk+11113872 1113873
(26)
By the triangle inequality and Lemma 2 we obtain
4 Mathematical Problems in Engineering
1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 AδtU
n δtU
k+11113872 1113873 + Mk Aψ δtU
k+11113872 1113873
le 1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δtUn
2A
+ δtUk+1
2
A1113874 1113875
+12Mk ψ
2A + δtU
k+1
2
A1113874 1113875
1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δtUn
2A
+12
M0 minus Mk( 1113857
middot δtUk+1
2
A+12Mk ψ
2A + δtU
k+1
2
A1113874 1113875
1113944k
n1
12Mkminus n δtU
n
2A
minus 1113944k
n1
12Mkminus n+1 δtU
n
2A
+12M0
δtUk+1
2
A+12Mkψ
2A
(27)Combining equation (25) with (26)sim(27) we get
1113944
k+1
n1Mkminus n+1 δtU
n
2A
+(α minus 1) δxUk+1
11138681113868111386811138681113868
111386811138681113868111386811138682
1
le 1113944k
n1Mkminus n δtU
n
2A
+(α minus 1) δxUk
11138681113868111386811138681113868
111386811138681113868111386811138682
1+ Mkψ
2A
+(α minus 1)Δt fk+(12)
2
A
(28)
Let
Q Uk
1113872 1113873 1113944k
n1Mkminus n δtU
n
2A
+(α minus 1) δxUk
11138681113868111386811138681113868
111386811138681113868111386811138682
1 (29)
Summing up with respect to k from 0 to N minus 1 leads to
Q UN
1113872 1113873leQ U0
1113872 1113873 + 1113944Nminus 1
k0Mkψ
2A +(α minus 1)Δt 1113944
Nminus 1
k0f
k+(12)
2
A
(30)
)e initial condition U0 ϕ implies that Q(U0)
(α minus 1)| δxϕ| 21 and then
Q UN
1113872 1113873le (α minus 1) δxϕ1113868111386811138681113868
111386811138681113868111386821 + 1113944
Nminus 1
k0Mkψ
2A
+(α minus 1)Δt 1113944Nminus 1
k0f
k+(12)
2
A
(31)
Theorem 1 For scheme (24) we have the following stableconclusion
Q Um
( 1113857le (α minus 1) δxϕ1113868111386811138681113868
111386811138681113868111386821 + 1113944
mminus 1
k0Mkψ
2A
+(α minus 1)Δt 1113944mminus 1
k0f
k+(12)
2
A forall0lemleN
(32)
322 Optimal Error Estimates Combining equations (21)and (23) with (24) we get an error equation as follows
(α minus 1)ΔtAδtek+1i + M0Aδte
k+1i minus
(α minus 1)Δt2
δ2xek+1
(α minus 1)Δt
2δ2xe
k+ 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857Aδte
ni
+(α minus 1)ΔtRk+(12)i
(33)
where Rk+(12)i O(Δt2 + h4) and ek
i uki minus uk
i forallkge 0Multiplying hδte
k+1i on both sides of equation (33) and
summing up with respect to i from 1 to M minus 1 we get
(α minus 1)Δt δtek+1
2
A+ M0 δte
k+1
2
A+α minus 12
δxek+1
11138681113868111386811138681113868
111386811138681113868111386811138682
1
α minus 12
δxek
11138681113868111386811138681113868
111386811138681113868111386811138682
1+ 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 Aδte
n δte
k+11113872 1113873
+(α minus 1)Δt Rk+(12)
δtek+1
1113872 1113873
(34)
By the triangle inequality and Lemma 2 we obtain
1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 Aδte
n δte
k+11113872 1113873
le 1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δten
2A
+ δtek+1
2
A1113874 1113875
1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δten
2A
+12
M0 minus Mk( 1113857 δtek+1
2
A
le 1113944k
n1
12Mkminus n δte
n
2A
minus 1113944k
n1
12Mkminus n+1 δte
n
2A
+12M0 δte
k+1
2
A
(35)
Combining equation (34) with (35) we have
1113944
k+1
n1Mkminus n+1 δte
n
2A
+(α minus 1) δxek+1
11138681113868111386811138681113868
111386811138681113868111386811138682
1le 1113944
k
n1Mkminus n δte
n
2A
+(α minus 1) δxek
11138681113868111386811138681113868
111386811138681113868111386811138682
1+(α minus 1)Δt R
k+(12)
2
(36)
By the definition of Q in stability analysis the inequality(36) can be rearranged as
Mathematical Problems in Engineering 5
Q ek+1
1113872 1113873leQ ek
1113872 1113873 +(α minus 1)Δt Rk+(12)
2 (37)
Summing up with respect to k from 0 to N minus 1 we get
Q eN
1113872 1113873leQ e0
1113872 1113873 + C Δt2 + h4
1113872 11138732 (38)
Observing that the initial error e0 0 implies Q(e0) 0)en we have
(α minus 1) δxeN
11138681113868111386811138681113868
111386811138681113868111386811138682
1leQ e
N1113872 1113873leC Δt2 + h
41113872 1113873
2
1113944
N
n1MNminus n δte
n
2AleQ e
N1113872 1113873leC Δt2 + h
41113872 1113873
2
(39)
Theorem 2 Suppose that the exact solution of the fractionalCattaneo equation is smooth sufficiently then there exists apositive constant C independent of h k and Δt such that
ek
11138681113868111386811138681113868
111386811138681113868111386811138681leC Δt2 + h
41113872 1113873 forall0le kleN (40)
where eki uk
i minus uki
4 Compact Finite Difference Scheme inTwo Dimensions
In this section the following fractional Cattaneo equation intwo dimensions will be considered
zu(x y t)
zt+
zαu(x y t)
ztα
z2u(x y t)
zx2 +z2u(x y t)
zy2 + f(x y t) (x y t) isin (a b) times(c d) times(0 T]
u(x y 0) ϕ(x y)zu
zt
1113868111386811138681113868111386811138681113868 t0 ψ(x y) ale xle b cleyled
u(a y t) u(b y t) u(x c t) u(x d t) 0 tgt 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(41)
where 1lt αlt 2 f(x y t) is the source termϕ(x y) andψ(x y) are the given functions and zαuztα isdefined by the new Caputo fractional derivative withoutsingular kernel
In order to construct the finite difference schemes therectangle [a b] times [c d] is discretized with xi
a + ihx (0le ileMx) and yj c + jhy (0le jleMy) andthe time interval [0 T] is discretized with tk kΔt (0le kleN)where hx (b minus a)Mx hy (d minus c) My and Δt TN arethe spatial grid and temporal step sizes respectively DenoteΩh
(xi yj) 0le ileMx 0le1113966 jleMy and ΩΔt tk 0le kleN1113864 1113865then Ωh timesΩΔt is a discretization of the physical computationaldomain [a b] times [c d] times [0 T] uk
ij u(xi yj tk) denotes thevalues of function u at the grid points and uk
ij denotes the valuesof the numerical solution at the point (xi yj tk)
Denote Vh v| v (v00 v01 vMxMy)1113882 1113883 and V0
h
v| v isin Vh v0j vMxj vi0 viMy 0 0 le i leMx 0le j1113882
leMy For any mesh function v isin Vh we use the followingnotations
δxviminus (12)j vij minus viminus 1j
hx
δ2xvij vi+1j minus 2vij + viminus 1j
h2x
δyvijminus (12) vij minus vijminus 1
hy
δ2yvij vij+1 minus 2vij + vijminus 1
h2y
1le ileMx minus 1 1le jleMy minus 1
(42)
and define the average operator
Axvij
112
vi+1j + 10vij + viminus 1j1113872 1113873 1le ileMx minus 1 0le jleMy
vij i 0 Mx 0le jleMy
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
Ayvij
112
vij+1 + 10vij + vijminus 11113872 1113873 1le jleMy minus 1 0le ileMx
vij j 0 My 0le ileMx
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(43)
6 Mathematical Problems in Engineering
It is clear that
Axvij I +h2
x
12δ2x1113888 1113889vij
Ayvij I +h2
y
12δ2y⎛⎝ ⎞⎠vij
(44)
We also denote AxAy(u v) (AxAyu v) It is easy to seethat AxAy AyAx
For any gird function u v isin V0h the discrete inner
product and norms are defined as follows
(u v) hxhy 1113944
Mxminus 1
i11113944
My minus 1
j1uijvij
u22 (u u)
(45)
For the average operator AxAy define
AxAy(v v)≜ AxAyv v1113872 1113873 v2A (46)
41 Compact Finite Difference Scheme At the node(xi yj tk+(12)) the differential equation is rewritten as
zu
ztxi yj tk+(12)1113872 1113873 +
zαu
ztαxi yj tk+(12)1113872 1113873
z2u
zx2 xi yj tk+(12)1113872 1113873 +z2u
zy2 xi yj tk+(12)1113872 1113873
+ f xi yj tk+(12)1113872 1113873
1le ileMx minus 1 1le jleMy minus 1 1le kleN minus 1
(47)
For the approximation of the time-fractional derivativewe have the following approximation [45]
CF0 D
αt u xi yj tk+(12)1113872 1113873
1(α minus 1)Δt
M0δtuk+1ij minus 1113944
k
n1Mkminus n(⎡⎣
minus Mkminus n+11113857δtunij minus Mkψij
⎤⎦
+ Rk+(12)ij
(48)
where the truncation error Rk+(12)ij O(Δt2) and
Mn exp1 minus α2 minus αΔtn1113876 1113877 minus exp
1 minus α2 minus αΔt(n + 1)1113876 1113877 (49)
Furthermore we also have
z2u
zx2 xi yj tk+(12)1113872 1113873 12
δ2xuk+1ij + δ2xuk
ij1113872 1113873
Ax
+ O Δt2 + h4x1113872 1113873
(50)
z2u
zy2 xi yj tk+(12)1113872 1113873 12
δ2yuk+1ij + δ2yuk
ij1113872 1113873
Ay
+ O Δt2 + h4y1113872 1113873
(51)
zu
ztxi yj tk+(12)1113872 1113873 δtu
k+1ij + O Δt21113872 1113873 (52)
Substituting (48) and (50)sim(52) into (47) leads to
AxAyδtuk+1ij +
1(α minus 1)τ
AxAy M0δtuk+1ij1113890
minus 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857δtu
nij minus Mkψij
⎤⎦
12
δ2xuk+1ij + δ2xu
kij1113872 1113873 +
12
δ2yuk+1ij + δ2yu
kij1113872 1113873 + AxAyf
k+(12)ij
+ Rk+(12)ij
1le ileMx minus 1 1le jleMy minus 1 0le kleN minus 1
(53)
and there exists a constant C depending on the function uand its derivatives such that
Rk+(12)ij leC Δt2 + h
4x + h
4y1113872 1113873 (54)
By the initial and boundary conditions we have
u0ij ϕij 1le ileMx minus 1 1le jleMy minus 1
uk0j u
kMxj u
ki0 u
kiMy
0 0le kleN(55)
Omitting the truncation error Rk+(12)ij and replacing the
true solution ukij with numerical solution uk
ij a compactfinite difference scheme can be obtained as follows
(α minus 1)ΔtAxAyδtUk+1ij + M0AxAyδtU
k+1ij
minus Ay
(α minus 1)Δt2
δ2xUk+1ij minus Ax
(α minus 1)Δt2
δ2yUk+1ij
Ay
(α minus 1)Δt2
δ2xukij + Ax
(α minus 1)Δt2
δ2yukij
+ 1113944k
n1Mkminus n minus Mkminus n+1( 1113857AxAyδtu
nij + MkAxAyψij
+ (α minus 1)ΔtAxAyfk+(12)ij
1le ileMx minus 1 1le jleMy minus 1 1le kleN minus 1
u0ij ϕij 1le ileMx minus 1 1le jleMy minus 1
uk0j u
kMxj u
ki0 u
kiMy
0 0le kleN
(56)
Mathematical Problems in Engineering 7
42 Stability Analysis and Optimal Error Estimates
421 Stability Analysis
Definition 4 (see [46]) For any gird function u isin V0h define
the norm
⦀nablahu⦀2A δxu
2
minush2
y
12δyδxu
2
⎛⎝ ⎞⎠ + δyu
2
minush2
x
12δxδyu
2
1113888 1113889
(57)
)e lemmas below is useful in the subsequent analysis ofstability
Lemma 3 (see [46]) For any gird function u isin V0h the
following equation is held13u
2 le u2A le u
2 (58)
Lemma 4 (see [46]) For any gird function u isin V0h the
following equation is held23nablahu
2 le⦀nablahu⦀2A le nablahu
2
Ayδ2xu
nminus (12)+ Axδ
2yu
nminus (12) δtu
nminus (12)1113872 1113873
minus12Δt⦀nablahu
n⦀2A minus ⦀nablahunminus 1⦀
2A1113874 1113875
(59)
where nablahu2 δxu22 + δyu22
Multiplying hxhyδtUk+1ij on both sides of equation (56)
and summing up wrt i j from 1 to (Mx minus 1) and from 1 to(My minus 1) respectively the following equation is obtained
(α minus 1)Δt δtUk+1
2
A+ M0 δtU
k+1
2
A
(α minus 1)Δt Ayδ2xU
k+(12)+ Axδ
2yU
k+(12) δtU
k+11113872 1113873 + 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 AyAxδtU
n δtU
k+11113872 1113873
+ Mk AyAxψ δtUk+1
1113872 1113873 +(α minus 1)Δt AyAxfk+(12)
δtUk+1
1113872 1113873
(60)
Observing Lemma 4 we have
Ayδ2xUk+(12) + Axδ
2yUk+(12) δtU
k+11113872 1113873 minus12Δt⦀nablahu
k+1⦀2A minus ⦀nablahu
k⦀2A1113874 1113875 (61)
(α minus 1)Δt AyAxfk+(12) δtUk+11113872 1113873le
(α minus 1)Δt2
fk+(12)
2
A+ δtu
k+1
2
A1113874 1113875 (62)
By the triangle inequality and Lemma 2 we obtain
1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 AyAxδtU
n δtU
k+11113872 1113873 + Mk AyAxψ δtU
k+11113872 1113873
le 1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δtUn
2A
+ δtUk+1
2
A1113874 1113875 +
12Mk ψ
2A + δtU
k+1
2
A1113874 1113875
1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δtUn
2A
+12
M0 minus Mk( 1113857 δtUk+1
2
A+12Mk ψ
2A + δtU
k+1
2
A1113874 1113875
1113944k
n1
12Mkminus n δtU
n
2A
minus 1113944k
n1
12Mkminus n+1 δtU
n
2A
+12M0 δtU
k+1
2
A+12Mkψ
2A
(63)
Combining equation (60) with (61)sim(63)we get
1113944
k+1
n1Mkminus n+1 δtU
n
2A
+(α minus 1)⦀nablahUk+1⦀
2A
le 1113944k
n1Mkminus n δtU
n
2A
+(α minus 1) nablahUk
2
A+ Mkψ
2A +(α minus 1)Δt⦀fk+(12)⦀
2A
(64)
8 Mathematical Problems in Engineering
Let
Q Uk
1113872 1113873 1113944k
n1Mkminus n δtU
n
2A
+(α minus 1)⦀nablahUk⦀
2A (65)
Summing up with respect to k from 0 to N minus 1 we get
Q UN
1113872 1113873leQ U0
1113872 1113873 + 1113944Nminus 1
k0Mkψ
2A +(α minus 1)Δt 1113944
Nminus 1
k0f
k+(12)
2
A
(66)
Noting that U0 ϕ we have Q(U0) (α minus 1)nablahϕ2A Itfollows that
Q UN
1113872 1113873le (α minus 1)⦀nablahϕ⦀2A + 1113944
Nminus 1
k0Mkψ
2A
+(α minus 1)Δt 1113944Nminus 1
k0f
k+(12)
2
A
(67)
Theorem 3 For the compact finite difference scheme (56)the following stability inequality holds
Q Um
( 1113857le (α minus 1) nablahϕ1113868111386811138681113868
1113868111386811138681113868
2A
+ 1113944
mminus 1
k0Mkψ
2A
+(α minus 1)Δt 1113944
mminus 1
k0f
k+(12)
2
A forall0lemleN
(68)
Similar to the stability the convergence can also beanalyzed
Theorem 4 Suppose that the exact solution of the fractionalCattaneo equation is sufficiently smooth then there exists apositive constant C independent of h k and Δt such that
ek
11138681113868111386811138681113868
111386811138681113868111386811138681leC Δt2 + h
41113872 1113873 forall0le kleN (69)
where ekij uk
ij minus ukij and h max hx hy1113966 1113967
5 Efficient Storage and Fast Evaluation of theCaputondashFabrizio Fractional Derivative
Since time-fractional derivative operator is nonlocal thetraditional direct method for numerically solving the frac-tional Cattaneo equations generally requires total O(MN)
memory units and O(MN2) computational complexitywhere N andM are the total number of time steps and spacesteps respectively
In this section we develop a fast solution method for thefinite difference scheme of the time-fractional Cattaneoequation
Let
Nkn Mkminus n minus Mkminus n+1
exp1 minus α2 minus α
(k minus n)Δt1113874 1113875
middot 1 minus 2exp1 minus α2 minus αΔt1113874 1113875 + exp
1 minus α2 minus α
2Δt1113874 11138751113876 1113877
1le nle k
(70)
then
Nk+1n Mk+1minus n minus Mk+1minus n+1
exp1 minus α2 minus α
(k minus n)Δt1113874 1113875
middot 1 minus 2exp1 minus α2 minus αΔt1113874 1113875 + exp
1 minus α2 minus α
2Δt1113874 11138751113876 1113877exp1 minus α2 minus αΔt1113874 1113875
exp1 minus α2 minus αΔt1113874 1113875N
kn
Nk+1k+1 M0 minus M1 1 minus 2exp
1 minus α2 minus αΔt1113874 1113875 + exp
1 minus α2 minus α
2Δt1113874 1113875
(71)
So we have
1113944
k+1
n1N
k+1n Aδtu
ni exp
1 minus α2 minus αΔt1113874 1113875 1113944
k
n1N
knAδtu
ni + N
k+1k+1Aδtu
k+1i
1113944
k+1
n1N
k+1n AyAxδtu
nij exp
1 minus α2 minus αΔt1113874 1113875 1113944
k
n1N
knAyAxδtu
nij
+ Nk+1k+1AyAxδtU
k+1ij
(72)
Remark We find that at the k-th level only O(1) operationsare needed to compute the k-th level since the (k minus 1)-thlevel is known at that point )us the total operations arereduced from O(N2) to O(N) and the memory require-ment decreases from O(N) to O(1) We conclude that thatthis fast method significantly reduces the total computa-tional cost from O(MN2) to O(MN) and the memoryrequirement from O(MN) to O(M)
6 Numerical Experiments
In this section we carry out several numerical experiments tocheck the effectiveness of the proposed scheme )e conver-gence rate and CPU consumption are all compared in thesimulations We take the space-time domainΩ [0 1] T 1for one-dimensional case and Ω [0 1] times [0 1] T 1 fortwo-dimensional case )ese simulations are implemented inMatlab and the numerical experiments are run on a computerwith 4GB memory )e time-fractional Cattaneo equation ofthe following forms is considered
Example 1 We provide the exact solution u(x t)
etsin(πx) and for different α we have different f(x t)
accordingly
Mathematical Problems in Engineering 9
Table 1 Considering Δt h2 the discrete linfin error and convergence rates of u with different α for Example 1
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 15320e minus 04 mdash 15941e minus 04 mdash 18076e minus 04 mdash2minus 4 95109e minus 06 40097 98981e minus 06 40094 11230e minus 05 400862minus 5 59343e minus 07 40024 61762e minus 07 40024 70079e minus 07 400222minus 6 37074e minus 08 40006 38585e minus 08 40006 43783e minus 08 40005
Table 2 Considering h Δt
radic the discrete l2 error and convergence rates of u with different α for Example 1
Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
2minus 4 00017 mdash 00018 mdash 00021 mdash2minus 6 10832e minus 04 40382 11271e minus 04 40373 12781e minus 04 403442minus 8 67252e minus 06 40097 69990e minus 06 40094 79405e minus 06 400862minus 10 41961e minus 07 40024 43672e minus 07 40023 49553e minus 07 400212minus 12 26215e minus 08 40006 27284e minus 08 40005 30959e minus 08 40005
Table 3 )e CPU time consumption of the fast compact difference scheme and direct difference scheme for Example 1
Δtα 125 α 15 α 175
DCD FCD DCD FCD DCD FCD11000 34008 18876 34632 19032 33540 1918812500 142740 46800 141960 47580 138840 4742415000 470967 93600 463010 94692 461450 9375617500 994194 139464 956754 140088 967206 138684110 000 1699318 186733 1641130 186421 1658290 184393125 000 9816362 466910 9509820 469251 9591409 460046150 000 38269853 934133 36923096 940062 37547412 919625
N times104
0
500
1000
1500
2000
2500
3000
3500
4000
CPU
tim
e
Fast evaluationDirect evaluation
0 05 1 15 2 25 3 35 4 45 5
(a)
Fast evaluationDirect evaluation
log10 (N)
ndash05
0
05
1
15
2
25
3
35
4
log1
0 (C
PU ti
me)
25 3 35 4 45 5
(b)
Figure 1 )e CPU time (a) and the log-log CPU time (b) versus the total number of time steps N for Example 1
10 Mathematical Problems in Engineering
Table 4 Considering Δt 2minus 13 the discrete linfin error and convergence rates of u with different α for Example 1
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 22486e minus 04 mdash 22733e minus 04 mdash 23705e minus 04 mdash2minus 4 13985e minus 05 40071 14140e minus 05 40069 14745e minus 05 400692minus 5 86897e minus 07 40084 87882e minus 07 40081 91726e minus 07 400682minus 6 50194e minus 08 41137 51024e minus 08 41063 54092e minus 08 40838
Table 5 Considering h 0001 the discrete linfin error and convergence rates of u with different α for Example 1
Δtα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 00046 mdash 00043 mdash 00036 mdash2minus 4 00011 20641 00011 19668 90063e minus 04 199902minus 5 28662e minus 04 19403 27171e minus 04 20174 22518e minus 04 199992minus 6 71658e minus 05 19999 67928e minus 05 20000 56295e minus 05 200002minus 7 17915e minus 05 20000 16982e minus 05 20000 14074e minus 05 20000
x
ndash02
0
02
04
06
08
1
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(a)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(b)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(c)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(d)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(e)
ndash02
0
02
04
06
08
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(f)
Figure 2 Considering c 0001 α 15 M N 100 real and numerical solutions of u at T 1 for Example 2 with the value of x0 equalto (a) 0 (b) 02 (c) 04 (d) 06 (e) 08 and (f) 1
Mathematical Problems in Engineering 11
zu(x t)
zt+
zαu(x t)
ztα
z2u(x t)
zx2 + f(x t)
u(x 0) sin(πx)zu
zt
1113868111386811138681113868111386811138681113868 t0 sin(πx)
u(0 t) u(1 t) 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(73)
In Tables 1 and 2 we take Δt h2 and h Δt
radicto ex-
amine the discrete linfin-norm (l2-norm) errors and
corresponding spatial and temporal convergence rates re-spectively We list the errors and convergence rates (order) ofthe proposed compact finite difference (CD) scheme which isalmost O(Δt2 + h4) for different α Additionally Table 3shows the CPU time (CPU) consumed by direct compact(DCD) scheme and fast compact difference (FCD) schemerespectively It is obvious that the FCD scheme has a sig-nificantly reduced CPU time over the DCD scheme Forinstance when α 15 we choose h 01 and Δt 150 000and observe that the FCD scheme consumes only 94 secondswhile the DCD scheme consumes 3692 seconds We can find
x
075
08
085
09
095
1
105
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(a)
0
01
02
03
04
05
06
07
08
09
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(b)
0
01
02
03
04
05
06
07
08
09
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(c)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(d)
Figure 3 Considering x0 05 α 15 M N 100 real and numerical solutions of u at T 1 for Example 2 with the value of c equal to(a) 1 (b) 01 (c) 001 and (d) 0001
12 Mathematical Problems in Engineering
that the performance of the FCD scheme will be moreconspicuous as the time step size Δt decreases
In Figure 1 we set h 01 and α 175 and change thetotal number of time steps N to plot out the CPU time (inseconds) of the FCD scheme and DCD scheme We canobserve that the CPU time increases almost linearly withrespect to N for the FCD scheme while the DCD schemescales like O(N2)
Tables 4 and 5 show the discrete linfin errors and con-vergence rates of the compact finite difference scheme forExample 1 )e space rates are almost O(h4) for fixedΔt 2minus 13 and the time convergence rates are always O(Δt2)for fixed h 0001 We can conclude that the numericalconvergence rates of our scheme approach almost toO(Δt2 + h4)
Example 2 )e example is described by
zu(x t)
zt+
zαu(x t)
ztα
z2u(x t)
zx2 + f(x t)
u(x 0) 0zu
zt
1113868111386811138681113868111386811138681113868t0 0
u(0 t) t2exp minusx20
c1113888 1113889 u(1 t) t2exp minus
1 minus x0( 11138572
c1113888 1113889
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(74)
Note that the exact solution of the above problem isu(x t) t2exp(minus (x minus x0)
2c) where x0 isin [0 1] and c is a
N times104
0
1000
2000
3000
4000
5000
6000
7000
8000CP
U ti
me
Fast evaluationDirect evaluation
0 05 1 15 2 25 3 35 4 45 5
(a)
Fast evaluationDirect evaluation
log10 (N)
0
05
1
15
2
25
3
35
4
log1
0 (C
PU ti
me)
25 3 35 4 45 5
(b)
Figure 4 Considering x0 05 c 001 α 15 andM 10 the CPU time (a) and the log-log CPU time (b) versus the total number oftime steps N for Example 2
Table 6 Considering Δt h2 andx0 05 the discrete linfin error and convergence rates of u with different α and c for Example 2
c hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
1
2minus 3 73224e minus 05 mdash 55643e minus 05 mdash 38478e minus 05 mdash2minus 4 45707e-06 40018 34695e minus 06 40034 23900e minus 06 400902minus 5 28558e minus 07 40004 21671e minus 07 40009 14913e minus 07 400242minus 6 17847e minus 08 40001 13543e minus 08 40001 93169e minus 09 40006
01
2minus 3 00012 mdash 00012 mdash 00012 mdash2minus 4 70462e minus 05 40900 69731e minus 05 41051 69007e minus 05 412012minus 5 43571e minus 06 40154 43110e minus 06 40157 42650e minus 06 401612minus 6 27160e minus 07 40038 26871e minus 07 40039 26583e minus 07 40040
001
2minus 4 00090 mdash 00090 mdash 00090 mdash2minus 5 49257e minus 04 41915 49235e minus 04 41922 49232e minus 04 419232minus 6 29868e minus 05 40437 29855e minus 05 40436 29854e minus 05 404362minus 7 18531e minus 06 40106 18523e minus 06 40106 18522e minus 06 40106
Mathematical Problems in Engineering 13
Table 7 Considering h Δt
radicand c 001 the discrete l2 error and convergence rates of u with different α andx0 for Example 2
x0 Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
02minus 8 00044 mdash 00044 mdash 00042 mdash2minus 10 24205e minus 04 41841 23721e minus 04 42133 23098e minus 04 418452minus 12 14684e minus 05 40430 14390e minus 05 40430 14013e minus 05 40429
0252minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14902e minus 04 41794 14894e minus 04 41802 14892e minus 04 418032minus 12 90680e minus 06 40386 90633e minus 06 40386 90616e minus 06 40386
052minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14848e minus 04 41846 14842e minus 04 41852 14842e minus 04 418522minus 12 90323e minus 06 40390 90288e minus 06 40390 90286e minus 06 40390
0752minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14902e minus 04 41794 14894e minus 04 41802 14892e minus 04 418032minus 12 90680e minus 06 40386 90633e minus 06 40386 90616e minus 06 40386
12minus 8 00044 mdash 00044 mdash 00042 mdash2minus 10 24205e minus 04 41841 23721e minus 04 42133 23098e minus 04 418452minus 12 14684e minus 05 40430 14390e minus 05 40430 14013e minus 05 40429
0 2 4 6 8 10times104
0
2000
4000
6000
8000
10000
12000
14000
N
CPU
tim
e
Fast evaluationDirect evaluation
(a)
Fast evaluationDirect evaluation
36 38 4 42 44 46 48 515
2
25
3
35
4
45
log10 (N)
log1
0 (C
PU ti
me)
(b)
Figure 5 )e CPU time (a) and the log-log CPU time (b) versus the total number of time steps N for Example 3
Table 8 Considering Δt h2 the discrete linfin error and convergence rates of u with different α for Example 3
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 2 00028 mdash 00028 mdash 00030 mdash2minus 3 16948e minus 04 40462 17328e minus 04 40142 18411e minus 04 402632minus 4 10522e minus 05 40096 10759e minus 05 40095 11434e minus 05 400922minus 5 65652e minus 07 40024 67133e minus 07 40024 71346e minus 07 40024
14 Mathematical Problems in Engineering
constant and for different α we have different f(x t)
accordinglyWe apply the fast compact difference scheme to dis-
cretize the equation In Figure 2 we setc 0001 α 15 andM N 100 and plot exact andnumerical solutions at time T 1 for Example 2 with dif-ferent x0 For x0 05 α 15 andM N 100 we alsoplot exact and numerical solutions with the different c inFigure 3 In Figure 4 for h 01 and α 15 we vary thetotal number of time steps N to plot out the CPU time (inseconds) of the FCD scheme and DCD scheme )e nu-merical experiments verified our theoretical results In Ta-ble 6 by equating Δt h2 and fixing x0 05 we compute
the discrete linfin error and convergence rates with differentfractional derivative orders α and different c It shows thatthe compact finite difference scheme has space accuracy offourth order and temporal accuracy of second order We seth
Δt
radicand fix c 001 and the discrete l2 error and
convergence rates with different α and x0 are displayed inTable 7 )ese numerical convergence rates are almostapproaching O(Δt2 + h4) as Example 1
Example 3 If the exact solution is given byu(x t) etsin(πx)sin(πy) we have different f(x y t) fordifferent α accordingly
zu(x y t)
zt+
zαu(x y t)
ztα
z2u(x y t)
zx2 +z2u(x y t)
zy2 + f(x y t)
u(x 0) sin(πx)sin(πy)
zu
zt | t0 sin(πx)sin(πy)
u(0 y t) u(1 y t) u(x 0 t) u(x 1 t) 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(75)
In Figure 5 h 01 and α 175 are fixed and the totalnumber of time steps N vary to plot out the CPU time (inseconds) of the FCD procedure and DCD procedure and itpresents an approximately linear computation complexityfor FCD procedure We set Δt h2 in Table 8 and h
Δt
radic
in Table 9 the discrete linfin error discrete l2 error andconvergence rates with different derivative orders α arepresented )e fourth-order space accuracy and second-order temporal accuracy can be observed clearly
7 Conclusion
In this paper we develop and analyze a fast compact finitedifference procedure for the Cattaneo equation equipped withtime-fractional derivative without singular kernel )e time-fractional derivative is of CaputondashFabrizio type with the orderof α(1lt αlt 2) Compact difference discretization is appliedto obtain a high-order approximation for spatial derivatives ofinteger order in the partial differential equation and theCaputondashFabrizio fractional derivative is discretized by means
of CrankndashNicolson approximation It has been proved thatthe proposed compact finite difference scheme has spatialaccuracy of fourth order and temporal accuracy of secondorder Since the fractional derivatives are history dependentand nonlocal huge memory for storage and computationalcost are required )is means extreme difficulty especially fora long-time simulation Enlightened by the treatment forCaputo fractional derivative [32] we develop an effective fastevaluation procedure for the new CaputondashFabrizio fractionalderivative for the compact finite difference scheme Severalnumerical experiments have been carried out to show theconvergence orders and applicability of the scheme
Inspired by the work [43] the topic about modelling andnumerical solutions of porous media flow equipped withfractional derivatives is very interesting and challenging andwill be our main research direction in the future
Data Availability
All data generated or analyzed during this study are includedin this article
Table 9 Considering h Δt
radic the discrete l2 error and convergence rates of u with different α for Example 3
Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
2minus 4 00014 mdash 00014 mdash 00015 mdash2minus 6 84738e minus 05 40463 86640e minus 05 40142 92056e minus 05 402632minus 8 52610e minus 06 40096 53795e minus 06 40095 57168e minus 06 400922minus 10 32826e minus 07 40024 33566e minus 07 40024 35673e minus 07 40023
Mathematical Problems in Engineering 15
Conflicts of Interest
)e authors declare that they have no conflicts of interest
Acknowledgments
)is work was supported in part by the National NaturalScience Foundation of China under Grant nos 91630207and 11971272
References
[1] A Chaves ldquoA fractional diffusion equation to describe Levyflightsrdquo Physics Letters A vol 239 no 1-2 pp 13ndash16 1998
[2] M Giona and H E Roman ldquoFractional diffusion equation fortransport phenomena in random mediardquo Physica A Statis-tical Mechanics and its Applications vol 185 no 1ndash4pp 87ndash97 1992
[3] R Metzler and J Klafter ldquo)e random walkrsquos guide toanomalous diffusion a fractional dynamics approachrdquoPhysics Reports vol 339 no 1 pp 1ndash77 2000
[4] R L Magin Fractional Calculus in Bioengineering BegellHouse Redding Redding CT USA 2006
[5] R L Magin O Abdullah D Baleanu and X J ZhouldquoAnomalous diffusion expressed through fractional orderdifferential operators in the Bloch-Torrey equationrdquo Journalof Magnetic Resonance vol 190 no 2 pp 255ndash270 2008
[6] I Podlubny Fractional Differential Equations An Introduc-tion to Fractional Derivatives Fractional Differential Equa-tions to Methods of Beir Solution and Some of BeirApplications Vol 198 Elsevier Amsterdam Netherlands1998
[7] M Ran and C Zhang ldquoCompact difference scheme for a classof fractional-in-space nonlinear damped wave equations intwo space dimensionsrdquo Computers amp Mathematics withApplications vol 71 no 5 pp 1151ndash1162 2016
[8] A H Bhrawy and M A Zaky ldquoHighly accurate numericalschemes for multi-dimensional space variable-order frac-tional Schrodinger equationsrdquo Computers amp Mathematicswith Applications vol 73 no 6 pp 1100ndash1117 2017
[9] S B Yuste ldquoWeighted average finite difference methods forfractional diffusion equationsrdquo Journal of ComputationalPhysics vol 216 no 1 pp 264ndash274 2006
[10] T A M Langlands and B I Henry ldquo)e accuracy andstability of an implicit solution method for the fractionaldiffusion equationrdquo Journal of Computational Physicsvol 205 no 2 pp 719ndash736 2005
[11] C-M Chen F Liu I Turner and V Anh ldquoA Fourier method forthe fractional diffusion equation describing sub-diffusionrdquo Journalof Computational Physics vol 227 no 2 pp 886ndash897 2007
[12] Y Lin and C Xu ldquoFinite differencespectral approximationsfor the time-fractional diffusion equationrdquo Journal of Com-putational Physics vol 225 no 2 pp 1533ndash1552 2007
[13] G-h Gao and Z-z Sun ldquoA compact finite difference schemefor the fractional sub-diffusion equationsrdquo Journal of Com-putational Physics vol 230 no 3 pp 586ndash595 2011
[14] F Zeng C Li F Liu and I Turner ldquoNumerical algorithms fortime-fractional subdiffusion equation with second-order ac-curacyrdquo SIAM Journal on Scientific Computing vol 37 no 1pp A55ndashA78 2015
[15] G-H Gao Z-Z Sun and H-W Zhang ldquoA new fractionalnumerical differentiation formula to approximate the Caputofractional derivative and its applicationsrdquo Journal of Com-putational Physics vol 259 no 2 pp 33ndash50 2014
[16] H Li J Cao and C Li ldquoHigh-order approximation to Caputoderivatives and Caputo-type advection-diffusion equations(III)rdquo Journal of Computational and Applied Mathematicsvol 299 pp 159ndash175 2016
[17] A A Alikhanov ldquoA new difference scheme for the timefractional diffusion equationrdquo Journal of ComputationalPhysics vol 280 pp 424ndash438 2015
[18] Z Wang and S Vong ldquoCompact difference schemes for themodified anomalous fractional sub-diffusion equation and thefractional diffusion-wave equationrdquo Journal of ComputationalPhysics vol 277 pp 1ndash15 2014
[19] H-K Pang and H-W Sun ldquoFourth order finite differenceschemes for time-space fractional sub-diffusion equationsrdquoComputers amp Mathematics with Applications vol 71 no 6pp 1287ndash1302 2016
[20] M Dehghan M Abbaszadeh and W Deng ldquoFourth-ordernumerical method for the space-time tempered fractionaldiffusion-wave equationrdquo Applied Mathematics Lettersvol 73 no 11 pp 120ndash127 2017
[21] H Sun Z-Z Sun and G-H Gao ldquoSome high order dif-ference schemes for the space and time fractional Bloch-Torrey equationsrdquo Applied Mathematics and Computationvol 281 pp 356ndash380 2016
[22] S Arshad J Huang A Q M Khaliq and Y Tang ldquoTrape-zoidal scheme for time-space fractional diffusion equationwith Riesz derivativerdquo Journal of Computational Physicsvol 350 pp 1ndash15 2017
[23] H Wang K Wang and T Sircar ldquoA direct O(Nlog2N) finitedifferencemethod for fractional diffusion equationsrdquo Journal ofComputational Physics vol 229 no 21 pp 8095ndash8104 2010
[24] H Wang and T S Basu ldquoA fast finite difference method fortwo-dimensional space-fractional diffusion equationsrdquo SIAMJournal on Scientific Computing vol 34 no 5 pp A2444ndashA2458 2012
[25] H Wang and N Du ldquoA superfast-preconditioned iterativemethod for steady-state space-fractional diffusion equationsrdquoJournal of Computational Physics vol 240 pp 49ndash57 2013
[26] H Wang and N Du ldquoA fast finite difference method forthree-dimensional time-dependent space-fractional diffusionequations and its efficient implementationrdquo Journal ofComputational Physics vol 253 pp 50ndash63 2013
[27] H Wang and N Du ldquoFast alternating-direction finite dif-ference methods for three-dimensional space-fractional dif-fusion equationsrdquo Journal of Computational Physics vol 258pp 305ndash318 2014
[28] K Wang and H Wang ldquoA fast characteristic finite differencemethod for fractional advection-diffusion equationsrdquo Ad-vances in Water Resources vol 34 no 7 pp 810ndash816 2011
[29] H Wang and H Tian ldquoA fast Galerkin method with efficientmatrix assembly and storage for a peridynamic modelrdquoJournal of Computational Physics vol 231 no 23pp 7730ndash7738 2012
[30] H Tian H Wang and W Wang ldquoAn efficient collocationmethod for a non-local diffusion modelrdquo InternationalJournal of Numerical Analysis amp Modeling vol 10 no 4pp 815ndash825 2013
[31] H Wang and H Tian ldquoA fast and faithful collocation methodwith efficient matrix assembly for a two-dimensional nonlocaldiffusion modelrdquo Computer Methods in Applied Mechanicsand Engineering vol 273 no 5 pp 19ndash36 2014
[32] S Jiang J Zhang Q Zhang and Z Zhang ldquoFast evaluation ofthe Caputo fractional derivative and its applications tofractional diffusion equationsrdquo Communications in Compu-tational Physics vol 21 no 3 pp 650ndash678 2017
16 Mathematical Problems in Engineering
[33] Y Yan Z-Z Sun and J Zhang ldquoFast evaluation of the Caputofractional derivative and its applications to fractional diffusionequations a second-order schemerdquo Communications inComputational Physics vol 22 no 4 pp 1028ndash1048 2017
[34] X Lu H-K Pang and H-W Sun ldquoFast approximate in-version of a block triangular Toeplitz matrix with applicationsto fractional sub-diffusion equationsrdquo Numerical Linear Al-gebra with Applications vol 22 no 5 pp 866ndash882 2015
[35] M Caputo and M Fabrizio ldquoA new definition of fractionalderivative without singular kernelrdquo Progress in FractionalDifferentiation and Applications vol 1 no 2 pp 73ndash85 2015
[36] A Atangana ldquoOn the new fractional derivative and appli-cation to nonlinear Fisherrsquos reaction-diffusion equationrdquoAppliedMathematics and Computation vol 273 pp 948ndash9562016
[37] M D Ortigueira and J Tenreiro Machado ldquoA critical analysisof the Caputo-Fabrizio operatorrdquo Communications in Non-linear Science and Numerical Simulation vol 59 pp 608ndash6112018
[38] V E Tarasov ldquoNo nonlocality no fractional derivativerdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 62 pp 157ndash163 2018
[39] V Tarasov ldquoCaputo-Fabrizio operator in terms of integerderivatives memory or distributed lagrdquo Computational andApplied Mathematics vol 38 p 113 2019
[40] A Giusti ldquoA comment on some new definitions of fractionalderivativerdquoNonlinear Dynamics vol 93 no 3 pp 1757ndash17632018
[41] A Atangana and J F Gomez-Aguilar ldquoFractional derivativeswith no-index law property application to chaos and sta-tisticsrdquo Chaos Solitons amp Fractals vol 114 pp 516ndash535 2018
[42] J Hristov ldquoResponse functions in linear viscoelastic consti-tutive equations and related fractional operatorsrdquo Mathe-matical Modelling of Natural Phenomena vol 14 no 3pp 1ndash34 2019
[43] L X Vivas-Cruz A Gonzalez-Calderon M A Taneco-Hernandez and D P Luis ldquo)eoretical analysis of a model offluid flow in a reservoir with the Caputo-Fabrizio operatorrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 84 Article ID 105186 2020
[44] H-L Liao and Z-Z Sun ldquoMaximum norm error bounds ofADI and compact ADI methods for solving parabolicequationsrdquo Numerical Methods for Partial DifferentialEquations vol 26 no 1 pp 37ndash60 2010
[45] Z Liu A Cheng and X Li ldquoA second order Crank-Nicolsonscheme for fractional Cattaneo equation based on newfractional derivativerdquoAppliedMathematics and Computationvol 311 pp 361ndash374 2017
[46] Z Sun and G Gao Finite Difference Method for FractionalDifferential Equations Science Press Beijing China 2015
Mathematical Problems in Engineering 17
A time-fractional Cattaneo equation is considered withthe following form
zu(x t)
zt+
zαu(x t)
ztα
z2u(x t)
zx2+ f(x t) (x t) isin Ω times(0 T]
u(x 0) ϕ(x) zuzt
1113868111386811138681113868t0 ψ(x) x isin Ω
u(x t) 0 x isin zΩ tgt 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(1)
where 1lt αlt 2 Ω (a b) for one-dimensional case andΩ (a b) times (c d) for two-dimensional case f(x t) is thesource term ϕ(x) andψ(x) are the prescribed functions forinitial conditions and zαuztα is a new Caputo fractionalderivative without singular kernel which is defined in thenext section
Our purpose is to establish a fast finite difference scheme ofhigh order for this equation We will extract the recursive re-lation between the (k + 1) time step and the k time step of thefinite difference solution )e computational work is signifi-cantly reduced from O(MN2) to O(MN) and the memoryrequirement from O(MN) to O(M) where M and N are thetotal numbers of points for space steps and time steps re-spectively For improving the accuracy a compact finite dif-ference scheme is established)eoretical analysis shows that thefast compact difference scheme has spatial accuracy of fourthorder and temporal accuracy of second order Several numericalexperiments are implemented which verify the effectivenessapplicability and convergence rate of the proposed scheme
)is paper is organized as follows some definitions andnotations are prepared in Section 2 )e compact finitedifference scheme is described and then the stability andconvergence rates are rigorously analyzed for the scheme inSection 3 )e compact finite difference scheme is extendedto the case of two space dimensions in Section 4 Fastevaluation and efficient storage are established skillfully inSection 5 Some numerical experiments are carried out inSection 6 In the end we summarize the major contributionof this paper in Section 7
2 Some Notations and Definitions
We provide some definitions which will be used in thefollowing analysis
First let us recall the usual Caputo fractional derivativeof order α with respect to time variable t which is given by
Ca D
αt u(x t)
1Γ(n minus α)
1113944
t
a
(t minus s)nminus αminus 1
u(n)
(x s)ds
n minus 1lt αlt n
(2)
By replacing the kernel function (t minus s)minus α with the ex-ponential function exp(minus α(t minus s1 minus α)) and 1(Γ(1 minus α))
with M(α)1 minus α Caputo and Fabrizio [35] proposed thefollowing definition of fractional time derivative
Definition 1 (see [35]) Let u(middot t) isin H1(a b) bgt aα isin (0 1) then the new Caputo derivative of the fractionalorder is defined as
Dαt u(x t)
M(α)
1 minus α1113946
t
auprime(x s)exp minus α
t minus s
1 minus α1113876 1113877ds (3)
where M(α) is a normalization function satisfyingM(0) M(1) 1 When the function u does not belong toH1(a b) then this derivative can be reformulated as
Dαt u(x t)
αM(α)
1 minus α1113946
t
a(u(x t) minus u(x s))exp minus α
t minus s
1 minus α1113876 1113877ds
(4)
Definition 2 (see [36]) )e above new Caputo derivative oforder 0lt αlt 1 can also be reformulated as
CF0 D
αt u(x t)
11 minus α
1113944
t
0uprime(x s)exp minus α
t minus s
1 minus α1113876 1113877ds (5)
Definition 3 (see [35 36]) Let u(middot t) isin H1(a b) if nge 1 andα isin [0 1] the fractional time derivative CF
0 Dαt u(x t) of order
(n + α) is defined byCF0 D
n+αt u(x t)
CF0 D
αt
CF0 D
nt u(x t)1113872 1113873 (6)
Particularly for 1lt αlt 2 we have
CF0 D
αt u(x t)
M(α)
2 minus α1113944
t
0uPrime(x s)exp (1 minus α)
t minus s
2 minus α1113876 1113877ds
(7)
Remark 1 )e CaputondashFabrizio (CF) operator was pro-posed with a nonsingular kernel for describing materialheterogeneities that do not exhibit power-law behavior [35]
Remark 2 An open discussion is ongoing about themathematical construction of the CF operator Ortigueiraand Tenreiro Machado [37] indicated that the CF fractionalderivative is neither a fractional operator nor a derivativeoperator the authors of [38 39] showed that this operatorcannot describe dynamic memory and Giusti [40] indicatedthat this operator can be expressed as an infinite linearcombination of RiemannndashLiouville integrals with integer
2 Mathematical Problems in Engineering
powers As responses to these criticisms Atangana andGomez-Aguilar [41] pointed out the need to account for afractional calculus approach without an imposed index lawand with nonsingular kernels Furthermore Hristov [42]indicated that the CF operator is not applicable forexplaining the physical examples in [37 40] instead hesuggested that the CF operator can be used for the analysis ofmaterials that do not follow a power-law behavior )eauthors of [43] believe that models with CF operatorsproduce a better representation of physical behaviors thando integer-order models providing a way to model theintermediate (between elliptic and parabolic or betweenparabolic and hyperbolic) behaviors
To obtain the accuracy of the fourth order in spatialdirections the following lemma is necessary
Lemma 1 (see [44]) Denote θ(s) (1 minus s)3[5 minus 3(1 minus s)2]If f(x) isin C6[a b] h (b minus a)M xi a + ih(0le ileM)then it holds that
112
fPrime ximinus 1( 1113857 + 10fPrime xi( 1113857 + fPrime xi+1( 11138571113858 1113859
1h2 f ximinus 1( 11138571113890 minus 2f xi( 1113857 + f xi+1( 11138571113891
+h4
3601113946
1
0 f(6)
xi minus sh( 1113857 + f(6)
1113960
middot xi + sh( 1113857]θ(s)ds 1le ileM minus 1
(8)
3 Compact Finite Difference Scheme for One-Dimensional Fractional Cattaneo Equation
In order to construct the finite difference schemes the in-terval [a b] is divided into subintervals withxi a + ih (0le ileM) and [0 T] is discretized withtk kΔt (0le kleN) where h (b minus a)M and Δt TNare the spatial grid size and temporal step size respectivelyDenote Ωh xi 0le ileM1113864 1113865 ΩΔt tk 0le kleN1113864 1113865 thenΩh timesΩΔt becomes a discretization of the practical compu-tational domain [a b] times [0 T] )e values of the function uat the grid points are denoted as uk
j u(xj tk) and theapproximate solution at the point (xj tk) is denoted as uk
j Denote Vh v|v (v0 v1 vM)1113864 1113865 We also introduce
the following notations for any mesh function v isin Vh
δxvjminus (12) vj minus vjminus 1
h
δ2xvj vj+1 minus 2vj + vjminus 1
h2 1le jleM minus 1
(9)
and define the average operator
Avj
112
vj+1 + 10vj + vjminus 11113872 1113873 1le jleM minus 1
vj j 0 M
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(10)
It is easy to see that
Avj I +h2
12δ2x1113888 1113889vj (11)
where I is the identical operator We also denoteAv (Av1Av2 AvM) for vector v (v1 v2 vM)and A(u v) (Au v)
For any two gird functions u v isin V0h
v|v isin Vh v0 vM 01113864 1113865 the discrete inner products andnorms are defined as
(u v) h 1113944Mminus 1
j1ujvj
langu vrang h 1113944Mminus 1
j1δxujminus (12)δxvjminus (12)
u22 (u u)
|u|21 langu urang
uinfin max1lejleMminus 1
uj
11138681113868111386811138681113868
11138681113868111386811138681113868
(12)
By summation by parts it is easy to see that
δ2xu v1113872 1113873 minus δxu δxv( 1113857 minus langu vrang u δ2xv1113872 1113873 (13)
For the average operator A define
A(v v)≜ (Av v) v2A (14)
Additionally let VΔt v|v (v0 v1 middot middot middot vN)1113864 1113865 be thespace of grid function defined on ΩΔt For any functionv isin VΔt a difference operator is introduced as follows
δtvk
vk minus vkminus 1
Δt (15)
31 Compact Finite Difference Scheme We will consider thetime-fractional Cattaneo equation equipped with theCaputondashFabrizio derivative Vivas-Cruz et al [43] gave thetheoretical analysis of a model of fluid flow in a reservoirwith the CaputondashFabrizio operator )ey proved that thismodel cannot be used to describe nonlocal processes since itcan be represented as an equivalent differential equationwith a finite number of integer-order derivatives
)e finite difference methods usually lead to stencilsthrough the whole history passed by the solution whichconsume too much computational work In this paper wewill establish a high-order finite difference scheme andpropose a procedure to reduce the computational cost In[43] the authors proposed a recurrence formula of dis-cretized CF operator and obtained an algorithm which canbe considered a stencil with a one-step expression withoutthe need of integrals over the whole history It seems that theprocedure in our paper and the algorithm in [43] are dif-ferent in approach but equally satisfactory in result
For obtaining effective approximation with high orderwe introduce the numerical discretization for the fractional
Mathematical Problems in Engineering 3
Cattaneo equation by means of compact finite differencemethods
At the node (xi tk+(12)) the differential equation takesthe following form
zu
ztxi tk+(12)1113872 1113873 +
zαu
ztαxi tk+(12)1113872 1113873
z2u
zx2 xi tk+(12)1113872 1113873 + f xi tk+(12)1113872 1113873
1le ileM minus 1 1le kleN minus 1
(16)
)e approximation of the fractional derivative is givenby [45]
CF0 D
αt u xi tk+(12)1113872 1113873
1(α minus 1)Δt
M0δtuk+1i minus 1113944
k
n1Mkminus n(⎡⎣
minus Mkminus n+11113857δtuji minus Mkψi
⎤⎦ + Rk+(12)i
(17)
with truncation error Rk+(12)i O(Δt2) and
Mn exp1 minus α2 minus αΔtn1113876 1113877 minus exp
1 minus α2 minus αΔt(n + 1)1113876 1113877 (18)
Furthermore by Lemma 1 and equation (15) the spaceand time derivative are approximated by
z2u
zx2 xi tk+ (12)1113872 1113873 12
δ2xuk+1i + δ2xuk
i1113872 1113873
A+ O Δt2 + h
41113872 1113873
(19)
zu
ztxi tk+(12)1113872 1113873 δtu
k+1i + O Δt21113872 1113873 (20)
Substituting (17) and (19)sim(20) into (16) we get
Aδtuk+1i +
1(α minus 1)Δt
A M0δtuk+1i minus 1113944
k
n1Mkminus n(⎡⎣
minus Mkminus n+11113857δtuni minus Mkψi1113891
12
δ2xuk+1i + δ2xu
ki1113872 1113873 + Af
k+(12)i + R
k+(12)i
1le ileM minus 1 0le kleN minus 1
(21)
and there exists a constant C depending on the function uand its derivatives such that
Rk+(12)i leC Δt2 + h
41113872 1113873 (22)
By the initial and boundary value conditions we have
u0i ϕi 1le ileM minus 1
uk0 u
kM 0 0le kleN
(23)
A compact finite difference scheme can be established byomitting the truncation term R
k+(12)i and replacing the exact
solution uki in equation (21) with numerical solution uk
i
(α minus 1)ΔtAδtuk+1i + M0Aδtu
k+1i minus
(α minus 1)Δt2
δ2xUk+1
(α minus 1)Δt
2δ2xU
k+ 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857Aδtu
ni + MkAψi
+(α minus 1)ΔtAfk+ (12)i
1le ileM minus 1 1le kleN minus 1
u0i ϕi 1le ileM minus 1
uk0 u
kM 0 0le kleN
(24)
32 Stability Analysis and Optimal Error Estimates
321 Stability Analysis )e following Lemma about Mn isuseful for the analysis of stability
Lemma 2 (see [45]) For the definition of Mn Mn gt 0 andMn+1 ltMn forallnle k are held
Multiplying hδtuk+1i on both sides of equation (24) and
summing up with respect to i from 1 to M minus 1 the followingequation is obtained
(α minus 1)Δt δtUk+1
2
A+ M0 δtU
k+1
2
Aminusα minus 12
δ2xUk+1
Uk+1
1113872 1113873
+α minus 12
δ2xUk+1
Uk
1113872 1113873
α minus 12
δ2xUk U
k+11113872 1113873 minus
α minus 12
δ2xUk U
k1113872 1113873
+ 1113944k
n1Mkminus n minus Mkminus n+1( 1113857 AδtU
n δtU
k+11113872 1113873
+ Mk Aψ δtUk+1
1113872 1113873 +(α minus 1)Δt Afk+(12)
δtUk+1
1113872 1113873
(25)
Observing equation (13) we have
δ2xUk+1 Uk+11113872 1113873 minus δxUk+1 δxUk+1( 1113857 minus δxUk+11113868111386811138681113868
111386811138681113868111386821 le 0
δ2xUk Uk1113872 1113873 minus δxUk δxUk( 1113857 minus δxUk1113868111386811138681113868
111386811138681113868111386821 le 0
δ2xUk+1 Uk1113872 1113873 minus δxUk+1 δxUk( 1113857 minus langUk+1 Ukrang δ2xUk Uk+11113872 1113873
(26)
By the triangle inequality and Lemma 2 we obtain
4 Mathematical Problems in Engineering
1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 AδtU
n δtU
k+11113872 1113873 + Mk Aψ δtU
k+11113872 1113873
le 1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δtUn
2A
+ δtUk+1
2
A1113874 1113875
+12Mk ψ
2A + δtU
k+1
2
A1113874 1113875
1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δtUn
2A
+12
M0 minus Mk( 1113857
middot δtUk+1
2
A+12Mk ψ
2A + δtU
k+1
2
A1113874 1113875
1113944k
n1
12Mkminus n δtU
n
2A
minus 1113944k
n1
12Mkminus n+1 δtU
n
2A
+12M0
δtUk+1
2
A+12Mkψ
2A
(27)Combining equation (25) with (26)sim(27) we get
1113944
k+1
n1Mkminus n+1 δtU
n
2A
+(α minus 1) δxUk+1
11138681113868111386811138681113868
111386811138681113868111386811138682
1
le 1113944k
n1Mkminus n δtU
n
2A
+(α minus 1) δxUk
11138681113868111386811138681113868
111386811138681113868111386811138682
1+ Mkψ
2A
+(α minus 1)Δt fk+(12)
2
A
(28)
Let
Q Uk
1113872 1113873 1113944k
n1Mkminus n δtU
n
2A
+(α minus 1) δxUk
11138681113868111386811138681113868
111386811138681113868111386811138682
1 (29)
Summing up with respect to k from 0 to N minus 1 leads to
Q UN
1113872 1113873leQ U0
1113872 1113873 + 1113944Nminus 1
k0Mkψ
2A +(α minus 1)Δt 1113944
Nminus 1
k0f
k+(12)
2
A
(30)
)e initial condition U0 ϕ implies that Q(U0)
(α minus 1)| δxϕ| 21 and then
Q UN
1113872 1113873le (α minus 1) δxϕ1113868111386811138681113868
111386811138681113868111386821 + 1113944
Nminus 1
k0Mkψ
2A
+(α minus 1)Δt 1113944Nminus 1
k0f
k+(12)
2
A
(31)
Theorem 1 For scheme (24) we have the following stableconclusion
Q Um
( 1113857le (α minus 1) δxϕ1113868111386811138681113868
111386811138681113868111386821 + 1113944
mminus 1
k0Mkψ
2A
+(α minus 1)Δt 1113944mminus 1
k0f
k+(12)
2
A forall0lemleN
(32)
322 Optimal Error Estimates Combining equations (21)and (23) with (24) we get an error equation as follows
(α minus 1)ΔtAδtek+1i + M0Aδte
k+1i minus
(α minus 1)Δt2
δ2xek+1
(α minus 1)Δt
2δ2xe
k+ 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857Aδte
ni
+(α minus 1)ΔtRk+(12)i
(33)
where Rk+(12)i O(Δt2 + h4) and ek
i uki minus uk
i forallkge 0Multiplying hδte
k+1i on both sides of equation (33) and
summing up with respect to i from 1 to M minus 1 we get
(α minus 1)Δt δtek+1
2
A+ M0 δte
k+1
2
A+α minus 12
δxek+1
11138681113868111386811138681113868
111386811138681113868111386811138682
1
α minus 12
δxek
11138681113868111386811138681113868
111386811138681113868111386811138682
1+ 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 Aδte
n δte
k+11113872 1113873
+(α minus 1)Δt Rk+(12)
δtek+1
1113872 1113873
(34)
By the triangle inequality and Lemma 2 we obtain
1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 Aδte
n δte
k+11113872 1113873
le 1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δten
2A
+ δtek+1
2
A1113874 1113875
1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δten
2A
+12
M0 minus Mk( 1113857 δtek+1
2
A
le 1113944k
n1
12Mkminus n δte
n
2A
minus 1113944k
n1
12Mkminus n+1 δte
n
2A
+12M0 δte
k+1
2
A
(35)
Combining equation (34) with (35) we have
1113944
k+1
n1Mkminus n+1 δte
n
2A
+(α minus 1) δxek+1
11138681113868111386811138681113868
111386811138681113868111386811138682
1le 1113944
k
n1Mkminus n δte
n
2A
+(α minus 1) δxek
11138681113868111386811138681113868
111386811138681113868111386811138682
1+(α minus 1)Δt R
k+(12)
2
(36)
By the definition of Q in stability analysis the inequality(36) can be rearranged as
Mathematical Problems in Engineering 5
Q ek+1
1113872 1113873leQ ek
1113872 1113873 +(α minus 1)Δt Rk+(12)
2 (37)
Summing up with respect to k from 0 to N minus 1 we get
Q eN
1113872 1113873leQ e0
1113872 1113873 + C Δt2 + h4
1113872 11138732 (38)
Observing that the initial error e0 0 implies Q(e0) 0)en we have
(α minus 1) δxeN
11138681113868111386811138681113868
111386811138681113868111386811138682
1leQ e
N1113872 1113873leC Δt2 + h
41113872 1113873
2
1113944
N
n1MNminus n δte
n
2AleQ e
N1113872 1113873leC Δt2 + h
41113872 1113873
2
(39)
Theorem 2 Suppose that the exact solution of the fractionalCattaneo equation is smooth sufficiently then there exists apositive constant C independent of h k and Δt such that
ek
11138681113868111386811138681113868
111386811138681113868111386811138681leC Δt2 + h
41113872 1113873 forall0le kleN (40)
where eki uk
i minus uki
4 Compact Finite Difference Scheme inTwo Dimensions
In this section the following fractional Cattaneo equation intwo dimensions will be considered
zu(x y t)
zt+
zαu(x y t)
ztα
z2u(x y t)
zx2 +z2u(x y t)
zy2 + f(x y t) (x y t) isin (a b) times(c d) times(0 T]
u(x y 0) ϕ(x y)zu
zt
1113868111386811138681113868111386811138681113868 t0 ψ(x y) ale xle b cleyled
u(a y t) u(b y t) u(x c t) u(x d t) 0 tgt 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(41)
where 1lt αlt 2 f(x y t) is the source termϕ(x y) andψ(x y) are the given functions and zαuztα isdefined by the new Caputo fractional derivative withoutsingular kernel
In order to construct the finite difference schemes therectangle [a b] times [c d] is discretized with xi
a + ihx (0le ileMx) and yj c + jhy (0le jleMy) andthe time interval [0 T] is discretized with tk kΔt (0le kleN)where hx (b minus a)Mx hy (d minus c) My and Δt TN arethe spatial grid and temporal step sizes respectively DenoteΩh
(xi yj) 0le ileMx 0le1113966 jleMy and ΩΔt tk 0le kleN1113864 1113865then Ωh timesΩΔt is a discretization of the physical computationaldomain [a b] times [c d] times [0 T] uk
ij u(xi yj tk) denotes thevalues of function u at the grid points and uk
ij denotes the valuesof the numerical solution at the point (xi yj tk)
Denote Vh v| v (v00 v01 vMxMy)1113882 1113883 and V0
h
v| v isin Vh v0j vMxj vi0 viMy 0 0 le i leMx 0le j1113882
leMy For any mesh function v isin Vh we use the followingnotations
δxviminus (12)j vij minus viminus 1j
hx
δ2xvij vi+1j minus 2vij + viminus 1j
h2x
δyvijminus (12) vij minus vijminus 1
hy
δ2yvij vij+1 minus 2vij + vijminus 1
h2y
1le ileMx minus 1 1le jleMy minus 1
(42)
and define the average operator
Axvij
112
vi+1j + 10vij + viminus 1j1113872 1113873 1le ileMx minus 1 0le jleMy
vij i 0 Mx 0le jleMy
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
Ayvij
112
vij+1 + 10vij + vijminus 11113872 1113873 1le jleMy minus 1 0le ileMx
vij j 0 My 0le ileMx
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(43)
6 Mathematical Problems in Engineering
It is clear that
Axvij I +h2
x
12δ2x1113888 1113889vij
Ayvij I +h2
y
12δ2y⎛⎝ ⎞⎠vij
(44)
We also denote AxAy(u v) (AxAyu v) It is easy to seethat AxAy AyAx
For any gird function u v isin V0h the discrete inner
product and norms are defined as follows
(u v) hxhy 1113944
Mxminus 1
i11113944
My minus 1
j1uijvij
u22 (u u)
(45)
For the average operator AxAy define
AxAy(v v)≜ AxAyv v1113872 1113873 v2A (46)
41 Compact Finite Difference Scheme At the node(xi yj tk+(12)) the differential equation is rewritten as
zu
ztxi yj tk+(12)1113872 1113873 +
zαu
ztαxi yj tk+(12)1113872 1113873
z2u
zx2 xi yj tk+(12)1113872 1113873 +z2u
zy2 xi yj tk+(12)1113872 1113873
+ f xi yj tk+(12)1113872 1113873
1le ileMx minus 1 1le jleMy minus 1 1le kleN minus 1
(47)
For the approximation of the time-fractional derivativewe have the following approximation [45]
CF0 D
αt u xi yj tk+(12)1113872 1113873
1(α minus 1)Δt
M0δtuk+1ij minus 1113944
k
n1Mkminus n(⎡⎣
minus Mkminus n+11113857δtunij minus Mkψij
⎤⎦
+ Rk+(12)ij
(48)
where the truncation error Rk+(12)ij O(Δt2) and
Mn exp1 minus α2 minus αΔtn1113876 1113877 minus exp
1 minus α2 minus αΔt(n + 1)1113876 1113877 (49)
Furthermore we also have
z2u
zx2 xi yj tk+(12)1113872 1113873 12
δ2xuk+1ij + δ2xuk
ij1113872 1113873
Ax
+ O Δt2 + h4x1113872 1113873
(50)
z2u
zy2 xi yj tk+(12)1113872 1113873 12
δ2yuk+1ij + δ2yuk
ij1113872 1113873
Ay
+ O Δt2 + h4y1113872 1113873
(51)
zu
ztxi yj tk+(12)1113872 1113873 δtu
k+1ij + O Δt21113872 1113873 (52)
Substituting (48) and (50)sim(52) into (47) leads to
AxAyδtuk+1ij +
1(α minus 1)τ
AxAy M0δtuk+1ij1113890
minus 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857δtu
nij minus Mkψij
⎤⎦
12
δ2xuk+1ij + δ2xu
kij1113872 1113873 +
12
δ2yuk+1ij + δ2yu
kij1113872 1113873 + AxAyf
k+(12)ij
+ Rk+(12)ij
1le ileMx minus 1 1le jleMy minus 1 0le kleN minus 1
(53)
and there exists a constant C depending on the function uand its derivatives such that
Rk+(12)ij leC Δt2 + h
4x + h
4y1113872 1113873 (54)
By the initial and boundary conditions we have
u0ij ϕij 1le ileMx minus 1 1le jleMy minus 1
uk0j u
kMxj u
ki0 u
kiMy
0 0le kleN(55)
Omitting the truncation error Rk+(12)ij and replacing the
true solution ukij with numerical solution uk
ij a compactfinite difference scheme can be obtained as follows
(α minus 1)ΔtAxAyδtUk+1ij + M0AxAyδtU
k+1ij
minus Ay
(α minus 1)Δt2
δ2xUk+1ij minus Ax
(α minus 1)Δt2
δ2yUk+1ij
Ay
(α minus 1)Δt2
δ2xukij + Ax
(α minus 1)Δt2
δ2yukij
+ 1113944k
n1Mkminus n minus Mkminus n+1( 1113857AxAyδtu
nij + MkAxAyψij
+ (α minus 1)ΔtAxAyfk+(12)ij
1le ileMx minus 1 1le jleMy minus 1 1le kleN minus 1
u0ij ϕij 1le ileMx minus 1 1le jleMy minus 1
uk0j u
kMxj u
ki0 u
kiMy
0 0le kleN
(56)
Mathematical Problems in Engineering 7
42 Stability Analysis and Optimal Error Estimates
421 Stability Analysis
Definition 4 (see [46]) For any gird function u isin V0h define
the norm
⦀nablahu⦀2A δxu
2
minush2
y
12δyδxu
2
⎛⎝ ⎞⎠ + δyu
2
minush2
x
12δxδyu
2
1113888 1113889
(57)
)e lemmas below is useful in the subsequent analysis ofstability
Lemma 3 (see [46]) For any gird function u isin V0h the
following equation is held13u
2 le u2A le u
2 (58)
Lemma 4 (see [46]) For any gird function u isin V0h the
following equation is held23nablahu
2 le⦀nablahu⦀2A le nablahu
2
Ayδ2xu
nminus (12)+ Axδ
2yu
nminus (12) δtu
nminus (12)1113872 1113873
minus12Δt⦀nablahu
n⦀2A minus ⦀nablahunminus 1⦀
2A1113874 1113875
(59)
where nablahu2 δxu22 + δyu22
Multiplying hxhyδtUk+1ij on both sides of equation (56)
and summing up wrt i j from 1 to (Mx minus 1) and from 1 to(My minus 1) respectively the following equation is obtained
(α minus 1)Δt δtUk+1
2
A+ M0 δtU
k+1
2
A
(α minus 1)Δt Ayδ2xU
k+(12)+ Axδ
2yU
k+(12) δtU
k+11113872 1113873 + 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 AyAxδtU
n δtU
k+11113872 1113873
+ Mk AyAxψ δtUk+1
1113872 1113873 +(α minus 1)Δt AyAxfk+(12)
δtUk+1
1113872 1113873
(60)
Observing Lemma 4 we have
Ayδ2xUk+(12) + Axδ
2yUk+(12) δtU
k+11113872 1113873 minus12Δt⦀nablahu
k+1⦀2A minus ⦀nablahu
k⦀2A1113874 1113875 (61)
(α minus 1)Δt AyAxfk+(12) δtUk+11113872 1113873le
(α minus 1)Δt2
fk+(12)
2
A+ δtu
k+1
2
A1113874 1113875 (62)
By the triangle inequality and Lemma 2 we obtain
1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 AyAxδtU
n δtU
k+11113872 1113873 + Mk AyAxψ δtU
k+11113872 1113873
le 1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δtUn
2A
+ δtUk+1
2
A1113874 1113875 +
12Mk ψ
2A + δtU
k+1
2
A1113874 1113875
1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δtUn
2A
+12
M0 minus Mk( 1113857 δtUk+1
2
A+12Mk ψ
2A + δtU
k+1
2
A1113874 1113875
1113944k
n1
12Mkminus n δtU
n
2A
minus 1113944k
n1
12Mkminus n+1 δtU
n
2A
+12M0 δtU
k+1
2
A+12Mkψ
2A
(63)
Combining equation (60) with (61)sim(63)we get
1113944
k+1
n1Mkminus n+1 δtU
n
2A
+(α minus 1)⦀nablahUk+1⦀
2A
le 1113944k
n1Mkminus n δtU
n
2A
+(α minus 1) nablahUk
2
A+ Mkψ
2A +(α minus 1)Δt⦀fk+(12)⦀
2A
(64)
8 Mathematical Problems in Engineering
Let
Q Uk
1113872 1113873 1113944k
n1Mkminus n δtU
n
2A
+(α minus 1)⦀nablahUk⦀
2A (65)
Summing up with respect to k from 0 to N minus 1 we get
Q UN
1113872 1113873leQ U0
1113872 1113873 + 1113944Nminus 1
k0Mkψ
2A +(α minus 1)Δt 1113944
Nminus 1
k0f
k+(12)
2
A
(66)
Noting that U0 ϕ we have Q(U0) (α minus 1)nablahϕ2A Itfollows that
Q UN
1113872 1113873le (α minus 1)⦀nablahϕ⦀2A + 1113944
Nminus 1
k0Mkψ
2A
+(α minus 1)Δt 1113944Nminus 1
k0f
k+(12)
2
A
(67)
Theorem 3 For the compact finite difference scheme (56)the following stability inequality holds
Q Um
( 1113857le (α minus 1) nablahϕ1113868111386811138681113868
1113868111386811138681113868
2A
+ 1113944
mminus 1
k0Mkψ
2A
+(α minus 1)Δt 1113944
mminus 1
k0f
k+(12)
2
A forall0lemleN
(68)
Similar to the stability the convergence can also beanalyzed
Theorem 4 Suppose that the exact solution of the fractionalCattaneo equation is sufficiently smooth then there exists apositive constant C independent of h k and Δt such that
ek
11138681113868111386811138681113868
111386811138681113868111386811138681leC Δt2 + h
41113872 1113873 forall0le kleN (69)
where ekij uk
ij minus ukij and h max hx hy1113966 1113967
5 Efficient Storage and Fast Evaluation of theCaputondashFabrizio Fractional Derivative
Since time-fractional derivative operator is nonlocal thetraditional direct method for numerically solving the frac-tional Cattaneo equations generally requires total O(MN)
memory units and O(MN2) computational complexitywhere N andM are the total number of time steps and spacesteps respectively
In this section we develop a fast solution method for thefinite difference scheme of the time-fractional Cattaneoequation
Let
Nkn Mkminus n minus Mkminus n+1
exp1 minus α2 minus α
(k minus n)Δt1113874 1113875
middot 1 minus 2exp1 minus α2 minus αΔt1113874 1113875 + exp
1 minus α2 minus α
2Δt1113874 11138751113876 1113877
1le nle k
(70)
then
Nk+1n Mk+1minus n minus Mk+1minus n+1
exp1 minus α2 minus α
(k minus n)Δt1113874 1113875
middot 1 minus 2exp1 minus α2 minus αΔt1113874 1113875 + exp
1 minus α2 minus α
2Δt1113874 11138751113876 1113877exp1 minus α2 minus αΔt1113874 1113875
exp1 minus α2 minus αΔt1113874 1113875N
kn
Nk+1k+1 M0 minus M1 1 minus 2exp
1 minus α2 minus αΔt1113874 1113875 + exp
1 minus α2 minus α
2Δt1113874 1113875
(71)
So we have
1113944
k+1
n1N
k+1n Aδtu
ni exp
1 minus α2 minus αΔt1113874 1113875 1113944
k
n1N
knAδtu
ni + N
k+1k+1Aδtu
k+1i
1113944
k+1
n1N
k+1n AyAxδtu
nij exp
1 minus α2 minus αΔt1113874 1113875 1113944
k
n1N
knAyAxδtu
nij
+ Nk+1k+1AyAxδtU
k+1ij
(72)
Remark We find that at the k-th level only O(1) operationsare needed to compute the k-th level since the (k minus 1)-thlevel is known at that point )us the total operations arereduced from O(N2) to O(N) and the memory require-ment decreases from O(N) to O(1) We conclude that thatthis fast method significantly reduces the total computa-tional cost from O(MN2) to O(MN) and the memoryrequirement from O(MN) to O(M)
6 Numerical Experiments
In this section we carry out several numerical experiments tocheck the effectiveness of the proposed scheme )e conver-gence rate and CPU consumption are all compared in thesimulations We take the space-time domainΩ [0 1] T 1for one-dimensional case and Ω [0 1] times [0 1] T 1 fortwo-dimensional case )ese simulations are implemented inMatlab and the numerical experiments are run on a computerwith 4GB memory )e time-fractional Cattaneo equation ofthe following forms is considered
Example 1 We provide the exact solution u(x t)
etsin(πx) and for different α we have different f(x t)
accordingly
Mathematical Problems in Engineering 9
Table 1 Considering Δt h2 the discrete linfin error and convergence rates of u with different α for Example 1
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 15320e minus 04 mdash 15941e minus 04 mdash 18076e minus 04 mdash2minus 4 95109e minus 06 40097 98981e minus 06 40094 11230e minus 05 400862minus 5 59343e minus 07 40024 61762e minus 07 40024 70079e minus 07 400222minus 6 37074e minus 08 40006 38585e minus 08 40006 43783e minus 08 40005
Table 2 Considering h Δt
radic the discrete l2 error and convergence rates of u with different α for Example 1
Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
2minus 4 00017 mdash 00018 mdash 00021 mdash2minus 6 10832e minus 04 40382 11271e minus 04 40373 12781e minus 04 403442minus 8 67252e minus 06 40097 69990e minus 06 40094 79405e minus 06 400862minus 10 41961e minus 07 40024 43672e minus 07 40023 49553e minus 07 400212minus 12 26215e minus 08 40006 27284e minus 08 40005 30959e minus 08 40005
Table 3 )e CPU time consumption of the fast compact difference scheme and direct difference scheme for Example 1
Δtα 125 α 15 α 175
DCD FCD DCD FCD DCD FCD11000 34008 18876 34632 19032 33540 1918812500 142740 46800 141960 47580 138840 4742415000 470967 93600 463010 94692 461450 9375617500 994194 139464 956754 140088 967206 138684110 000 1699318 186733 1641130 186421 1658290 184393125 000 9816362 466910 9509820 469251 9591409 460046150 000 38269853 934133 36923096 940062 37547412 919625
N times104
0
500
1000
1500
2000
2500
3000
3500
4000
CPU
tim
e
Fast evaluationDirect evaluation
0 05 1 15 2 25 3 35 4 45 5
(a)
Fast evaluationDirect evaluation
log10 (N)
ndash05
0
05
1
15
2
25
3
35
4
log1
0 (C
PU ti
me)
25 3 35 4 45 5
(b)
Figure 1 )e CPU time (a) and the log-log CPU time (b) versus the total number of time steps N for Example 1
10 Mathematical Problems in Engineering
Table 4 Considering Δt 2minus 13 the discrete linfin error and convergence rates of u with different α for Example 1
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 22486e minus 04 mdash 22733e minus 04 mdash 23705e minus 04 mdash2minus 4 13985e minus 05 40071 14140e minus 05 40069 14745e minus 05 400692minus 5 86897e minus 07 40084 87882e minus 07 40081 91726e minus 07 400682minus 6 50194e minus 08 41137 51024e minus 08 41063 54092e minus 08 40838
Table 5 Considering h 0001 the discrete linfin error and convergence rates of u with different α for Example 1
Δtα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 00046 mdash 00043 mdash 00036 mdash2minus 4 00011 20641 00011 19668 90063e minus 04 199902minus 5 28662e minus 04 19403 27171e minus 04 20174 22518e minus 04 199992minus 6 71658e minus 05 19999 67928e minus 05 20000 56295e minus 05 200002minus 7 17915e minus 05 20000 16982e minus 05 20000 14074e minus 05 20000
x
ndash02
0
02
04
06
08
1
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(a)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(b)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(c)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(d)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(e)
ndash02
0
02
04
06
08
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(f)
Figure 2 Considering c 0001 α 15 M N 100 real and numerical solutions of u at T 1 for Example 2 with the value of x0 equalto (a) 0 (b) 02 (c) 04 (d) 06 (e) 08 and (f) 1
Mathematical Problems in Engineering 11
zu(x t)
zt+
zαu(x t)
ztα
z2u(x t)
zx2 + f(x t)
u(x 0) sin(πx)zu
zt
1113868111386811138681113868111386811138681113868 t0 sin(πx)
u(0 t) u(1 t) 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(73)
In Tables 1 and 2 we take Δt h2 and h Δt
radicto ex-
amine the discrete linfin-norm (l2-norm) errors and
corresponding spatial and temporal convergence rates re-spectively We list the errors and convergence rates (order) ofthe proposed compact finite difference (CD) scheme which isalmost O(Δt2 + h4) for different α Additionally Table 3shows the CPU time (CPU) consumed by direct compact(DCD) scheme and fast compact difference (FCD) schemerespectively It is obvious that the FCD scheme has a sig-nificantly reduced CPU time over the DCD scheme Forinstance when α 15 we choose h 01 and Δt 150 000and observe that the FCD scheme consumes only 94 secondswhile the DCD scheme consumes 3692 seconds We can find
x
075
08
085
09
095
1
105
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(a)
0
01
02
03
04
05
06
07
08
09
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(b)
0
01
02
03
04
05
06
07
08
09
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(c)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(d)
Figure 3 Considering x0 05 α 15 M N 100 real and numerical solutions of u at T 1 for Example 2 with the value of c equal to(a) 1 (b) 01 (c) 001 and (d) 0001
12 Mathematical Problems in Engineering
that the performance of the FCD scheme will be moreconspicuous as the time step size Δt decreases
In Figure 1 we set h 01 and α 175 and change thetotal number of time steps N to plot out the CPU time (inseconds) of the FCD scheme and DCD scheme We canobserve that the CPU time increases almost linearly withrespect to N for the FCD scheme while the DCD schemescales like O(N2)
Tables 4 and 5 show the discrete linfin errors and con-vergence rates of the compact finite difference scheme forExample 1 )e space rates are almost O(h4) for fixedΔt 2minus 13 and the time convergence rates are always O(Δt2)for fixed h 0001 We can conclude that the numericalconvergence rates of our scheme approach almost toO(Δt2 + h4)
Example 2 )e example is described by
zu(x t)
zt+
zαu(x t)
ztα
z2u(x t)
zx2 + f(x t)
u(x 0) 0zu
zt
1113868111386811138681113868111386811138681113868t0 0
u(0 t) t2exp minusx20
c1113888 1113889 u(1 t) t2exp minus
1 minus x0( 11138572
c1113888 1113889
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(74)
Note that the exact solution of the above problem isu(x t) t2exp(minus (x minus x0)
2c) where x0 isin [0 1] and c is a
N times104
0
1000
2000
3000
4000
5000
6000
7000
8000CP
U ti
me
Fast evaluationDirect evaluation
0 05 1 15 2 25 3 35 4 45 5
(a)
Fast evaluationDirect evaluation
log10 (N)
0
05
1
15
2
25
3
35
4
log1
0 (C
PU ti
me)
25 3 35 4 45 5
(b)
Figure 4 Considering x0 05 c 001 α 15 andM 10 the CPU time (a) and the log-log CPU time (b) versus the total number oftime steps N for Example 2
Table 6 Considering Δt h2 andx0 05 the discrete linfin error and convergence rates of u with different α and c for Example 2
c hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
1
2minus 3 73224e minus 05 mdash 55643e minus 05 mdash 38478e minus 05 mdash2minus 4 45707e-06 40018 34695e minus 06 40034 23900e minus 06 400902minus 5 28558e minus 07 40004 21671e minus 07 40009 14913e minus 07 400242minus 6 17847e minus 08 40001 13543e minus 08 40001 93169e minus 09 40006
01
2minus 3 00012 mdash 00012 mdash 00012 mdash2minus 4 70462e minus 05 40900 69731e minus 05 41051 69007e minus 05 412012minus 5 43571e minus 06 40154 43110e minus 06 40157 42650e minus 06 401612minus 6 27160e minus 07 40038 26871e minus 07 40039 26583e minus 07 40040
001
2minus 4 00090 mdash 00090 mdash 00090 mdash2minus 5 49257e minus 04 41915 49235e minus 04 41922 49232e minus 04 419232minus 6 29868e minus 05 40437 29855e minus 05 40436 29854e minus 05 404362minus 7 18531e minus 06 40106 18523e minus 06 40106 18522e minus 06 40106
Mathematical Problems in Engineering 13
Table 7 Considering h Δt
radicand c 001 the discrete l2 error and convergence rates of u with different α andx0 for Example 2
x0 Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
02minus 8 00044 mdash 00044 mdash 00042 mdash2minus 10 24205e minus 04 41841 23721e minus 04 42133 23098e minus 04 418452minus 12 14684e minus 05 40430 14390e minus 05 40430 14013e minus 05 40429
0252minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14902e minus 04 41794 14894e minus 04 41802 14892e minus 04 418032minus 12 90680e minus 06 40386 90633e minus 06 40386 90616e minus 06 40386
052minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14848e minus 04 41846 14842e minus 04 41852 14842e minus 04 418522minus 12 90323e minus 06 40390 90288e minus 06 40390 90286e minus 06 40390
0752minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14902e minus 04 41794 14894e minus 04 41802 14892e minus 04 418032minus 12 90680e minus 06 40386 90633e minus 06 40386 90616e minus 06 40386
12minus 8 00044 mdash 00044 mdash 00042 mdash2minus 10 24205e minus 04 41841 23721e minus 04 42133 23098e minus 04 418452minus 12 14684e minus 05 40430 14390e minus 05 40430 14013e minus 05 40429
0 2 4 6 8 10times104
0
2000
4000
6000
8000
10000
12000
14000
N
CPU
tim
e
Fast evaluationDirect evaluation
(a)
Fast evaluationDirect evaluation
36 38 4 42 44 46 48 515
2
25
3
35
4
45
log10 (N)
log1
0 (C
PU ti
me)
(b)
Figure 5 )e CPU time (a) and the log-log CPU time (b) versus the total number of time steps N for Example 3
Table 8 Considering Δt h2 the discrete linfin error and convergence rates of u with different α for Example 3
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 2 00028 mdash 00028 mdash 00030 mdash2minus 3 16948e minus 04 40462 17328e minus 04 40142 18411e minus 04 402632minus 4 10522e minus 05 40096 10759e minus 05 40095 11434e minus 05 400922minus 5 65652e minus 07 40024 67133e minus 07 40024 71346e minus 07 40024
14 Mathematical Problems in Engineering
constant and for different α we have different f(x t)
accordinglyWe apply the fast compact difference scheme to dis-
cretize the equation In Figure 2 we setc 0001 α 15 andM N 100 and plot exact andnumerical solutions at time T 1 for Example 2 with dif-ferent x0 For x0 05 α 15 andM N 100 we alsoplot exact and numerical solutions with the different c inFigure 3 In Figure 4 for h 01 and α 15 we vary thetotal number of time steps N to plot out the CPU time (inseconds) of the FCD scheme and DCD scheme )e nu-merical experiments verified our theoretical results In Ta-ble 6 by equating Δt h2 and fixing x0 05 we compute
the discrete linfin error and convergence rates with differentfractional derivative orders α and different c It shows thatthe compact finite difference scheme has space accuracy offourth order and temporal accuracy of second order We seth
Δt
radicand fix c 001 and the discrete l2 error and
convergence rates with different α and x0 are displayed inTable 7 )ese numerical convergence rates are almostapproaching O(Δt2 + h4) as Example 1
Example 3 If the exact solution is given byu(x t) etsin(πx)sin(πy) we have different f(x y t) fordifferent α accordingly
zu(x y t)
zt+
zαu(x y t)
ztα
z2u(x y t)
zx2 +z2u(x y t)
zy2 + f(x y t)
u(x 0) sin(πx)sin(πy)
zu
zt | t0 sin(πx)sin(πy)
u(0 y t) u(1 y t) u(x 0 t) u(x 1 t) 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(75)
In Figure 5 h 01 and α 175 are fixed and the totalnumber of time steps N vary to plot out the CPU time (inseconds) of the FCD procedure and DCD procedure and itpresents an approximately linear computation complexityfor FCD procedure We set Δt h2 in Table 8 and h
Δt
radic
in Table 9 the discrete linfin error discrete l2 error andconvergence rates with different derivative orders α arepresented )e fourth-order space accuracy and second-order temporal accuracy can be observed clearly
7 Conclusion
In this paper we develop and analyze a fast compact finitedifference procedure for the Cattaneo equation equipped withtime-fractional derivative without singular kernel )e time-fractional derivative is of CaputondashFabrizio type with the orderof α(1lt αlt 2) Compact difference discretization is appliedto obtain a high-order approximation for spatial derivatives ofinteger order in the partial differential equation and theCaputondashFabrizio fractional derivative is discretized by means
of CrankndashNicolson approximation It has been proved thatthe proposed compact finite difference scheme has spatialaccuracy of fourth order and temporal accuracy of secondorder Since the fractional derivatives are history dependentand nonlocal huge memory for storage and computationalcost are required )is means extreme difficulty especially fora long-time simulation Enlightened by the treatment forCaputo fractional derivative [32] we develop an effective fastevaluation procedure for the new CaputondashFabrizio fractionalderivative for the compact finite difference scheme Severalnumerical experiments have been carried out to show theconvergence orders and applicability of the scheme
Inspired by the work [43] the topic about modelling andnumerical solutions of porous media flow equipped withfractional derivatives is very interesting and challenging andwill be our main research direction in the future
Data Availability
All data generated or analyzed during this study are includedin this article
Table 9 Considering h Δt
radic the discrete l2 error and convergence rates of u with different α for Example 3
Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
2minus 4 00014 mdash 00014 mdash 00015 mdash2minus 6 84738e minus 05 40463 86640e minus 05 40142 92056e minus 05 402632minus 8 52610e minus 06 40096 53795e minus 06 40095 57168e minus 06 400922minus 10 32826e minus 07 40024 33566e minus 07 40024 35673e minus 07 40023
Mathematical Problems in Engineering 15
Conflicts of Interest
)e authors declare that they have no conflicts of interest
Acknowledgments
)is work was supported in part by the National NaturalScience Foundation of China under Grant nos 91630207and 11971272
References
[1] A Chaves ldquoA fractional diffusion equation to describe Levyflightsrdquo Physics Letters A vol 239 no 1-2 pp 13ndash16 1998
[2] M Giona and H E Roman ldquoFractional diffusion equation fortransport phenomena in random mediardquo Physica A Statis-tical Mechanics and its Applications vol 185 no 1ndash4pp 87ndash97 1992
[3] R Metzler and J Klafter ldquo)e random walkrsquos guide toanomalous diffusion a fractional dynamics approachrdquoPhysics Reports vol 339 no 1 pp 1ndash77 2000
[4] R L Magin Fractional Calculus in Bioengineering BegellHouse Redding Redding CT USA 2006
[5] R L Magin O Abdullah D Baleanu and X J ZhouldquoAnomalous diffusion expressed through fractional orderdifferential operators in the Bloch-Torrey equationrdquo Journalof Magnetic Resonance vol 190 no 2 pp 255ndash270 2008
[6] I Podlubny Fractional Differential Equations An Introduc-tion to Fractional Derivatives Fractional Differential Equa-tions to Methods of Beir Solution and Some of BeirApplications Vol 198 Elsevier Amsterdam Netherlands1998
[7] M Ran and C Zhang ldquoCompact difference scheme for a classof fractional-in-space nonlinear damped wave equations intwo space dimensionsrdquo Computers amp Mathematics withApplications vol 71 no 5 pp 1151ndash1162 2016
[8] A H Bhrawy and M A Zaky ldquoHighly accurate numericalschemes for multi-dimensional space variable-order frac-tional Schrodinger equationsrdquo Computers amp Mathematicswith Applications vol 73 no 6 pp 1100ndash1117 2017
[9] S B Yuste ldquoWeighted average finite difference methods forfractional diffusion equationsrdquo Journal of ComputationalPhysics vol 216 no 1 pp 264ndash274 2006
[10] T A M Langlands and B I Henry ldquo)e accuracy andstability of an implicit solution method for the fractionaldiffusion equationrdquo Journal of Computational Physicsvol 205 no 2 pp 719ndash736 2005
[11] C-M Chen F Liu I Turner and V Anh ldquoA Fourier method forthe fractional diffusion equation describing sub-diffusionrdquo Journalof Computational Physics vol 227 no 2 pp 886ndash897 2007
[12] Y Lin and C Xu ldquoFinite differencespectral approximationsfor the time-fractional diffusion equationrdquo Journal of Com-putational Physics vol 225 no 2 pp 1533ndash1552 2007
[13] G-h Gao and Z-z Sun ldquoA compact finite difference schemefor the fractional sub-diffusion equationsrdquo Journal of Com-putational Physics vol 230 no 3 pp 586ndash595 2011
[14] F Zeng C Li F Liu and I Turner ldquoNumerical algorithms fortime-fractional subdiffusion equation with second-order ac-curacyrdquo SIAM Journal on Scientific Computing vol 37 no 1pp A55ndashA78 2015
[15] G-H Gao Z-Z Sun and H-W Zhang ldquoA new fractionalnumerical differentiation formula to approximate the Caputofractional derivative and its applicationsrdquo Journal of Com-putational Physics vol 259 no 2 pp 33ndash50 2014
[16] H Li J Cao and C Li ldquoHigh-order approximation to Caputoderivatives and Caputo-type advection-diffusion equations(III)rdquo Journal of Computational and Applied Mathematicsvol 299 pp 159ndash175 2016
[17] A A Alikhanov ldquoA new difference scheme for the timefractional diffusion equationrdquo Journal of ComputationalPhysics vol 280 pp 424ndash438 2015
[18] Z Wang and S Vong ldquoCompact difference schemes for themodified anomalous fractional sub-diffusion equation and thefractional diffusion-wave equationrdquo Journal of ComputationalPhysics vol 277 pp 1ndash15 2014
[19] H-K Pang and H-W Sun ldquoFourth order finite differenceschemes for time-space fractional sub-diffusion equationsrdquoComputers amp Mathematics with Applications vol 71 no 6pp 1287ndash1302 2016
[20] M Dehghan M Abbaszadeh and W Deng ldquoFourth-ordernumerical method for the space-time tempered fractionaldiffusion-wave equationrdquo Applied Mathematics Lettersvol 73 no 11 pp 120ndash127 2017
[21] H Sun Z-Z Sun and G-H Gao ldquoSome high order dif-ference schemes for the space and time fractional Bloch-Torrey equationsrdquo Applied Mathematics and Computationvol 281 pp 356ndash380 2016
[22] S Arshad J Huang A Q M Khaliq and Y Tang ldquoTrape-zoidal scheme for time-space fractional diffusion equationwith Riesz derivativerdquo Journal of Computational Physicsvol 350 pp 1ndash15 2017
[23] H Wang K Wang and T Sircar ldquoA direct O(Nlog2N) finitedifferencemethod for fractional diffusion equationsrdquo Journal ofComputational Physics vol 229 no 21 pp 8095ndash8104 2010
[24] H Wang and T S Basu ldquoA fast finite difference method fortwo-dimensional space-fractional diffusion equationsrdquo SIAMJournal on Scientific Computing vol 34 no 5 pp A2444ndashA2458 2012
[25] H Wang and N Du ldquoA superfast-preconditioned iterativemethod for steady-state space-fractional diffusion equationsrdquoJournal of Computational Physics vol 240 pp 49ndash57 2013
[26] H Wang and N Du ldquoA fast finite difference method forthree-dimensional time-dependent space-fractional diffusionequations and its efficient implementationrdquo Journal ofComputational Physics vol 253 pp 50ndash63 2013
[27] H Wang and N Du ldquoFast alternating-direction finite dif-ference methods for three-dimensional space-fractional dif-fusion equationsrdquo Journal of Computational Physics vol 258pp 305ndash318 2014
[28] K Wang and H Wang ldquoA fast characteristic finite differencemethod for fractional advection-diffusion equationsrdquo Ad-vances in Water Resources vol 34 no 7 pp 810ndash816 2011
[29] H Wang and H Tian ldquoA fast Galerkin method with efficientmatrix assembly and storage for a peridynamic modelrdquoJournal of Computational Physics vol 231 no 23pp 7730ndash7738 2012
[30] H Tian H Wang and W Wang ldquoAn efficient collocationmethod for a non-local diffusion modelrdquo InternationalJournal of Numerical Analysis amp Modeling vol 10 no 4pp 815ndash825 2013
[31] H Wang and H Tian ldquoA fast and faithful collocation methodwith efficient matrix assembly for a two-dimensional nonlocaldiffusion modelrdquo Computer Methods in Applied Mechanicsand Engineering vol 273 no 5 pp 19ndash36 2014
[32] S Jiang J Zhang Q Zhang and Z Zhang ldquoFast evaluation ofthe Caputo fractional derivative and its applications tofractional diffusion equationsrdquo Communications in Compu-tational Physics vol 21 no 3 pp 650ndash678 2017
16 Mathematical Problems in Engineering
[33] Y Yan Z-Z Sun and J Zhang ldquoFast evaluation of the Caputofractional derivative and its applications to fractional diffusionequations a second-order schemerdquo Communications inComputational Physics vol 22 no 4 pp 1028ndash1048 2017
[34] X Lu H-K Pang and H-W Sun ldquoFast approximate in-version of a block triangular Toeplitz matrix with applicationsto fractional sub-diffusion equationsrdquo Numerical Linear Al-gebra with Applications vol 22 no 5 pp 866ndash882 2015
[35] M Caputo and M Fabrizio ldquoA new definition of fractionalderivative without singular kernelrdquo Progress in FractionalDifferentiation and Applications vol 1 no 2 pp 73ndash85 2015
[36] A Atangana ldquoOn the new fractional derivative and appli-cation to nonlinear Fisherrsquos reaction-diffusion equationrdquoAppliedMathematics and Computation vol 273 pp 948ndash9562016
[37] M D Ortigueira and J Tenreiro Machado ldquoA critical analysisof the Caputo-Fabrizio operatorrdquo Communications in Non-linear Science and Numerical Simulation vol 59 pp 608ndash6112018
[38] V E Tarasov ldquoNo nonlocality no fractional derivativerdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 62 pp 157ndash163 2018
[39] V Tarasov ldquoCaputo-Fabrizio operator in terms of integerderivatives memory or distributed lagrdquo Computational andApplied Mathematics vol 38 p 113 2019
[40] A Giusti ldquoA comment on some new definitions of fractionalderivativerdquoNonlinear Dynamics vol 93 no 3 pp 1757ndash17632018
[41] A Atangana and J F Gomez-Aguilar ldquoFractional derivativeswith no-index law property application to chaos and sta-tisticsrdquo Chaos Solitons amp Fractals vol 114 pp 516ndash535 2018
[42] J Hristov ldquoResponse functions in linear viscoelastic consti-tutive equations and related fractional operatorsrdquo Mathe-matical Modelling of Natural Phenomena vol 14 no 3pp 1ndash34 2019
[43] L X Vivas-Cruz A Gonzalez-Calderon M A Taneco-Hernandez and D P Luis ldquo)eoretical analysis of a model offluid flow in a reservoir with the Caputo-Fabrizio operatorrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 84 Article ID 105186 2020
[44] H-L Liao and Z-Z Sun ldquoMaximum norm error bounds ofADI and compact ADI methods for solving parabolicequationsrdquo Numerical Methods for Partial DifferentialEquations vol 26 no 1 pp 37ndash60 2010
[45] Z Liu A Cheng and X Li ldquoA second order Crank-Nicolsonscheme for fractional Cattaneo equation based on newfractional derivativerdquoAppliedMathematics and Computationvol 311 pp 361ndash374 2017
[46] Z Sun and G Gao Finite Difference Method for FractionalDifferential Equations Science Press Beijing China 2015
Mathematical Problems in Engineering 17
powers As responses to these criticisms Atangana andGomez-Aguilar [41] pointed out the need to account for afractional calculus approach without an imposed index lawand with nonsingular kernels Furthermore Hristov [42]indicated that the CF operator is not applicable forexplaining the physical examples in [37 40] instead hesuggested that the CF operator can be used for the analysis ofmaterials that do not follow a power-law behavior )eauthors of [43] believe that models with CF operatorsproduce a better representation of physical behaviors thando integer-order models providing a way to model theintermediate (between elliptic and parabolic or betweenparabolic and hyperbolic) behaviors
To obtain the accuracy of the fourth order in spatialdirections the following lemma is necessary
Lemma 1 (see [44]) Denote θ(s) (1 minus s)3[5 minus 3(1 minus s)2]If f(x) isin C6[a b] h (b minus a)M xi a + ih(0le ileM)then it holds that
112
fPrime ximinus 1( 1113857 + 10fPrime xi( 1113857 + fPrime xi+1( 11138571113858 1113859
1h2 f ximinus 1( 11138571113890 minus 2f xi( 1113857 + f xi+1( 11138571113891
+h4
3601113946
1
0 f(6)
xi minus sh( 1113857 + f(6)
1113960
middot xi + sh( 1113857]θ(s)ds 1le ileM minus 1
(8)
3 Compact Finite Difference Scheme for One-Dimensional Fractional Cattaneo Equation
In order to construct the finite difference schemes the in-terval [a b] is divided into subintervals withxi a + ih (0le ileM) and [0 T] is discretized withtk kΔt (0le kleN) where h (b minus a)M and Δt TNare the spatial grid size and temporal step size respectivelyDenote Ωh xi 0le ileM1113864 1113865 ΩΔt tk 0le kleN1113864 1113865 thenΩh timesΩΔt becomes a discretization of the practical compu-tational domain [a b] times [0 T] )e values of the function uat the grid points are denoted as uk
j u(xj tk) and theapproximate solution at the point (xj tk) is denoted as uk
j Denote Vh v|v (v0 v1 vM)1113864 1113865 We also introduce
the following notations for any mesh function v isin Vh
δxvjminus (12) vj minus vjminus 1
h
δ2xvj vj+1 minus 2vj + vjminus 1
h2 1le jleM minus 1
(9)
and define the average operator
Avj
112
vj+1 + 10vj + vjminus 11113872 1113873 1le jleM minus 1
vj j 0 M
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(10)
It is easy to see that
Avj I +h2
12δ2x1113888 1113889vj (11)
where I is the identical operator We also denoteAv (Av1Av2 AvM) for vector v (v1 v2 vM)and A(u v) (Au v)
For any two gird functions u v isin V0h
v|v isin Vh v0 vM 01113864 1113865 the discrete inner products andnorms are defined as
(u v) h 1113944Mminus 1
j1ujvj
langu vrang h 1113944Mminus 1
j1δxujminus (12)δxvjminus (12)
u22 (u u)
|u|21 langu urang
uinfin max1lejleMminus 1
uj
11138681113868111386811138681113868
11138681113868111386811138681113868
(12)
By summation by parts it is easy to see that
δ2xu v1113872 1113873 minus δxu δxv( 1113857 minus langu vrang u δ2xv1113872 1113873 (13)
For the average operator A define
A(v v)≜ (Av v) v2A (14)
Additionally let VΔt v|v (v0 v1 middot middot middot vN)1113864 1113865 be thespace of grid function defined on ΩΔt For any functionv isin VΔt a difference operator is introduced as follows
δtvk
vk minus vkminus 1
Δt (15)
31 Compact Finite Difference Scheme We will consider thetime-fractional Cattaneo equation equipped with theCaputondashFabrizio derivative Vivas-Cruz et al [43] gave thetheoretical analysis of a model of fluid flow in a reservoirwith the CaputondashFabrizio operator )ey proved that thismodel cannot be used to describe nonlocal processes since itcan be represented as an equivalent differential equationwith a finite number of integer-order derivatives
)e finite difference methods usually lead to stencilsthrough the whole history passed by the solution whichconsume too much computational work In this paper wewill establish a high-order finite difference scheme andpropose a procedure to reduce the computational cost In[43] the authors proposed a recurrence formula of dis-cretized CF operator and obtained an algorithm which canbe considered a stencil with a one-step expression withoutthe need of integrals over the whole history It seems that theprocedure in our paper and the algorithm in [43] are dif-ferent in approach but equally satisfactory in result
For obtaining effective approximation with high orderwe introduce the numerical discretization for the fractional
Mathematical Problems in Engineering 3
Cattaneo equation by means of compact finite differencemethods
At the node (xi tk+(12)) the differential equation takesthe following form
zu
ztxi tk+(12)1113872 1113873 +
zαu
ztαxi tk+(12)1113872 1113873
z2u
zx2 xi tk+(12)1113872 1113873 + f xi tk+(12)1113872 1113873
1le ileM minus 1 1le kleN minus 1
(16)
)e approximation of the fractional derivative is givenby [45]
CF0 D
αt u xi tk+(12)1113872 1113873
1(α minus 1)Δt
M0δtuk+1i minus 1113944
k
n1Mkminus n(⎡⎣
minus Mkminus n+11113857δtuji minus Mkψi
⎤⎦ + Rk+(12)i
(17)
with truncation error Rk+(12)i O(Δt2) and
Mn exp1 minus α2 minus αΔtn1113876 1113877 minus exp
1 minus α2 minus αΔt(n + 1)1113876 1113877 (18)
Furthermore by Lemma 1 and equation (15) the spaceand time derivative are approximated by
z2u
zx2 xi tk+ (12)1113872 1113873 12
δ2xuk+1i + δ2xuk
i1113872 1113873
A+ O Δt2 + h
41113872 1113873
(19)
zu
ztxi tk+(12)1113872 1113873 δtu
k+1i + O Δt21113872 1113873 (20)
Substituting (17) and (19)sim(20) into (16) we get
Aδtuk+1i +
1(α minus 1)Δt
A M0δtuk+1i minus 1113944
k
n1Mkminus n(⎡⎣
minus Mkminus n+11113857δtuni minus Mkψi1113891
12
δ2xuk+1i + δ2xu
ki1113872 1113873 + Af
k+(12)i + R
k+(12)i
1le ileM minus 1 0le kleN minus 1
(21)
and there exists a constant C depending on the function uand its derivatives such that
Rk+(12)i leC Δt2 + h
41113872 1113873 (22)
By the initial and boundary value conditions we have
u0i ϕi 1le ileM minus 1
uk0 u
kM 0 0le kleN
(23)
A compact finite difference scheme can be established byomitting the truncation term R
k+(12)i and replacing the exact
solution uki in equation (21) with numerical solution uk
i
(α minus 1)ΔtAδtuk+1i + M0Aδtu
k+1i minus
(α minus 1)Δt2
δ2xUk+1
(α minus 1)Δt
2δ2xU
k+ 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857Aδtu
ni + MkAψi
+(α minus 1)ΔtAfk+ (12)i
1le ileM minus 1 1le kleN minus 1
u0i ϕi 1le ileM minus 1
uk0 u
kM 0 0le kleN
(24)
32 Stability Analysis and Optimal Error Estimates
321 Stability Analysis )e following Lemma about Mn isuseful for the analysis of stability
Lemma 2 (see [45]) For the definition of Mn Mn gt 0 andMn+1 ltMn forallnle k are held
Multiplying hδtuk+1i on both sides of equation (24) and
summing up with respect to i from 1 to M minus 1 the followingequation is obtained
(α minus 1)Δt δtUk+1
2
A+ M0 δtU
k+1
2
Aminusα minus 12
δ2xUk+1
Uk+1
1113872 1113873
+α minus 12
δ2xUk+1
Uk
1113872 1113873
α minus 12
δ2xUk U
k+11113872 1113873 minus
α minus 12
δ2xUk U
k1113872 1113873
+ 1113944k
n1Mkminus n minus Mkminus n+1( 1113857 AδtU
n δtU
k+11113872 1113873
+ Mk Aψ δtUk+1
1113872 1113873 +(α minus 1)Δt Afk+(12)
δtUk+1
1113872 1113873
(25)
Observing equation (13) we have
δ2xUk+1 Uk+11113872 1113873 minus δxUk+1 δxUk+1( 1113857 minus δxUk+11113868111386811138681113868
111386811138681113868111386821 le 0
δ2xUk Uk1113872 1113873 minus δxUk δxUk( 1113857 minus δxUk1113868111386811138681113868
111386811138681113868111386821 le 0
δ2xUk+1 Uk1113872 1113873 minus δxUk+1 δxUk( 1113857 minus langUk+1 Ukrang δ2xUk Uk+11113872 1113873
(26)
By the triangle inequality and Lemma 2 we obtain
4 Mathematical Problems in Engineering
1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 AδtU
n δtU
k+11113872 1113873 + Mk Aψ δtU
k+11113872 1113873
le 1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δtUn
2A
+ δtUk+1
2
A1113874 1113875
+12Mk ψ
2A + δtU
k+1
2
A1113874 1113875
1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δtUn
2A
+12
M0 minus Mk( 1113857
middot δtUk+1
2
A+12Mk ψ
2A + δtU
k+1
2
A1113874 1113875
1113944k
n1
12Mkminus n δtU
n
2A
minus 1113944k
n1
12Mkminus n+1 δtU
n
2A
+12M0
δtUk+1
2
A+12Mkψ
2A
(27)Combining equation (25) with (26)sim(27) we get
1113944
k+1
n1Mkminus n+1 δtU
n
2A
+(α minus 1) δxUk+1
11138681113868111386811138681113868
111386811138681113868111386811138682
1
le 1113944k
n1Mkminus n δtU
n
2A
+(α minus 1) δxUk
11138681113868111386811138681113868
111386811138681113868111386811138682
1+ Mkψ
2A
+(α minus 1)Δt fk+(12)
2
A
(28)
Let
Q Uk
1113872 1113873 1113944k
n1Mkminus n δtU
n
2A
+(α minus 1) δxUk
11138681113868111386811138681113868
111386811138681113868111386811138682
1 (29)
Summing up with respect to k from 0 to N minus 1 leads to
Q UN
1113872 1113873leQ U0
1113872 1113873 + 1113944Nminus 1
k0Mkψ
2A +(α minus 1)Δt 1113944
Nminus 1
k0f
k+(12)
2
A
(30)
)e initial condition U0 ϕ implies that Q(U0)
(α minus 1)| δxϕ| 21 and then
Q UN
1113872 1113873le (α minus 1) δxϕ1113868111386811138681113868
111386811138681113868111386821 + 1113944
Nminus 1
k0Mkψ
2A
+(α minus 1)Δt 1113944Nminus 1
k0f
k+(12)
2
A
(31)
Theorem 1 For scheme (24) we have the following stableconclusion
Q Um
( 1113857le (α minus 1) δxϕ1113868111386811138681113868
111386811138681113868111386821 + 1113944
mminus 1
k0Mkψ
2A
+(α minus 1)Δt 1113944mminus 1
k0f
k+(12)
2
A forall0lemleN
(32)
322 Optimal Error Estimates Combining equations (21)and (23) with (24) we get an error equation as follows
(α minus 1)ΔtAδtek+1i + M0Aδte
k+1i minus
(α minus 1)Δt2
δ2xek+1
(α minus 1)Δt
2δ2xe
k+ 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857Aδte
ni
+(α minus 1)ΔtRk+(12)i
(33)
where Rk+(12)i O(Δt2 + h4) and ek
i uki minus uk
i forallkge 0Multiplying hδte
k+1i on both sides of equation (33) and
summing up with respect to i from 1 to M minus 1 we get
(α minus 1)Δt δtek+1
2
A+ M0 δte
k+1
2
A+α minus 12
δxek+1
11138681113868111386811138681113868
111386811138681113868111386811138682
1
α minus 12
δxek
11138681113868111386811138681113868
111386811138681113868111386811138682
1+ 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 Aδte
n δte
k+11113872 1113873
+(α minus 1)Δt Rk+(12)
δtek+1
1113872 1113873
(34)
By the triangle inequality and Lemma 2 we obtain
1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 Aδte
n δte
k+11113872 1113873
le 1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δten
2A
+ δtek+1
2
A1113874 1113875
1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δten
2A
+12
M0 minus Mk( 1113857 δtek+1
2
A
le 1113944k
n1
12Mkminus n δte
n
2A
minus 1113944k
n1
12Mkminus n+1 δte
n
2A
+12M0 δte
k+1
2
A
(35)
Combining equation (34) with (35) we have
1113944
k+1
n1Mkminus n+1 δte
n
2A
+(α minus 1) δxek+1
11138681113868111386811138681113868
111386811138681113868111386811138682
1le 1113944
k
n1Mkminus n δte
n
2A
+(α minus 1) δxek
11138681113868111386811138681113868
111386811138681113868111386811138682
1+(α minus 1)Δt R
k+(12)
2
(36)
By the definition of Q in stability analysis the inequality(36) can be rearranged as
Mathematical Problems in Engineering 5
Q ek+1
1113872 1113873leQ ek
1113872 1113873 +(α minus 1)Δt Rk+(12)
2 (37)
Summing up with respect to k from 0 to N minus 1 we get
Q eN
1113872 1113873leQ e0
1113872 1113873 + C Δt2 + h4
1113872 11138732 (38)
Observing that the initial error e0 0 implies Q(e0) 0)en we have
(α minus 1) δxeN
11138681113868111386811138681113868
111386811138681113868111386811138682
1leQ e
N1113872 1113873leC Δt2 + h
41113872 1113873
2
1113944
N
n1MNminus n δte
n
2AleQ e
N1113872 1113873leC Δt2 + h
41113872 1113873
2
(39)
Theorem 2 Suppose that the exact solution of the fractionalCattaneo equation is smooth sufficiently then there exists apositive constant C independent of h k and Δt such that
ek
11138681113868111386811138681113868
111386811138681113868111386811138681leC Δt2 + h
41113872 1113873 forall0le kleN (40)
where eki uk
i minus uki
4 Compact Finite Difference Scheme inTwo Dimensions
In this section the following fractional Cattaneo equation intwo dimensions will be considered
zu(x y t)
zt+
zαu(x y t)
ztα
z2u(x y t)
zx2 +z2u(x y t)
zy2 + f(x y t) (x y t) isin (a b) times(c d) times(0 T]
u(x y 0) ϕ(x y)zu
zt
1113868111386811138681113868111386811138681113868 t0 ψ(x y) ale xle b cleyled
u(a y t) u(b y t) u(x c t) u(x d t) 0 tgt 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(41)
where 1lt αlt 2 f(x y t) is the source termϕ(x y) andψ(x y) are the given functions and zαuztα isdefined by the new Caputo fractional derivative withoutsingular kernel
In order to construct the finite difference schemes therectangle [a b] times [c d] is discretized with xi
a + ihx (0le ileMx) and yj c + jhy (0le jleMy) andthe time interval [0 T] is discretized with tk kΔt (0le kleN)where hx (b minus a)Mx hy (d minus c) My and Δt TN arethe spatial grid and temporal step sizes respectively DenoteΩh
(xi yj) 0le ileMx 0le1113966 jleMy and ΩΔt tk 0le kleN1113864 1113865then Ωh timesΩΔt is a discretization of the physical computationaldomain [a b] times [c d] times [0 T] uk
ij u(xi yj tk) denotes thevalues of function u at the grid points and uk
ij denotes the valuesof the numerical solution at the point (xi yj tk)
Denote Vh v| v (v00 v01 vMxMy)1113882 1113883 and V0
h
v| v isin Vh v0j vMxj vi0 viMy 0 0 le i leMx 0le j1113882
leMy For any mesh function v isin Vh we use the followingnotations
δxviminus (12)j vij minus viminus 1j
hx
δ2xvij vi+1j minus 2vij + viminus 1j
h2x
δyvijminus (12) vij minus vijminus 1
hy
δ2yvij vij+1 minus 2vij + vijminus 1
h2y
1le ileMx minus 1 1le jleMy minus 1
(42)
and define the average operator
Axvij
112
vi+1j + 10vij + viminus 1j1113872 1113873 1le ileMx minus 1 0le jleMy
vij i 0 Mx 0le jleMy
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
Ayvij
112
vij+1 + 10vij + vijminus 11113872 1113873 1le jleMy minus 1 0le ileMx
vij j 0 My 0le ileMx
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(43)
6 Mathematical Problems in Engineering
It is clear that
Axvij I +h2
x
12δ2x1113888 1113889vij
Ayvij I +h2
y
12δ2y⎛⎝ ⎞⎠vij
(44)
We also denote AxAy(u v) (AxAyu v) It is easy to seethat AxAy AyAx
For any gird function u v isin V0h the discrete inner
product and norms are defined as follows
(u v) hxhy 1113944
Mxminus 1
i11113944
My minus 1
j1uijvij
u22 (u u)
(45)
For the average operator AxAy define
AxAy(v v)≜ AxAyv v1113872 1113873 v2A (46)
41 Compact Finite Difference Scheme At the node(xi yj tk+(12)) the differential equation is rewritten as
zu
ztxi yj tk+(12)1113872 1113873 +
zαu
ztαxi yj tk+(12)1113872 1113873
z2u
zx2 xi yj tk+(12)1113872 1113873 +z2u
zy2 xi yj tk+(12)1113872 1113873
+ f xi yj tk+(12)1113872 1113873
1le ileMx minus 1 1le jleMy minus 1 1le kleN minus 1
(47)
For the approximation of the time-fractional derivativewe have the following approximation [45]
CF0 D
αt u xi yj tk+(12)1113872 1113873
1(α minus 1)Δt
M0δtuk+1ij minus 1113944
k
n1Mkminus n(⎡⎣
minus Mkminus n+11113857δtunij minus Mkψij
⎤⎦
+ Rk+(12)ij
(48)
where the truncation error Rk+(12)ij O(Δt2) and
Mn exp1 minus α2 minus αΔtn1113876 1113877 minus exp
1 minus α2 minus αΔt(n + 1)1113876 1113877 (49)
Furthermore we also have
z2u
zx2 xi yj tk+(12)1113872 1113873 12
δ2xuk+1ij + δ2xuk
ij1113872 1113873
Ax
+ O Δt2 + h4x1113872 1113873
(50)
z2u
zy2 xi yj tk+(12)1113872 1113873 12
δ2yuk+1ij + δ2yuk
ij1113872 1113873
Ay
+ O Δt2 + h4y1113872 1113873
(51)
zu
ztxi yj tk+(12)1113872 1113873 δtu
k+1ij + O Δt21113872 1113873 (52)
Substituting (48) and (50)sim(52) into (47) leads to
AxAyδtuk+1ij +
1(α minus 1)τ
AxAy M0δtuk+1ij1113890
minus 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857δtu
nij minus Mkψij
⎤⎦
12
δ2xuk+1ij + δ2xu
kij1113872 1113873 +
12
δ2yuk+1ij + δ2yu
kij1113872 1113873 + AxAyf
k+(12)ij
+ Rk+(12)ij
1le ileMx minus 1 1le jleMy minus 1 0le kleN minus 1
(53)
and there exists a constant C depending on the function uand its derivatives such that
Rk+(12)ij leC Δt2 + h
4x + h
4y1113872 1113873 (54)
By the initial and boundary conditions we have
u0ij ϕij 1le ileMx minus 1 1le jleMy minus 1
uk0j u
kMxj u
ki0 u
kiMy
0 0le kleN(55)
Omitting the truncation error Rk+(12)ij and replacing the
true solution ukij with numerical solution uk
ij a compactfinite difference scheme can be obtained as follows
(α minus 1)ΔtAxAyδtUk+1ij + M0AxAyδtU
k+1ij
minus Ay
(α minus 1)Δt2
δ2xUk+1ij minus Ax
(α minus 1)Δt2
δ2yUk+1ij
Ay
(α minus 1)Δt2
δ2xukij + Ax
(α minus 1)Δt2
δ2yukij
+ 1113944k
n1Mkminus n minus Mkminus n+1( 1113857AxAyδtu
nij + MkAxAyψij
+ (α minus 1)ΔtAxAyfk+(12)ij
1le ileMx minus 1 1le jleMy minus 1 1le kleN minus 1
u0ij ϕij 1le ileMx minus 1 1le jleMy minus 1
uk0j u
kMxj u
ki0 u
kiMy
0 0le kleN
(56)
Mathematical Problems in Engineering 7
42 Stability Analysis and Optimal Error Estimates
421 Stability Analysis
Definition 4 (see [46]) For any gird function u isin V0h define
the norm
⦀nablahu⦀2A δxu
2
minush2
y
12δyδxu
2
⎛⎝ ⎞⎠ + δyu
2
minush2
x
12δxδyu
2
1113888 1113889
(57)
)e lemmas below is useful in the subsequent analysis ofstability
Lemma 3 (see [46]) For any gird function u isin V0h the
following equation is held13u
2 le u2A le u
2 (58)
Lemma 4 (see [46]) For any gird function u isin V0h the
following equation is held23nablahu
2 le⦀nablahu⦀2A le nablahu
2
Ayδ2xu
nminus (12)+ Axδ
2yu
nminus (12) δtu
nminus (12)1113872 1113873
minus12Δt⦀nablahu
n⦀2A minus ⦀nablahunminus 1⦀
2A1113874 1113875
(59)
where nablahu2 δxu22 + δyu22
Multiplying hxhyδtUk+1ij on both sides of equation (56)
and summing up wrt i j from 1 to (Mx minus 1) and from 1 to(My minus 1) respectively the following equation is obtained
(α minus 1)Δt δtUk+1
2
A+ M0 δtU
k+1
2
A
(α minus 1)Δt Ayδ2xU
k+(12)+ Axδ
2yU
k+(12) δtU
k+11113872 1113873 + 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 AyAxδtU
n δtU
k+11113872 1113873
+ Mk AyAxψ δtUk+1
1113872 1113873 +(α minus 1)Δt AyAxfk+(12)
δtUk+1
1113872 1113873
(60)
Observing Lemma 4 we have
Ayδ2xUk+(12) + Axδ
2yUk+(12) δtU
k+11113872 1113873 minus12Δt⦀nablahu
k+1⦀2A minus ⦀nablahu
k⦀2A1113874 1113875 (61)
(α minus 1)Δt AyAxfk+(12) δtUk+11113872 1113873le
(α minus 1)Δt2
fk+(12)
2
A+ δtu
k+1
2
A1113874 1113875 (62)
By the triangle inequality and Lemma 2 we obtain
1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 AyAxδtU
n δtU
k+11113872 1113873 + Mk AyAxψ δtU
k+11113872 1113873
le 1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δtUn
2A
+ δtUk+1
2
A1113874 1113875 +
12Mk ψ
2A + δtU
k+1
2
A1113874 1113875
1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δtUn
2A
+12
M0 minus Mk( 1113857 δtUk+1
2
A+12Mk ψ
2A + δtU
k+1
2
A1113874 1113875
1113944k
n1
12Mkminus n δtU
n
2A
minus 1113944k
n1
12Mkminus n+1 δtU
n
2A
+12M0 δtU
k+1
2
A+12Mkψ
2A
(63)
Combining equation (60) with (61)sim(63)we get
1113944
k+1
n1Mkminus n+1 δtU
n
2A
+(α minus 1)⦀nablahUk+1⦀
2A
le 1113944k
n1Mkminus n δtU
n
2A
+(α minus 1) nablahUk
2
A+ Mkψ
2A +(α minus 1)Δt⦀fk+(12)⦀
2A
(64)
8 Mathematical Problems in Engineering
Let
Q Uk
1113872 1113873 1113944k
n1Mkminus n δtU
n
2A
+(α minus 1)⦀nablahUk⦀
2A (65)
Summing up with respect to k from 0 to N minus 1 we get
Q UN
1113872 1113873leQ U0
1113872 1113873 + 1113944Nminus 1
k0Mkψ
2A +(α minus 1)Δt 1113944
Nminus 1
k0f
k+(12)
2
A
(66)
Noting that U0 ϕ we have Q(U0) (α minus 1)nablahϕ2A Itfollows that
Q UN
1113872 1113873le (α minus 1)⦀nablahϕ⦀2A + 1113944
Nminus 1
k0Mkψ
2A
+(α minus 1)Δt 1113944Nminus 1
k0f
k+(12)
2
A
(67)
Theorem 3 For the compact finite difference scheme (56)the following stability inequality holds
Q Um
( 1113857le (α minus 1) nablahϕ1113868111386811138681113868
1113868111386811138681113868
2A
+ 1113944
mminus 1
k0Mkψ
2A
+(α minus 1)Δt 1113944
mminus 1
k0f
k+(12)
2
A forall0lemleN
(68)
Similar to the stability the convergence can also beanalyzed
Theorem 4 Suppose that the exact solution of the fractionalCattaneo equation is sufficiently smooth then there exists apositive constant C independent of h k and Δt such that
ek
11138681113868111386811138681113868
111386811138681113868111386811138681leC Δt2 + h
41113872 1113873 forall0le kleN (69)
where ekij uk
ij minus ukij and h max hx hy1113966 1113967
5 Efficient Storage and Fast Evaluation of theCaputondashFabrizio Fractional Derivative
Since time-fractional derivative operator is nonlocal thetraditional direct method for numerically solving the frac-tional Cattaneo equations generally requires total O(MN)
memory units and O(MN2) computational complexitywhere N andM are the total number of time steps and spacesteps respectively
In this section we develop a fast solution method for thefinite difference scheme of the time-fractional Cattaneoequation
Let
Nkn Mkminus n minus Mkminus n+1
exp1 minus α2 minus α
(k minus n)Δt1113874 1113875
middot 1 minus 2exp1 minus α2 minus αΔt1113874 1113875 + exp
1 minus α2 minus α
2Δt1113874 11138751113876 1113877
1le nle k
(70)
then
Nk+1n Mk+1minus n minus Mk+1minus n+1
exp1 minus α2 minus α
(k minus n)Δt1113874 1113875
middot 1 minus 2exp1 minus α2 minus αΔt1113874 1113875 + exp
1 minus α2 minus α
2Δt1113874 11138751113876 1113877exp1 minus α2 minus αΔt1113874 1113875
exp1 minus α2 minus αΔt1113874 1113875N
kn
Nk+1k+1 M0 minus M1 1 minus 2exp
1 minus α2 minus αΔt1113874 1113875 + exp
1 minus α2 minus α
2Δt1113874 1113875
(71)
So we have
1113944
k+1
n1N
k+1n Aδtu
ni exp
1 minus α2 minus αΔt1113874 1113875 1113944
k
n1N
knAδtu
ni + N
k+1k+1Aδtu
k+1i
1113944
k+1
n1N
k+1n AyAxδtu
nij exp
1 minus α2 minus αΔt1113874 1113875 1113944
k
n1N
knAyAxδtu
nij
+ Nk+1k+1AyAxδtU
k+1ij
(72)
Remark We find that at the k-th level only O(1) operationsare needed to compute the k-th level since the (k minus 1)-thlevel is known at that point )us the total operations arereduced from O(N2) to O(N) and the memory require-ment decreases from O(N) to O(1) We conclude that thatthis fast method significantly reduces the total computa-tional cost from O(MN2) to O(MN) and the memoryrequirement from O(MN) to O(M)
6 Numerical Experiments
In this section we carry out several numerical experiments tocheck the effectiveness of the proposed scheme )e conver-gence rate and CPU consumption are all compared in thesimulations We take the space-time domainΩ [0 1] T 1for one-dimensional case and Ω [0 1] times [0 1] T 1 fortwo-dimensional case )ese simulations are implemented inMatlab and the numerical experiments are run on a computerwith 4GB memory )e time-fractional Cattaneo equation ofthe following forms is considered
Example 1 We provide the exact solution u(x t)
etsin(πx) and for different α we have different f(x t)
accordingly
Mathematical Problems in Engineering 9
Table 1 Considering Δt h2 the discrete linfin error and convergence rates of u with different α for Example 1
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 15320e minus 04 mdash 15941e minus 04 mdash 18076e minus 04 mdash2minus 4 95109e minus 06 40097 98981e minus 06 40094 11230e minus 05 400862minus 5 59343e minus 07 40024 61762e minus 07 40024 70079e minus 07 400222minus 6 37074e minus 08 40006 38585e minus 08 40006 43783e minus 08 40005
Table 2 Considering h Δt
radic the discrete l2 error and convergence rates of u with different α for Example 1
Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
2minus 4 00017 mdash 00018 mdash 00021 mdash2minus 6 10832e minus 04 40382 11271e minus 04 40373 12781e minus 04 403442minus 8 67252e minus 06 40097 69990e minus 06 40094 79405e minus 06 400862minus 10 41961e minus 07 40024 43672e minus 07 40023 49553e minus 07 400212minus 12 26215e minus 08 40006 27284e minus 08 40005 30959e minus 08 40005
Table 3 )e CPU time consumption of the fast compact difference scheme and direct difference scheme for Example 1
Δtα 125 α 15 α 175
DCD FCD DCD FCD DCD FCD11000 34008 18876 34632 19032 33540 1918812500 142740 46800 141960 47580 138840 4742415000 470967 93600 463010 94692 461450 9375617500 994194 139464 956754 140088 967206 138684110 000 1699318 186733 1641130 186421 1658290 184393125 000 9816362 466910 9509820 469251 9591409 460046150 000 38269853 934133 36923096 940062 37547412 919625
N times104
0
500
1000
1500
2000
2500
3000
3500
4000
CPU
tim
e
Fast evaluationDirect evaluation
0 05 1 15 2 25 3 35 4 45 5
(a)
Fast evaluationDirect evaluation
log10 (N)
ndash05
0
05
1
15
2
25
3
35
4
log1
0 (C
PU ti
me)
25 3 35 4 45 5
(b)
Figure 1 )e CPU time (a) and the log-log CPU time (b) versus the total number of time steps N for Example 1
10 Mathematical Problems in Engineering
Table 4 Considering Δt 2minus 13 the discrete linfin error and convergence rates of u with different α for Example 1
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 22486e minus 04 mdash 22733e minus 04 mdash 23705e minus 04 mdash2minus 4 13985e minus 05 40071 14140e minus 05 40069 14745e minus 05 400692minus 5 86897e minus 07 40084 87882e minus 07 40081 91726e minus 07 400682minus 6 50194e minus 08 41137 51024e minus 08 41063 54092e minus 08 40838
Table 5 Considering h 0001 the discrete linfin error and convergence rates of u with different α for Example 1
Δtα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 00046 mdash 00043 mdash 00036 mdash2minus 4 00011 20641 00011 19668 90063e minus 04 199902minus 5 28662e minus 04 19403 27171e minus 04 20174 22518e minus 04 199992minus 6 71658e minus 05 19999 67928e minus 05 20000 56295e minus 05 200002minus 7 17915e minus 05 20000 16982e minus 05 20000 14074e minus 05 20000
x
ndash02
0
02
04
06
08
1
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(a)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(b)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(c)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(d)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(e)
ndash02
0
02
04
06
08
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(f)
Figure 2 Considering c 0001 α 15 M N 100 real and numerical solutions of u at T 1 for Example 2 with the value of x0 equalto (a) 0 (b) 02 (c) 04 (d) 06 (e) 08 and (f) 1
Mathematical Problems in Engineering 11
zu(x t)
zt+
zαu(x t)
ztα
z2u(x t)
zx2 + f(x t)
u(x 0) sin(πx)zu
zt
1113868111386811138681113868111386811138681113868 t0 sin(πx)
u(0 t) u(1 t) 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(73)
In Tables 1 and 2 we take Δt h2 and h Δt
radicto ex-
amine the discrete linfin-norm (l2-norm) errors and
corresponding spatial and temporal convergence rates re-spectively We list the errors and convergence rates (order) ofthe proposed compact finite difference (CD) scheme which isalmost O(Δt2 + h4) for different α Additionally Table 3shows the CPU time (CPU) consumed by direct compact(DCD) scheme and fast compact difference (FCD) schemerespectively It is obvious that the FCD scheme has a sig-nificantly reduced CPU time over the DCD scheme Forinstance when α 15 we choose h 01 and Δt 150 000and observe that the FCD scheme consumes only 94 secondswhile the DCD scheme consumes 3692 seconds We can find
x
075
08
085
09
095
1
105
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(a)
0
01
02
03
04
05
06
07
08
09
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(b)
0
01
02
03
04
05
06
07
08
09
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(c)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(d)
Figure 3 Considering x0 05 α 15 M N 100 real and numerical solutions of u at T 1 for Example 2 with the value of c equal to(a) 1 (b) 01 (c) 001 and (d) 0001
12 Mathematical Problems in Engineering
that the performance of the FCD scheme will be moreconspicuous as the time step size Δt decreases
In Figure 1 we set h 01 and α 175 and change thetotal number of time steps N to plot out the CPU time (inseconds) of the FCD scheme and DCD scheme We canobserve that the CPU time increases almost linearly withrespect to N for the FCD scheme while the DCD schemescales like O(N2)
Tables 4 and 5 show the discrete linfin errors and con-vergence rates of the compact finite difference scheme forExample 1 )e space rates are almost O(h4) for fixedΔt 2minus 13 and the time convergence rates are always O(Δt2)for fixed h 0001 We can conclude that the numericalconvergence rates of our scheme approach almost toO(Δt2 + h4)
Example 2 )e example is described by
zu(x t)
zt+
zαu(x t)
ztα
z2u(x t)
zx2 + f(x t)
u(x 0) 0zu
zt
1113868111386811138681113868111386811138681113868t0 0
u(0 t) t2exp minusx20
c1113888 1113889 u(1 t) t2exp minus
1 minus x0( 11138572
c1113888 1113889
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(74)
Note that the exact solution of the above problem isu(x t) t2exp(minus (x minus x0)
2c) where x0 isin [0 1] and c is a
N times104
0
1000
2000
3000
4000
5000
6000
7000
8000CP
U ti
me
Fast evaluationDirect evaluation
0 05 1 15 2 25 3 35 4 45 5
(a)
Fast evaluationDirect evaluation
log10 (N)
0
05
1
15
2
25
3
35
4
log1
0 (C
PU ti
me)
25 3 35 4 45 5
(b)
Figure 4 Considering x0 05 c 001 α 15 andM 10 the CPU time (a) and the log-log CPU time (b) versus the total number oftime steps N for Example 2
Table 6 Considering Δt h2 andx0 05 the discrete linfin error and convergence rates of u with different α and c for Example 2
c hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
1
2minus 3 73224e minus 05 mdash 55643e minus 05 mdash 38478e minus 05 mdash2minus 4 45707e-06 40018 34695e minus 06 40034 23900e minus 06 400902minus 5 28558e minus 07 40004 21671e minus 07 40009 14913e minus 07 400242minus 6 17847e minus 08 40001 13543e minus 08 40001 93169e minus 09 40006
01
2minus 3 00012 mdash 00012 mdash 00012 mdash2minus 4 70462e minus 05 40900 69731e minus 05 41051 69007e minus 05 412012minus 5 43571e minus 06 40154 43110e minus 06 40157 42650e minus 06 401612minus 6 27160e minus 07 40038 26871e minus 07 40039 26583e minus 07 40040
001
2minus 4 00090 mdash 00090 mdash 00090 mdash2minus 5 49257e minus 04 41915 49235e minus 04 41922 49232e minus 04 419232minus 6 29868e minus 05 40437 29855e minus 05 40436 29854e minus 05 404362minus 7 18531e minus 06 40106 18523e minus 06 40106 18522e minus 06 40106
Mathematical Problems in Engineering 13
Table 7 Considering h Δt
radicand c 001 the discrete l2 error and convergence rates of u with different α andx0 for Example 2
x0 Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
02minus 8 00044 mdash 00044 mdash 00042 mdash2minus 10 24205e minus 04 41841 23721e minus 04 42133 23098e minus 04 418452minus 12 14684e minus 05 40430 14390e minus 05 40430 14013e minus 05 40429
0252minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14902e minus 04 41794 14894e minus 04 41802 14892e minus 04 418032minus 12 90680e minus 06 40386 90633e minus 06 40386 90616e minus 06 40386
052minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14848e minus 04 41846 14842e minus 04 41852 14842e minus 04 418522minus 12 90323e minus 06 40390 90288e minus 06 40390 90286e minus 06 40390
0752minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14902e minus 04 41794 14894e minus 04 41802 14892e minus 04 418032minus 12 90680e minus 06 40386 90633e minus 06 40386 90616e minus 06 40386
12minus 8 00044 mdash 00044 mdash 00042 mdash2minus 10 24205e minus 04 41841 23721e minus 04 42133 23098e minus 04 418452minus 12 14684e minus 05 40430 14390e minus 05 40430 14013e minus 05 40429
0 2 4 6 8 10times104
0
2000
4000
6000
8000
10000
12000
14000
N
CPU
tim
e
Fast evaluationDirect evaluation
(a)
Fast evaluationDirect evaluation
36 38 4 42 44 46 48 515
2
25
3
35
4
45
log10 (N)
log1
0 (C
PU ti
me)
(b)
Figure 5 )e CPU time (a) and the log-log CPU time (b) versus the total number of time steps N for Example 3
Table 8 Considering Δt h2 the discrete linfin error and convergence rates of u with different α for Example 3
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 2 00028 mdash 00028 mdash 00030 mdash2minus 3 16948e minus 04 40462 17328e minus 04 40142 18411e minus 04 402632minus 4 10522e minus 05 40096 10759e minus 05 40095 11434e minus 05 400922minus 5 65652e minus 07 40024 67133e minus 07 40024 71346e minus 07 40024
14 Mathematical Problems in Engineering
constant and for different α we have different f(x t)
accordinglyWe apply the fast compact difference scheme to dis-
cretize the equation In Figure 2 we setc 0001 α 15 andM N 100 and plot exact andnumerical solutions at time T 1 for Example 2 with dif-ferent x0 For x0 05 α 15 andM N 100 we alsoplot exact and numerical solutions with the different c inFigure 3 In Figure 4 for h 01 and α 15 we vary thetotal number of time steps N to plot out the CPU time (inseconds) of the FCD scheme and DCD scheme )e nu-merical experiments verified our theoretical results In Ta-ble 6 by equating Δt h2 and fixing x0 05 we compute
the discrete linfin error and convergence rates with differentfractional derivative orders α and different c It shows thatthe compact finite difference scheme has space accuracy offourth order and temporal accuracy of second order We seth
Δt
radicand fix c 001 and the discrete l2 error and
convergence rates with different α and x0 are displayed inTable 7 )ese numerical convergence rates are almostapproaching O(Δt2 + h4) as Example 1
Example 3 If the exact solution is given byu(x t) etsin(πx)sin(πy) we have different f(x y t) fordifferent α accordingly
zu(x y t)
zt+
zαu(x y t)
ztα
z2u(x y t)
zx2 +z2u(x y t)
zy2 + f(x y t)
u(x 0) sin(πx)sin(πy)
zu
zt | t0 sin(πx)sin(πy)
u(0 y t) u(1 y t) u(x 0 t) u(x 1 t) 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(75)
In Figure 5 h 01 and α 175 are fixed and the totalnumber of time steps N vary to plot out the CPU time (inseconds) of the FCD procedure and DCD procedure and itpresents an approximately linear computation complexityfor FCD procedure We set Δt h2 in Table 8 and h
Δt
radic
in Table 9 the discrete linfin error discrete l2 error andconvergence rates with different derivative orders α arepresented )e fourth-order space accuracy and second-order temporal accuracy can be observed clearly
7 Conclusion
In this paper we develop and analyze a fast compact finitedifference procedure for the Cattaneo equation equipped withtime-fractional derivative without singular kernel )e time-fractional derivative is of CaputondashFabrizio type with the orderof α(1lt αlt 2) Compact difference discretization is appliedto obtain a high-order approximation for spatial derivatives ofinteger order in the partial differential equation and theCaputondashFabrizio fractional derivative is discretized by means
of CrankndashNicolson approximation It has been proved thatthe proposed compact finite difference scheme has spatialaccuracy of fourth order and temporal accuracy of secondorder Since the fractional derivatives are history dependentand nonlocal huge memory for storage and computationalcost are required )is means extreme difficulty especially fora long-time simulation Enlightened by the treatment forCaputo fractional derivative [32] we develop an effective fastevaluation procedure for the new CaputondashFabrizio fractionalderivative for the compact finite difference scheme Severalnumerical experiments have been carried out to show theconvergence orders and applicability of the scheme
Inspired by the work [43] the topic about modelling andnumerical solutions of porous media flow equipped withfractional derivatives is very interesting and challenging andwill be our main research direction in the future
Data Availability
All data generated or analyzed during this study are includedin this article
Table 9 Considering h Δt
radic the discrete l2 error and convergence rates of u with different α for Example 3
Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
2minus 4 00014 mdash 00014 mdash 00015 mdash2minus 6 84738e minus 05 40463 86640e minus 05 40142 92056e minus 05 402632minus 8 52610e minus 06 40096 53795e minus 06 40095 57168e minus 06 400922minus 10 32826e minus 07 40024 33566e minus 07 40024 35673e minus 07 40023
Mathematical Problems in Engineering 15
Conflicts of Interest
)e authors declare that they have no conflicts of interest
Acknowledgments
)is work was supported in part by the National NaturalScience Foundation of China under Grant nos 91630207and 11971272
References
[1] A Chaves ldquoA fractional diffusion equation to describe Levyflightsrdquo Physics Letters A vol 239 no 1-2 pp 13ndash16 1998
[2] M Giona and H E Roman ldquoFractional diffusion equation fortransport phenomena in random mediardquo Physica A Statis-tical Mechanics and its Applications vol 185 no 1ndash4pp 87ndash97 1992
[3] R Metzler and J Klafter ldquo)e random walkrsquos guide toanomalous diffusion a fractional dynamics approachrdquoPhysics Reports vol 339 no 1 pp 1ndash77 2000
[4] R L Magin Fractional Calculus in Bioengineering BegellHouse Redding Redding CT USA 2006
[5] R L Magin O Abdullah D Baleanu and X J ZhouldquoAnomalous diffusion expressed through fractional orderdifferential operators in the Bloch-Torrey equationrdquo Journalof Magnetic Resonance vol 190 no 2 pp 255ndash270 2008
[6] I Podlubny Fractional Differential Equations An Introduc-tion to Fractional Derivatives Fractional Differential Equa-tions to Methods of Beir Solution and Some of BeirApplications Vol 198 Elsevier Amsterdam Netherlands1998
[7] M Ran and C Zhang ldquoCompact difference scheme for a classof fractional-in-space nonlinear damped wave equations intwo space dimensionsrdquo Computers amp Mathematics withApplications vol 71 no 5 pp 1151ndash1162 2016
[8] A H Bhrawy and M A Zaky ldquoHighly accurate numericalschemes for multi-dimensional space variable-order frac-tional Schrodinger equationsrdquo Computers amp Mathematicswith Applications vol 73 no 6 pp 1100ndash1117 2017
[9] S B Yuste ldquoWeighted average finite difference methods forfractional diffusion equationsrdquo Journal of ComputationalPhysics vol 216 no 1 pp 264ndash274 2006
[10] T A M Langlands and B I Henry ldquo)e accuracy andstability of an implicit solution method for the fractionaldiffusion equationrdquo Journal of Computational Physicsvol 205 no 2 pp 719ndash736 2005
[11] C-M Chen F Liu I Turner and V Anh ldquoA Fourier method forthe fractional diffusion equation describing sub-diffusionrdquo Journalof Computational Physics vol 227 no 2 pp 886ndash897 2007
[12] Y Lin and C Xu ldquoFinite differencespectral approximationsfor the time-fractional diffusion equationrdquo Journal of Com-putational Physics vol 225 no 2 pp 1533ndash1552 2007
[13] G-h Gao and Z-z Sun ldquoA compact finite difference schemefor the fractional sub-diffusion equationsrdquo Journal of Com-putational Physics vol 230 no 3 pp 586ndash595 2011
[14] F Zeng C Li F Liu and I Turner ldquoNumerical algorithms fortime-fractional subdiffusion equation with second-order ac-curacyrdquo SIAM Journal on Scientific Computing vol 37 no 1pp A55ndashA78 2015
[15] G-H Gao Z-Z Sun and H-W Zhang ldquoA new fractionalnumerical differentiation formula to approximate the Caputofractional derivative and its applicationsrdquo Journal of Com-putational Physics vol 259 no 2 pp 33ndash50 2014
[16] H Li J Cao and C Li ldquoHigh-order approximation to Caputoderivatives and Caputo-type advection-diffusion equations(III)rdquo Journal of Computational and Applied Mathematicsvol 299 pp 159ndash175 2016
[17] A A Alikhanov ldquoA new difference scheme for the timefractional diffusion equationrdquo Journal of ComputationalPhysics vol 280 pp 424ndash438 2015
[18] Z Wang and S Vong ldquoCompact difference schemes for themodified anomalous fractional sub-diffusion equation and thefractional diffusion-wave equationrdquo Journal of ComputationalPhysics vol 277 pp 1ndash15 2014
[19] H-K Pang and H-W Sun ldquoFourth order finite differenceschemes for time-space fractional sub-diffusion equationsrdquoComputers amp Mathematics with Applications vol 71 no 6pp 1287ndash1302 2016
[20] M Dehghan M Abbaszadeh and W Deng ldquoFourth-ordernumerical method for the space-time tempered fractionaldiffusion-wave equationrdquo Applied Mathematics Lettersvol 73 no 11 pp 120ndash127 2017
[21] H Sun Z-Z Sun and G-H Gao ldquoSome high order dif-ference schemes for the space and time fractional Bloch-Torrey equationsrdquo Applied Mathematics and Computationvol 281 pp 356ndash380 2016
[22] S Arshad J Huang A Q M Khaliq and Y Tang ldquoTrape-zoidal scheme for time-space fractional diffusion equationwith Riesz derivativerdquo Journal of Computational Physicsvol 350 pp 1ndash15 2017
[23] H Wang K Wang and T Sircar ldquoA direct O(Nlog2N) finitedifferencemethod for fractional diffusion equationsrdquo Journal ofComputational Physics vol 229 no 21 pp 8095ndash8104 2010
[24] H Wang and T S Basu ldquoA fast finite difference method fortwo-dimensional space-fractional diffusion equationsrdquo SIAMJournal on Scientific Computing vol 34 no 5 pp A2444ndashA2458 2012
[25] H Wang and N Du ldquoA superfast-preconditioned iterativemethod for steady-state space-fractional diffusion equationsrdquoJournal of Computational Physics vol 240 pp 49ndash57 2013
[26] H Wang and N Du ldquoA fast finite difference method forthree-dimensional time-dependent space-fractional diffusionequations and its efficient implementationrdquo Journal ofComputational Physics vol 253 pp 50ndash63 2013
[27] H Wang and N Du ldquoFast alternating-direction finite dif-ference methods for three-dimensional space-fractional dif-fusion equationsrdquo Journal of Computational Physics vol 258pp 305ndash318 2014
[28] K Wang and H Wang ldquoA fast characteristic finite differencemethod for fractional advection-diffusion equationsrdquo Ad-vances in Water Resources vol 34 no 7 pp 810ndash816 2011
[29] H Wang and H Tian ldquoA fast Galerkin method with efficientmatrix assembly and storage for a peridynamic modelrdquoJournal of Computational Physics vol 231 no 23pp 7730ndash7738 2012
[30] H Tian H Wang and W Wang ldquoAn efficient collocationmethod for a non-local diffusion modelrdquo InternationalJournal of Numerical Analysis amp Modeling vol 10 no 4pp 815ndash825 2013
[31] H Wang and H Tian ldquoA fast and faithful collocation methodwith efficient matrix assembly for a two-dimensional nonlocaldiffusion modelrdquo Computer Methods in Applied Mechanicsand Engineering vol 273 no 5 pp 19ndash36 2014
[32] S Jiang J Zhang Q Zhang and Z Zhang ldquoFast evaluation ofthe Caputo fractional derivative and its applications tofractional diffusion equationsrdquo Communications in Compu-tational Physics vol 21 no 3 pp 650ndash678 2017
16 Mathematical Problems in Engineering
[33] Y Yan Z-Z Sun and J Zhang ldquoFast evaluation of the Caputofractional derivative and its applications to fractional diffusionequations a second-order schemerdquo Communications inComputational Physics vol 22 no 4 pp 1028ndash1048 2017
[34] X Lu H-K Pang and H-W Sun ldquoFast approximate in-version of a block triangular Toeplitz matrix with applicationsto fractional sub-diffusion equationsrdquo Numerical Linear Al-gebra with Applications vol 22 no 5 pp 866ndash882 2015
[35] M Caputo and M Fabrizio ldquoA new definition of fractionalderivative without singular kernelrdquo Progress in FractionalDifferentiation and Applications vol 1 no 2 pp 73ndash85 2015
[36] A Atangana ldquoOn the new fractional derivative and appli-cation to nonlinear Fisherrsquos reaction-diffusion equationrdquoAppliedMathematics and Computation vol 273 pp 948ndash9562016
[37] M D Ortigueira and J Tenreiro Machado ldquoA critical analysisof the Caputo-Fabrizio operatorrdquo Communications in Non-linear Science and Numerical Simulation vol 59 pp 608ndash6112018
[38] V E Tarasov ldquoNo nonlocality no fractional derivativerdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 62 pp 157ndash163 2018
[39] V Tarasov ldquoCaputo-Fabrizio operator in terms of integerderivatives memory or distributed lagrdquo Computational andApplied Mathematics vol 38 p 113 2019
[40] A Giusti ldquoA comment on some new definitions of fractionalderivativerdquoNonlinear Dynamics vol 93 no 3 pp 1757ndash17632018
[41] A Atangana and J F Gomez-Aguilar ldquoFractional derivativeswith no-index law property application to chaos and sta-tisticsrdquo Chaos Solitons amp Fractals vol 114 pp 516ndash535 2018
[42] J Hristov ldquoResponse functions in linear viscoelastic consti-tutive equations and related fractional operatorsrdquo Mathe-matical Modelling of Natural Phenomena vol 14 no 3pp 1ndash34 2019
[43] L X Vivas-Cruz A Gonzalez-Calderon M A Taneco-Hernandez and D P Luis ldquo)eoretical analysis of a model offluid flow in a reservoir with the Caputo-Fabrizio operatorrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 84 Article ID 105186 2020
[44] H-L Liao and Z-Z Sun ldquoMaximum norm error bounds ofADI and compact ADI methods for solving parabolicequationsrdquo Numerical Methods for Partial DifferentialEquations vol 26 no 1 pp 37ndash60 2010
[45] Z Liu A Cheng and X Li ldquoA second order Crank-Nicolsonscheme for fractional Cattaneo equation based on newfractional derivativerdquoAppliedMathematics and Computationvol 311 pp 361ndash374 2017
[46] Z Sun and G Gao Finite Difference Method for FractionalDifferential Equations Science Press Beijing China 2015
Mathematical Problems in Engineering 17
Cattaneo equation by means of compact finite differencemethods
At the node (xi tk+(12)) the differential equation takesthe following form
zu
ztxi tk+(12)1113872 1113873 +
zαu
ztαxi tk+(12)1113872 1113873
z2u
zx2 xi tk+(12)1113872 1113873 + f xi tk+(12)1113872 1113873
1le ileM minus 1 1le kleN minus 1
(16)
)e approximation of the fractional derivative is givenby [45]
CF0 D
αt u xi tk+(12)1113872 1113873
1(α minus 1)Δt
M0δtuk+1i minus 1113944
k
n1Mkminus n(⎡⎣
minus Mkminus n+11113857δtuji minus Mkψi
⎤⎦ + Rk+(12)i
(17)
with truncation error Rk+(12)i O(Δt2) and
Mn exp1 minus α2 minus αΔtn1113876 1113877 minus exp
1 minus α2 minus αΔt(n + 1)1113876 1113877 (18)
Furthermore by Lemma 1 and equation (15) the spaceand time derivative are approximated by
z2u
zx2 xi tk+ (12)1113872 1113873 12
δ2xuk+1i + δ2xuk
i1113872 1113873
A+ O Δt2 + h
41113872 1113873
(19)
zu
ztxi tk+(12)1113872 1113873 δtu
k+1i + O Δt21113872 1113873 (20)
Substituting (17) and (19)sim(20) into (16) we get
Aδtuk+1i +
1(α minus 1)Δt
A M0δtuk+1i minus 1113944
k
n1Mkminus n(⎡⎣
minus Mkminus n+11113857δtuni minus Mkψi1113891
12
δ2xuk+1i + δ2xu
ki1113872 1113873 + Af
k+(12)i + R
k+(12)i
1le ileM minus 1 0le kleN minus 1
(21)
and there exists a constant C depending on the function uand its derivatives such that
Rk+(12)i leC Δt2 + h
41113872 1113873 (22)
By the initial and boundary value conditions we have
u0i ϕi 1le ileM minus 1
uk0 u
kM 0 0le kleN
(23)
A compact finite difference scheme can be established byomitting the truncation term R
k+(12)i and replacing the exact
solution uki in equation (21) with numerical solution uk
i
(α minus 1)ΔtAδtuk+1i + M0Aδtu
k+1i minus
(α minus 1)Δt2
δ2xUk+1
(α minus 1)Δt
2δ2xU
k+ 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857Aδtu
ni + MkAψi
+(α minus 1)ΔtAfk+ (12)i
1le ileM minus 1 1le kleN minus 1
u0i ϕi 1le ileM minus 1
uk0 u
kM 0 0le kleN
(24)
32 Stability Analysis and Optimal Error Estimates
321 Stability Analysis )e following Lemma about Mn isuseful for the analysis of stability
Lemma 2 (see [45]) For the definition of Mn Mn gt 0 andMn+1 ltMn forallnle k are held
Multiplying hδtuk+1i on both sides of equation (24) and
summing up with respect to i from 1 to M minus 1 the followingequation is obtained
(α minus 1)Δt δtUk+1
2
A+ M0 δtU
k+1
2
Aminusα minus 12
δ2xUk+1
Uk+1
1113872 1113873
+α minus 12
δ2xUk+1
Uk
1113872 1113873
α minus 12
δ2xUk U
k+11113872 1113873 minus
α minus 12
δ2xUk U
k1113872 1113873
+ 1113944k
n1Mkminus n minus Mkminus n+1( 1113857 AδtU
n δtU
k+11113872 1113873
+ Mk Aψ δtUk+1
1113872 1113873 +(α minus 1)Δt Afk+(12)
δtUk+1
1113872 1113873
(25)
Observing equation (13) we have
δ2xUk+1 Uk+11113872 1113873 minus δxUk+1 δxUk+1( 1113857 minus δxUk+11113868111386811138681113868
111386811138681113868111386821 le 0
δ2xUk Uk1113872 1113873 minus δxUk δxUk( 1113857 minus δxUk1113868111386811138681113868
111386811138681113868111386821 le 0
δ2xUk+1 Uk1113872 1113873 minus δxUk+1 δxUk( 1113857 minus langUk+1 Ukrang δ2xUk Uk+11113872 1113873
(26)
By the triangle inequality and Lemma 2 we obtain
4 Mathematical Problems in Engineering
1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 AδtU
n δtU
k+11113872 1113873 + Mk Aψ δtU
k+11113872 1113873
le 1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δtUn
2A
+ δtUk+1
2
A1113874 1113875
+12Mk ψ
2A + δtU
k+1
2
A1113874 1113875
1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δtUn
2A
+12
M0 minus Mk( 1113857
middot δtUk+1
2
A+12Mk ψ
2A + δtU
k+1
2
A1113874 1113875
1113944k
n1
12Mkminus n δtU
n
2A
minus 1113944k
n1
12Mkminus n+1 δtU
n
2A
+12M0
δtUk+1
2
A+12Mkψ
2A
(27)Combining equation (25) with (26)sim(27) we get
1113944
k+1
n1Mkminus n+1 δtU
n
2A
+(α minus 1) δxUk+1
11138681113868111386811138681113868
111386811138681113868111386811138682
1
le 1113944k
n1Mkminus n δtU
n
2A
+(α minus 1) δxUk
11138681113868111386811138681113868
111386811138681113868111386811138682
1+ Mkψ
2A
+(α minus 1)Δt fk+(12)
2
A
(28)
Let
Q Uk
1113872 1113873 1113944k
n1Mkminus n δtU
n
2A
+(α minus 1) δxUk
11138681113868111386811138681113868
111386811138681113868111386811138682
1 (29)
Summing up with respect to k from 0 to N minus 1 leads to
Q UN
1113872 1113873leQ U0
1113872 1113873 + 1113944Nminus 1
k0Mkψ
2A +(α minus 1)Δt 1113944
Nminus 1
k0f
k+(12)
2
A
(30)
)e initial condition U0 ϕ implies that Q(U0)
(α minus 1)| δxϕ| 21 and then
Q UN
1113872 1113873le (α minus 1) δxϕ1113868111386811138681113868
111386811138681113868111386821 + 1113944
Nminus 1
k0Mkψ
2A
+(α minus 1)Δt 1113944Nminus 1
k0f
k+(12)
2
A
(31)
Theorem 1 For scheme (24) we have the following stableconclusion
Q Um
( 1113857le (α minus 1) δxϕ1113868111386811138681113868
111386811138681113868111386821 + 1113944
mminus 1
k0Mkψ
2A
+(α minus 1)Δt 1113944mminus 1
k0f
k+(12)
2
A forall0lemleN
(32)
322 Optimal Error Estimates Combining equations (21)and (23) with (24) we get an error equation as follows
(α minus 1)ΔtAδtek+1i + M0Aδte
k+1i minus
(α minus 1)Δt2
δ2xek+1
(α minus 1)Δt
2δ2xe
k+ 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857Aδte
ni
+(α minus 1)ΔtRk+(12)i
(33)
where Rk+(12)i O(Δt2 + h4) and ek
i uki minus uk
i forallkge 0Multiplying hδte
k+1i on both sides of equation (33) and
summing up with respect to i from 1 to M minus 1 we get
(α minus 1)Δt δtek+1
2
A+ M0 δte
k+1
2
A+α minus 12
δxek+1
11138681113868111386811138681113868
111386811138681113868111386811138682
1
α minus 12
δxek
11138681113868111386811138681113868
111386811138681113868111386811138682
1+ 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 Aδte
n δte
k+11113872 1113873
+(α minus 1)Δt Rk+(12)
δtek+1
1113872 1113873
(34)
By the triangle inequality and Lemma 2 we obtain
1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 Aδte
n δte
k+11113872 1113873
le 1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δten
2A
+ δtek+1
2
A1113874 1113875
1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δten
2A
+12
M0 minus Mk( 1113857 δtek+1
2
A
le 1113944k
n1
12Mkminus n δte
n
2A
minus 1113944k
n1
12Mkminus n+1 δte
n
2A
+12M0 δte
k+1
2
A
(35)
Combining equation (34) with (35) we have
1113944
k+1
n1Mkminus n+1 δte
n
2A
+(α minus 1) δxek+1
11138681113868111386811138681113868
111386811138681113868111386811138682
1le 1113944
k
n1Mkminus n δte
n
2A
+(α minus 1) δxek
11138681113868111386811138681113868
111386811138681113868111386811138682
1+(α minus 1)Δt R
k+(12)
2
(36)
By the definition of Q in stability analysis the inequality(36) can be rearranged as
Mathematical Problems in Engineering 5
Q ek+1
1113872 1113873leQ ek
1113872 1113873 +(α minus 1)Δt Rk+(12)
2 (37)
Summing up with respect to k from 0 to N minus 1 we get
Q eN
1113872 1113873leQ e0
1113872 1113873 + C Δt2 + h4
1113872 11138732 (38)
Observing that the initial error e0 0 implies Q(e0) 0)en we have
(α minus 1) δxeN
11138681113868111386811138681113868
111386811138681113868111386811138682
1leQ e
N1113872 1113873leC Δt2 + h
41113872 1113873
2
1113944
N
n1MNminus n δte
n
2AleQ e
N1113872 1113873leC Δt2 + h
41113872 1113873
2
(39)
Theorem 2 Suppose that the exact solution of the fractionalCattaneo equation is smooth sufficiently then there exists apositive constant C independent of h k and Δt such that
ek
11138681113868111386811138681113868
111386811138681113868111386811138681leC Δt2 + h
41113872 1113873 forall0le kleN (40)
where eki uk
i minus uki
4 Compact Finite Difference Scheme inTwo Dimensions
In this section the following fractional Cattaneo equation intwo dimensions will be considered
zu(x y t)
zt+
zαu(x y t)
ztα
z2u(x y t)
zx2 +z2u(x y t)
zy2 + f(x y t) (x y t) isin (a b) times(c d) times(0 T]
u(x y 0) ϕ(x y)zu
zt
1113868111386811138681113868111386811138681113868 t0 ψ(x y) ale xle b cleyled
u(a y t) u(b y t) u(x c t) u(x d t) 0 tgt 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(41)
where 1lt αlt 2 f(x y t) is the source termϕ(x y) andψ(x y) are the given functions and zαuztα isdefined by the new Caputo fractional derivative withoutsingular kernel
In order to construct the finite difference schemes therectangle [a b] times [c d] is discretized with xi
a + ihx (0le ileMx) and yj c + jhy (0le jleMy) andthe time interval [0 T] is discretized with tk kΔt (0le kleN)where hx (b minus a)Mx hy (d minus c) My and Δt TN arethe spatial grid and temporal step sizes respectively DenoteΩh
(xi yj) 0le ileMx 0le1113966 jleMy and ΩΔt tk 0le kleN1113864 1113865then Ωh timesΩΔt is a discretization of the physical computationaldomain [a b] times [c d] times [0 T] uk
ij u(xi yj tk) denotes thevalues of function u at the grid points and uk
ij denotes the valuesof the numerical solution at the point (xi yj tk)
Denote Vh v| v (v00 v01 vMxMy)1113882 1113883 and V0
h
v| v isin Vh v0j vMxj vi0 viMy 0 0 le i leMx 0le j1113882
leMy For any mesh function v isin Vh we use the followingnotations
δxviminus (12)j vij minus viminus 1j
hx
δ2xvij vi+1j minus 2vij + viminus 1j
h2x
δyvijminus (12) vij minus vijminus 1
hy
δ2yvij vij+1 minus 2vij + vijminus 1
h2y
1le ileMx minus 1 1le jleMy minus 1
(42)
and define the average operator
Axvij
112
vi+1j + 10vij + viminus 1j1113872 1113873 1le ileMx minus 1 0le jleMy
vij i 0 Mx 0le jleMy
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
Ayvij
112
vij+1 + 10vij + vijminus 11113872 1113873 1le jleMy minus 1 0le ileMx
vij j 0 My 0le ileMx
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(43)
6 Mathematical Problems in Engineering
It is clear that
Axvij I +h2
x
12δ2x1113888 1113889vij
Ayvij I +h2
y
12δ2y⎛⎝ ⎞⎠vij
(44)
We also denote AxAy(u v) (AxAyu v) It is easy to seethat AxAy AyAx
For any gird function u v isin V0h the discrete inner
product and norms are defined as follows
(u v) hxhy 1113944
Mxminus 1
i11113944
My minus 1
j1uijvij
u22 (u u)
(45)
For the average operator AxAy define
AxAy(v v)≜ AxAyv v1113872 1113873 v2A (46)
41 Compact Finite Difference Scheme At the node(xi yj tk+(12)) the differential equation is rewritten as
zu
ztxi yj tk+(12)1113872 1113873 +
zαu
ztαxi yj tk+(12)1113872 1113873
z2u
zx2 xi yj tk+(12)1113872 1113873 +z2u
zy2 xi yj tk+(12)1113872 1113873
+ f xi yj tk+(12)1113872 1113873
1le ileMx minus 1 1le jleMy minus 1 1le kleN minus 1
(47)
For the approximation of the time-fractional derivativewe have the following approximation [45]
CF0 D
αt u xi yj tk+(12)1113872 1113873
1(α minus 1)Δt
M0δtuk+1ij minus 1113944
k
n1Mkminus n(⎡⎣
minus Mkminus n+11113857δtunij minus Mkψij
⎤⎦
+ Rk+(12)ij
(48)
where the truncation error Rk+(12)ij O(Δt2) and
Mn exp1 minus α2 minus αΔtn1113876 1113877 minus exp
1 minus α2 minus αΔt(n + 1)1113876 1113877 (49)
Furthermore we also have
z2u
zx2 xi yj tk+(12)1113872 1113873 12
δ2xuk+1ij + δ2xuk
ij1113872 1113873
Ax
+ O Δt2 + h4x1113872 1113873
(50)
z2u
zy2 xi yj tk+(12)1113872 1113873 12
δ2yuk+1ij + δ2yuk
ij1113872 1113873
Ay
+ O Δt2 + h4y1113872 1113873
(51)
zu
ztxi yj tk+(12)1113872 1113873 δtu
k+1ij + O Δt21113872 1113873 (52)
Substituting (48) and (50)sim(52) into (47) leads to
AxAyδtuk+1ij +
1(α minus 1)τ
AxAy M0δtuk+1ij1113890
minus 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857δtu
nij minus Mkψij
⎤⎦
12
δ2xuk+1ij + δ2xu
kij1113872 1113873 +
12
δ2yuk+1ij + δ2yu
kij1113872 1113873 + AxAyf
k+(12)ij
+ Rk+(12)ij
1le ileMx minus 1 1le jleMy minus 1 0le kleN minus 1
(53)
and there exists a constant C depending on the function uand its derivatives such that
Rk+(12)ij leC Δt2 + h
4x + h
4y1113872 1113873 (54)
By the initial and boundary conditions we have
u0ij ϕij 1le ileMx minus 1 1le jleMy minus 1
uk0j u
kMxj u
ki0 u
kiMy
0 0le kleN(55)
Omitting the truncation error Rk+(12)ij and replacing the
true solution ukij with numerical solution uk
ij a compactfinite difference scheme can be obtained as follows
(α minus 1)ΔtAxAyδtUk+1ij + M0AxAyδtU
k+1ij
minus Ay
(α minus 1)Δt2
δ2xUk+1ij minus Ax
(α minus 1)Δt2
δ2yUk+1ij
Ay
(α minus 1)Δt2
δ2xukij + Ax
(α minus 1)Δt2
δ2yukij
+ 1113944k
n1Mkminus n minus Mkminus n+1( 1113857AxAyδtu
nij + MkAxAyψij
+ (α minus 1)ΔtAxAyfk+(12)ij
1le ileMx minus 1 1le jleMy minus 1 1le kleN minus 1
u0ij ϕij 1le ileMx minus 1 1le jleMy minus 1
uk0j u
kMxj u
ki0 u
kiMy
0 0le kleN
(56)
Mathematical Problems in Engineering 7
42 Stability Analysis and Optimal Error Estimates
421 Stability Analysis
Definition 4 (see [46]) For any gird function u isin V0h define
the norm
⦀nablahu⦀2A δxu
2
minush2
y
12δyδxu
2
⎛⎝ ⎞⎠ + δyu
2
minush2
x
12δxδyu
2
1113888 1113889
(57)
)e lemmas below is useful in the subsequent analysis ofstability
Lemma 3 (see [46]) For any gird function u isin V0h the
following equation is held13u
2 le u2A le u
2 (58)
Lemma 4 (see [46]) For any gird function u isin V0h the
following equation is held23nablahu
2 le⦀nablahu⦀2A le nablahu
2
Ayδ2xu
nminus (12)+ Axδ
2yu
nminus (12) δtu
nminus (12)1113872 1113873
minus12Δt⦀nablahu
n⦀2A minus ⦀nablahunminus 1⦀
2A1113874 1113875
(59)
where nablahu2 δxu22 + δyu22
Multiplying hxhyδtUk+1ij on both sides of equation (56)
and summing up wrt i j from 1 to (Mx minus 1) and from 1 to(My minus 1) respectively the following equation is obtained
(α minus 1)Δt δtUk+1
2
A+ M0 δtU
k+1
2
A
(α minus 1)Δt Ayδ2xU
k+(12)+ Axδ
2yU
k+(12) δtU
k+11113872 1113873 + 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 AyAxδtU
n δtU
k+11113872 1113873
+ Mk AyAxψ δtUk+1
1113872 1113873 +(α minus 1)Δt AyAxfk+(12)
δtUk+1
1113872 1113873
(60)
Observing Lemma 4 we have
Ayδ2xUk+(12) + Axδ
2yUk+(12) δtU
k+11113872 1113873 minus12Δt⦀nablahu
k+1⦀2A minus ⦀nablahu
k⦀2A1113874 1113875 (61)
(α minus 1)Δt AyAxfk+(12) δtUk+11113872 1113873le
(α minus 1)Δt2
fk+(12)
2
A+ δtu
k+1
2
A1113874 1113875 (62)
By the triangle inequality and Lemma 2 we obtain
1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 AyAxδtU
n δtU
k+11113872 1113873 + Mk AyAxψ δtU
k+11113872 1113873
le 1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δtUn
2A
+ δtUk+1
2
A1113874 1113875 +
12Mk ψ
2A + δtU
k+1
2
A1113874 1113875
1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δtUn
2A
+12
M0 minus Mk( 1113857 δtUk+1
2
A+12Mk ψ
2A + δtU
k+1
2
A1113874 1113875
1113944k
n1
12Mkminus n δtU
n
2A
minus 1113944k
n1
12Mkminus n+1 δtU
n
2A
+12M0 δtU
k+1
2
A+12Mkψ
2A
(63)
Combining equation (60) with (61)sim(63)we get
1113944
k+1
n1Mkminus n+1 δtU
n
2A
+(α minus 1)⦀nablahUk+1⦀
2A
le 1113944k
n1Mkminus n δtU
n
2A
+(α minus 1) nablahUk
2
A+ Mkψ
2A +(α minus 1)Δt⦀fk+(12)⦀
2A
(64)
8 Mathematical Problems in Engineering
Let
Q Uk
1113872 1113873 1113944k
n1Mkminus n δtU
n
2A
+(α minus 1)⦀nablahUk⦀
2A (65)
Summing up with respect to k from 0 to N minus 1 we get
Q UN
1113872 1113873leQ U0
1113872 1113873 + 1113944Nminus 1
k0Mkψ
2A +(α minus 1)Δt 1113944
Nminus 1
k0f
k+(12)
2
A
(66)
Noting that U0 ϕ we have Q(U0) (α minus 1)nablahϕ2A Itfollows that
Q UN
1113872 1113873le (α minus 1)⦀nablahϕ⦀2A + 1113944
Nminus 1
k0Mkψ
2A
+(α minus 1)Δt 1113944Nminus 1
k0f
k+(12)
2
A
(67)
Theorem 3 For the compact finite difference scheme (56)the following stability inequality holds
Q Um
( 1113857le (α minus 1) nablahϕ1113868111386811138681113868
1113868111386811138681113868
2A
+ 1113944
mminus 1
k0Mkψ
2A
+(α minus 1)Δt 1113944
mminus 1
k0f
k+(12)
2
A forall0lemleN
(68)
Similar to the stability the convergence can also beanalyzed
Theorem 4 Suppose that the exact solution of the fractionalCattaneo equation is sufficiently smooth then there exists apositive constant C independent of h k and Δt such that
ek
11138681113868111386811138681113868
111386811138681113868111386811138681leC Δt2 + h
41113872 1113873 forall0le kleN (69)
where ekij uk
ij minus ukij and h max hx hy1113966 1113967
5 Efficient Storage and Fast Evaluation of theCaputondashFabrizio Fractional Derivative
Since time-fractional derivative operator is nonlocal thetraditional direct method for numerically solving the frac-tional Cattaneo equations generally requires total O(MN)
memory units and O(MN2) computational complexitywhere N andM are the total number of time steps and spacesteps respectively
In this section we develop a fast solution method for thefinite difference scheme of the time-fractional Cattaneoequation
Let
Nkn Mkminus n minus Mkminus n+1
exp1 minus α2 minus α
(k minus n)Δt1113874 1113875
middot 1 minus 2exp1 minus α2 minus αΔt1113874 1113875 + exp
1 minus α2 minus α
2Δt1113874 11138751113876 1113877
1le nle k
(70)
then
Nk+1n Mk+1minus n minus Mk+1minus n+1
exp1 minus α2 minus α
(k minus n)Δt1113874 1113875
middot 1 minus 2exp1 minus α2 minus αΔt1113874 1113875 + exp
1 minus α2 minus α
2Δt1113874 11138751113876 1113877exp1 minus α2 minus αΔt1113874 1113875
exp1 minus α2 minus αΔt1113874 1113875N
kn
Nk+1k+1 M0 minus M1 1 minus 2exp
1 minus α2 minus αΔt1113874 1113875 + exp
1 minus α2 minus α
2Δt1113874 1113875
(71)
So we have
1113944
k+1
n1N
k+1n Aδtu
ni exp
1 minus α2 minus αΔt1113874 1113875 1113944
k
n1N
knAδtu
ni + N
k+1k+1Aδtu
k+1i
1113944
k+1
n1N
k+1n AyAxδtu
nij exp
1 minus α2 minus αΔt1113874 1113875 1113944
k
n1N
knAyAxδtu
nij
+ Nk+1k+1AyAxδtU
k+1ij
(72)
Remark We find that at the k-th level only O(1) operationsare needed to compute the k-th level since the (k minus 1)-thlevel is known at that point )us the total operations arereduced from O(N2) to O(N) and the memory require-ment decreases from O(N) to O(1) We conclude that thatthis fast method significantly reduces the total computa-tional cost from O(MN2) to O(MN) and the memoryrequirement from O(MN) to O(M)
6 Numerical Experiments
In this section we carry out several numerical experiments tocheck the effectiveness of the proposed scheme )e conver-gence rate and CPU consumption are all compared in thesimulations We take the space-time domainΩ [0 1] T 1for one-dimensional case and Ω [0 1] times [0 1] T 1 fortwo-dimensional case )ese simulations are implemented inMatlab and the numerical experiments are run on a computerwith 4GB memory )e time-fractional Cattaneo equation ofthe following forms is considered
Example 1 We provide the exact solution u(x t)
etsin(πx) and for different α we have different f(x t)
accordingly
Mathematical Problems in Engineering 9
Table 1 Considering Δt h2 the discrete linfin error and convergence rates of u with different α for Example 1
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 15320e minus 04 mdash 15941e minus 04 mdash 18076e minus 04 mdash2minus 4 95109e minus 06 40097 98981e minus 06 40094 11230e minus 05 400862minus 5 59343e minus 07 40024 61762e minus 07 40024 70079e minus 07 400222minus 6 37074e minus 08 40006 38585e minus 08 40006 43783e minus 08 40005
Table 2 Considering h Δt
radic the discrete l2 error and convergence rates of u with different α for Example 1
Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
2minus 4 00017 mdash 00018 mdash 00021 mdash2minus 6 10832e minus 04 40382 11271e minus 04 40373 12781e minus 04 403442minus 8 67252e minus 06 40097 69990e minus 06 40094 79405e minus 06 400862minus 10 41961e minus 07 40024 43672e minus 07 40023 49553e minus 07 400212minus 12 26215e minus 08 40006 27284e minus 08 40005 30959e minus 08 40005
Table 3 )e CPU time consumption of the fast compact difference scheme and direct difference scheme for Example 1
Δtα 125 α 15 α 175
DCD FCD DCD FCD DCD FCD11000 34008 18876 34632 19032 33540 1918812500 142740 46800 141960 47580 138840 4742415000 470967 93600 463010 94692 461450 9375617500 994194 139464 956754 140088 967206 138684110 000 1699318 186733 1641130 186421 1658290 184393125 000 9816362 466910 9509820 469251 9591409 460046150 000 38269853 934133 36923096 940062 37547412 919625
N times104
0
500
1000
1500
2000
2500
3000
3500
4000
CPU
tim
e
Fast evaluationDirect evaluation
0 05 1 15 2 25 3 35 4 45 5
(a)
Fast evaluationDirect evaluation
log10 (N)
ndash05
0
05
1
15
2
25
3
35
4
log1
0 (C
PU ti
me)
25 3 35 4 45 5
(b)
Figure 1 )e CPU time (a) and the log-log CPU time (b) versus the total number of time steps N for Example 1
10 Mathematical Problems in Engineering
Table 4 Considering Δt 2minus 13 the discrete linfin error and convergence rates of u with different α for Example 1
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 22486e minus 04 mdash 22733e minus 04 mdash 23705e minus 04 mdash2minus 4 13985e minus 05 40071 14140e minus 05 40069 14745e minus 05 400692minus 5 86897e minus 07 40084 87882e minus 07 40081 91726e minus 07 400682minus 6 50194e minus 08 41137 51024e minus 08 41063 54092e minus 08 40838
Table 5 Considering h 0001 the discrete linfin error and convergence rates of u with different α for Example 1
Δtα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 00046 mdash 00043 mdash 00036 mdash2minus 4 00011 20641 00011 19668 90063e minus 04 199902minus 5 28662e minus 04 19403 27171e minus 04 20174 22518e minus 04 199992minus 6 71658e minus 05 19999 67928e minus 05 20000 56295e minus 05 200002minus 7 17915e minus 05 20000 16982e minus 05 20000 14074e minus 05 20000
x
ndash02
0
02
04
06
08
1
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(a)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(b)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(c)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(d)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(e)
ndash02
0
02
04
06
08
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(f)
Figure 2 Considering c 0001 α 15 M N 100 real and numerical solutions of u at T 1 for Example 2 with the value of x0 equalto (a) 0 (b) 02 (c) 04 (d) 06 (e) 08 and (f) 1
Mathematical Problems in Engineering 11
zu(x t)
zt+
zαu(x t)
ztα
z2u(x t)
zx2 + f(x t)
u(x 0) sin(πx)zu
zt
1113868111386811138681113868111386811138681113868 t0 sin(πx)
u(0 t) u(1 t) 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(73)
In Tables 1 and 2 we take Δt h2 and h Δt
radicto ex-
amine the discrete linfin-norm (l2-norm) errors and
corresponding spatial and temporal convergence rates re-spectively We list the errors and convergence rates (order) ofthe proposed compact finite difference (CD) scheme which isalmost O(Δt2 + h4) for different α Additionally Table 3shows the CPU time (CPU) consumed by direct compact(DCD) scheme and fast compact difference (FCD) schemerespectively It is obvious that the FCD scheme has a sig-nificantly reduced CPU time over the DCD scheme Forinstance when α 15 we choose h 01 and Δt 150 000and observe that the FCD scheme consumes only 94 secondswhile the DCD scheme consumes 3692 seconds We can find
x
075
08
085
09
095
1
105
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(a)
0
01
02
03
04
05
06
07
08
09
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(b)
0
01
02
03
04
05
06
07
08
09
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(c)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(d)
Figure 3 Considering x0 05 α 15 M N 100 real and numerical solutions of u at T 1 for Example 2 with the value of c equal to(a) 1 (b) 01 (c) 001 and (d) 0001
12 Mathematical Problems in Engineering
that the performance of the FCD scheme will be moreconspicuous as the time step size Δt decreases
In Figure 1 we set h 01 and α 175 and change thetotal number of time steps N to plot out the CPU time (inseconds) of the FCD scheme and DCD scheme We canobserve that the CPU time increases almost linearly withrespect to N for the FCD scheme while the DCD schemescales like O(N2)
Tables 4 and 5 show the discrete linfin errors and con-vergence rates of the compact finite difference scheme forExample 1 )e space rates are almost O(h4) for fixedΔt 2minus 13 and the time convergence rates are always O(Δt2)for fixed h 0001 We can conclude that the numericalconvergence rates of our scheme approach almost toO(Δt2 + h4)
Example 2 )e example is described by
zu(x t)
zt+
zαu(x t)
ztα
z2u(x t)
zx2 + f(x t)
u(x 0) 0zu
zt
1113868111386811138681113868111386811138681113868t0 0
u(0 t) t2exp minusx20
c1113888 1113889 u(1 t) t2exp minus
1 minus x0( 11138572
c1113888 1113889
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(74)
Note that the exact solution of the above problem isu(x t) t2exp(minus (x minus x0)
2c) where x0 isin [0 1] and c is a
N times104
0
1000
2000
3000
4000
5000
6000
7000
8000CP
U ti
me
Fast evaluationDirect evaluation
0 05 1 15 2 25 3 35 4 45 5
(a)
Fast evaluationDirect evaluation
log10 (N)
0
05
1
15
2
25
3
35
4
log1
0 (C
PU ti
me)
25 3 35 4 45 5
(b)
Figure 4 Considering x0 05 c 001 α 15 andM 10 the CPU time (a) and the log-log CPU time (b) versus the total number oftime steps N for Example 2
Table 6 Considering Δt h2 andx0 05 the discrete linfin error and convergence rates of u with different α and c for Example 2
c hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
1
2minus 3 73224e minus 05 mdash 55643e minus 05 mdash 38478e minus 05 mdash2minus 4 45707e-06 40018 34695e minus 06 40034 23900e minus 06 400902minus 5 28558e minus 07 40004 21671e minus 07 40009 14913e minus 07 400242minus 6 17847e minus 08 40001 13543e minus 08 40001 93169e minus 09 40006
01
2minus 3 00012 mdash 00012 mdash 00012 mdash2minus 4 70462e minus 05 40900 69731e minus 05 41051 69007e minus 05 412012minus 5 43571e minus 06 40154 43110e minus 06 40157 42650e minus 06 401612minus 6 27160e minus 07 40038 26871e minus 07 40039 26583e minus 07 40040
001
2minus 4 00090 mdash 00090 mdash 00090 mdash2minus 5 49257e minus 04 41915 49235e minus 04 41922 49232e minus 04 419232minus 6 29868e minus 05 40437 29855e minus 05 40436 29854e minus 05 404362minus 7 18531e minus 06 40106 18523e minus 06 40106 18522e minus 06 40106
Mathematical Problems in Engineering 13
Table 7 Considering h Δt
radicand c 001 the discrete l2 error and convergence rates of u with different α andx0 for Example 2
x0 Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
02minus 8 00044 mdash 00044 mdash 00042 mdash2minus 10 24205e minus 04 41841 23721e minus 04 42133 23098e minus 04 418452minus 12 14684e minus 05 40430 14390e minus 05 40430 14013e minus 05 40429
0252minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14902e minus 04 41794 14894e minus 04 41802 14892e minus 04 418032minus 12 90680e minus 06 40386 90633e minus 06 40386 90616e minus 06 40386
052minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14848e minus 04 41846 14842e minus 04 41852 14842e minus 04 418522minus 12 90323e minus 06 40390 90288e minus 06 40390 90286e minus 06 40390
0752minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14902e minus 04 41794 14894e minus 04 41802 14892e minus 04 418032minus 12 90680e minus 06 40386 90633e minus 06 40386 90616e minus 06 40386
12minus 8 00044 mdash 00044 mdash 00042 mdash2minus 10 24205e minus 04 41841 23721e minus 04 42133 23098e minus 04 418452minus 12 14684e minus 05 40430 14390e minus 05 40430 14013e minus 05 40429
0 2 4 6 8 10times104
0
2000
4000
6000
8000
10000
12000
14000
N
CPU
tim
e
Fast evaluationDirect evaluation
(a)
Fast evaluationDirect evaluation
36 38 4 42 44 46 48 515
2
25
3
35
4
45
log10 (N)
log1
0 (C
PU ti
me)
(b)
Figure 5 )e CPU time (a) and the log-log CPU time (b) versus the total number of time steps N for Example 3
Table 8 Considering Δt h2 the discrete linfin error and convergence rates of u with different α for Example 3
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 2 00028 mdash 00028 mdash 00030 mdash2minus 3 16948e minus 04 40462 17328e minus 04 40142 18411e minus 04 402632minus 4 10522e minus 05 40096 10759e minus 05 40095 11434e minus 05 400922minus 5 65652e minus 07 40024 67133e minus 07 40024 71346e minus 07 40024
14 Mathematical Problems in Engineering
constant and for different α we have different f(x t)
accordinglyWe apply the fast compact difference scheme to dis-
cretize the equation In Figure 2 we setc 0001 α 15 andM N 100 and plot exact andnumerical solutions at time T 1 for Example 2 with dif-ferent x0 For x0 05 α 15 andM N 100 we alsoplot exact and numerical solutions with the different c inFigure 3 In Figure 4 for h 01 and α 15 we vary thetotal number of time steps N to plot out the CPU time (inseconds) of the FCD scheme and DCD scheme )e nu-merical experiments verified our theoretical results In Ta-ble 6 by equating Δt h2 and fixing x0 05 we compute
the discrete linfin error and convergence rates with differentfractional derivative orders α and different c It shows thatthe compact finite difference scheme has space accuracy offourth order and temporal accuracy of second order We seth
Δt
radicand fix c 001 and the discrete l2 error and
convergence rates with different α and x0 are displayed inTable 7 )ese numerical convergence rates are almostapproaching O(Δt2 + h4) as Example 1
Example 3 If the exact solution is given byu(x t) etsin(πx)sin(πy) we have different f(x y t) fordifferent α accordingly
zu(x y t)
zt+
zαu(x y t)
ztα
z2u(x y t)
zx2 +z2u(x y t)
zy2 + f(x y t)
u(x 0) sin(πx)sin(πy)
zu
zt | t0 sin(πx)sin(πy)
u(0 y t) u(1 y t) u(x 0 t) u(x 1 t) 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(75)
In Figure 5 h 01 and α 175 are fixed and the totalnumber of time steps N vary to plot out the CPU time (inseconds) of the FCD procedure and DCD procedure and itpresents an approximately linear computation complexityfor FCD procedure We set Δt h2 in Table 8 and h
Δt
radic
in Table 9 the discrete linfin error discrete l2 error andconvergence rates with different derivative orders α arepresented )e fourth-order space accuracy and second-order temporal accuracy can be observed clearly
7 Conclusion
In this paper we develop and analyze a fast compact finitedifference procedure for the Cattaneo equation equipped withtime-fractional derivative without singular kernel )e time-fractional derivative is of CaputondashFabrizio type with the orderof α(1lt αlt 2) Compact difference discretization is appliedto obtain a high-order approximation for spatial derivatives ofinteger order in the partial differential equation and theCaputondashFabrizio fractional derivative is discretized by means
of CrankndashNicolson approximation It has been proved thatthe proposed compact finite difference scheme has spatialaccuracy of fourth order and temporal accuracy of secondorder Since the fractional derivatives are history dependentand nonlocal huge memory for storage and computationalcost are required )is means extreme difficulty especially fora long-time simulation Enlightened by the treatment forCaputo fractional derivative [32] we develop an effective fastevaluation procedure for the new CaputondashFabrizio fractionalderivative for the compact finite difference scheme Severalnumerical experiments have been carried out to show theconvergence orders and applicability of the scheme
Inspired by the work [43] the topic about modelling andnumerical solutions of porous media flow equipped withfractional derivatives is very interesting and challenging andwill be our main research direction in the future
Data Availability
All data generated or analyzed during this study are includedin this article
Table 9 Considering h Δt
radic the discrete l2 error and convergence rates of u with different α for Example 3
Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
2minus 4 00014 mdash 00014 mdash 00015 mdash2minus 6 84738e minus 05 40463 86640e minus 05 40142 92056e minus 05 402632minus 8 52610e minus 06 40096 53795e minus 06 40095 57168e minus 06 400922minus 10 32826e minus 07 40024 33566e minus 07 40024 35673e minus 07 40023
Mathematical Problems in Engineering 15
Conflicts of Interest
)e authors declare that they have no conflicts of interest
Acknowledgments
)is work was supported in part by the National NaturalScience Foundation of China under Grant nos 91630207and 11971272
References
[1] A Chaves ldquoA fractional diffusion equation to describe Levyflightsrdquo Physics Letters A vol 239 no 1-2 pp 13ndash16 1998
[2] M Giona and H E Roman ldquoFractional diffusion equation fortransport phenomena in random mediardquo Physica A Statis-tical Mechanics and its Applications vol 185 no 1ndash4pp 87ndash97 1992
[3] R Metzler and J Klafter ldquo)e random walkrsquos guide toanomalous diffusion a fractional dynamics approachrdquoPhysics Reports vol 339 no 1 pp 1ndash77 2000
[4] R L Magin Fractional Calculus in Bioengineering BegellHouse Redding Redding CT USA 2006
[5] R L Magin O Abdullah D Baleanu and X J ZhouldquoAnomalous diffusion expressed through fractional orderdifferential operators in the Bloch-Torrey equationrdquo Journalof Magnetic Resonance vol 190 no 2 pp 255ndash270 2008
[6] I Podlubny Fractional Differential Equations An Introduc-tion to Fractional Derivatives Fractional Differential Equa-tions to Methods of Beir Solution and Some of BeirApplications Vol 198 Elsevier Amsterdam Netherlands1998
[7] M Ran and C Zhang ldquoCompact difference scheme for a classof fractional-in-space nonlinear damped wave equations intwo space dimensionsrdquo Computers amp Mathematics withApplications vol 71 no 5 pp 1151ndash1162 2016
[8] A H Bhrawy and M A Zaky ldquoHighly accurate numericalschemes for multi-dimensional space variable-order frac-tional Schrodinger equationsrdquo Computers amp Mathematicswith Applications vol 73 no 6 pp 1100ndash1117 2017
[9] S B Yuste ldquoWeighted average finite difference methods forfractional diffusion equationsrdquo Journal of ComputationalPhysics vol 216 no 1 pp 264ndash274 2006
[10] T A M Langlands and B I Henry ldquo)e accuracy andstability of an implicit solution method for the fractionaldiffusion equationrdquo Journal of Computational Physicsvol 205 no 2 pp 719ndash736 2005
[11] C-M Chen F Liu I Turner and V Anh ldquoA Fourier method forthe fractional diffusion equation describing sub-diffusionrdquo Journalof Computational Physics vol 227 no 2 pp 886ndash897 2007
[12] Y Lin and C Xu ldquoFinite differencespectral approximationsfor the time-fractional diffusion equationrdquo Journal of Com-putational Physics vol 225 no 2 pp 1533ndash1552 2007
[13] G-h Gao and Z-z Sun ldquoA compact finite difference schemefor the fractional sub-diffusion equationsrdquo Journal of Com-putational Physics vol 230 no 3 pp 586ndash595 2011
[14] F Zeng C Li F Liu and I Turner ldquoNumerical algorithms fortime-fractional subdiffusion equation with second-order ac-curacyrdquo SIAM Journal on Scientific Computing vol 37 no 1pp A55ndashA78 2015
[15] G-H Gao Z-Z Sun and H-W Zhang ldquoA new fractionalnumerical differentiation formula to approximate the Caputofractional derivative and its applicationsrdquo Journal of Com-putational Physics vol 259 no 2 pp 33ndash50 2014
[16] H Li J Cao and C Li ldquoHigh-order approximation to Caputoderivatives and Caputo-type advection-diffusion equations(III)rdquo Journal of Computational and Applied Mathematicsvol 299 pp 159ndash175 2016
[17] A A Alikhanov ldquoA new difference scheme for the timefractional diffusion equationrdquo Journal of ComputationalPhysics vol 280 pp 424ndash438 2015
[18] Z Wang and S Vong ldquoCompact difference schemes for themodified anomalous fractional sub-diffusion equation and thefractional diffusion-wave equationrdquo Journal of ComputationalPhysics vol 277 pp 1ndash15 2014
[19] H-K Pang and H-W Sun ldquoFourth order finite differenceschemes for time-space fractional sub-diffusion equationsrdquoComputers amp Mathematics with Applications vol 71 no 6pp 1287ndash1302 2016
[20] M Dehghan M Abbaszadeh and W Deng ldquoFourth-ordernumerical method for the space-time tempered fractionaldiffusion-wave equationrdquo Applied Mathematics Lettersvol 73 no 11 pp 120ndash127 2017
[21] H Sun Z-Z Sun and G-H Gao ldquoSome high order dif-ference schemes for the space and time fractional Bloch-Torrey equationsrdquo Applied Mathematics and Computationvol 281 pp 356ndash380 2016
[22] S Arshad J Huang A Q M Khaliq and Y Tang ldquoTrape-zoidal scheme for time-space fractional diffusion equationwith Riesz derivativerdquo Journal of Computational Physicsvol 350 pp 1ndash15 2017
[23] H Wang K Wang and T Sircar ldquoA direct O(Nlog2N) finitedifferencemethod for fractional diffusion equationsrdquo Journal ofComputational Physics vol 229 no 21 pp 8095ndash8104 2010
[24] H Wang and T S Basu ldquoA fast finite difference method fortwo-dimensional space-fractional diffusion equationsrdquo SIAMJournal on Scientific Computing vol 34 no 5 pp A2444ndashA2458 2012
[25] H Wang and N Du ldquoA superfast-preconditioned iterativemethod for steady-state space-fractional diffusion equationsrdquoJournal of Computational Physics vol 240 pp 49ndash57 2013
[26] H Wang and N Du ldquoA fast finite difference method forthree-dimensional time-dependent space-fractional diffusionequations and its efficient implementationrdquo Journal ofComputational Physics vol 253 pp 50ndash63 2013
[27] H Wang and N Du ldquoFast alternating-direction finite dif-ference methods for three-dimensional space-fractional dif-fusion equationsrdquo Journal of Computational Physics vol 258pp 305ndash318 2014
[28] K Wang and H Wang ldquoA fast characteristic finite differencemethod for fractional advection-diffusion equationsrdquo Ad-vances in Water Resources vol 34 no 7 pp 810ndash816 2011
[29] H Wang and H Tian ldquoA fast Galerkin method with efficientmatrix assembly and storage for a peridynamic modelrdquoJournal of Computational Physics vol 231 no 23pp 7730ndash7738 2012
[30] H Tian H Wang and W Wang ldquoAn efficient collocationmethod for a non-local diffusion modelrdquo InternationalJournal of Numerical Analysis amp Modeling vol 10 no 4pp 815ndash825 2013
[31] H Wang and H Tian ldquoA fast and faithful collocation methodwith efficient matrix assembly for a two-dimensional nonlocaldiffusion modelrdquo Computer Methods in Applied Mechanicsand Engineering vol 273 no 5 pp 19ndash36 2014
[32] S Jiang J Zhang Q Zhang and Z Zhang ldquoFast evaluation ofthe Caputo fractional derivative and its applications tofractional diffusion equationsrdquo Communications in Compu-tational Physics vol 21 no 3 pp 650ndash678 2017
16 Mathematical Problems in Engineering
[33] Y Yan Z-Z Sun and J Zhang ldquoFast evaluation of the Caputofractional derivative and its applications to fractional diffusionequations a second-order schemerdquo Communications inComputational Physics vol 22 no 4 pp 1028ndash1048 2017
[34] X Lu H-K Pang and H-W Sun ldquoFast approximate in-version of a block triangular Toeplitz matrix with applicationsto fractional sub-diffusion equationsrdquo Numerical Linear Al-gebra with Applications vol 22 no 5 pp 866ndash882 2015
[35] M Caputo and M Fabrizio ldquoA new definition of fractionalderivative without singular kernelrdquo Progress in FractionalDifferentiation and Applications vol 1 no 2 pp 73ndash85 2015
[36] A Atangana ldquoOn the new fractional derivative and appli-cation to nonlinear Fisherrsquos reaction-diffusion equationrdquoAppliedMathematics and Computation vol 273 pp 948ndash9562016
[37] M D Ortigueira and J Tenreiro Machado ldquoA critical analysisof the Caputo-Fabrizio operatorrdquo Communications in Non-linear Science and Numerical Simulation vol 59 pp 608ndash6112018
[38] V E Tarasov ldquoNo nonlocality no fractional derivativerdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 62 pp 157ndash163 2018
[39] V Tarasov ldquoCaputo-Fabrizio operator in terms of integerderivatives memory or distributed lagrdquo Computational andApplied Mathematics vol 38 p 113 2019
[40] A Giusti ldquoA comment on some new definitions of fractionalderivativerdquoNonlinear Dynamics vol 93 no 3 pp 1757ndash17632018
[41] A Atangana and J F Gomez-Aguilar ldquoFractional derivativeswith no-index law property application to chaos and sta-tisticsrdquo Chaos Solitons amp Fractals vol 114 pp 516ndash535 2018
[42] J Hristov ldquoResponse functions in linear viscoelastic consti-tutive equations and related fractional operatorsrdquo Mathe-matical Modelling of Natural Phenomena vol 14 no 3pp 1ndash34 2019
[43] L X Vivas-Cruz A Gonzalez-Calderon M A Taneco-Hernandez and D P Luis ldquo)eoretical analysis of a model offluid flow in a reservoir with the Caputo-Fabrizio operatorrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 84 Article ID 105186 2020
[44] H-L Liao and Z-Z Sun ldquoMaximum norm error bounds ofADI and compact ADI methods for solving parabolicequationsrdquo Numerical Methods for Partial DifferentialEquations vol 26 no 1 pp 37ndash60 2010
[45] Z Liu A Cheng and X Li ldquoA second order Crank-Nicolsonscheme for fractional Cattaneo equation based on newfractional derivativerdquoAppliedMathematics and Computationvol 311 pp 361ndash374 2017
[46] Z Sun and G Gao Finite Difference Method for FractionalDifferential Equations Science Press Beijing China 2015
Mathematical Problems in Engineering 17
1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 AδtU
n δtU
k+11113872 1113873 + Mk Aψ δtU
k+11113872 1113873
le 1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δtUn
2A
+ δtUk+1
2
A1113874 1113875
+12Mk ψ
2A + δtU
k+1
2
A1113874 1113875
1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δtUn
2A
+12
M0 minus Mk( 1113857
middot δtUk+1
2
A+12Mk ψ
2A + δtU
k+1
2
A1113874 1113875
1113944k
n1
12Mkminus n δtU
n
2A
minus 1113944k
n1
12Mkminus n+1 δtU
n
2A
+12M0
δtUk+1
2
A+12Mkψ
2A
(27)Combining equation (25) with (26)sim(27) we get
1113944
k+1
n1Mkminus n+1 δtU
n
2A
+(α minus 1) δxUk+1
11138681113868111386811138681113868
111386811138681113868111386811138682
1
le 1113944k
n1Mkminus n δtU
n
2A
+(α minus 1) δxUk
11138681113868111386811138681113868
111386811138681113868111386811138682
1+ Mkψ
2A
+(α minus 1)Δt fk+(12)
2
A
(28)
Let
Q Uk
1113872 1113873 1113944k
n1Mkminus n δtU
n
2A
+(α minus 1) δxUk
11138681113868111386811138681113868
111386811138681113868111386811138682
1 (29)
Summing up with respect to k from 0 to N minus 1 leads to
Q UN
1113872 1113873leQ U0
1113872 1113873 + 1113944Nminus 1
k0Mkψ
2A +(α minus 1)Δt 1113944
Nminus 1
k0f
k+(12)
2
A
(30)
)e initial condition U0 ϕ implies that Q(U0)
(α minus 1)| δxϕ| 21 and then
Q UN
1113872 1113873le (α minus 1) δxϕ1113868111386811138681113868
111386811138681113868111386821 + 1113944
Nminus 1
k0Mkψ
2A
+(α minus 1)Δt 1113944Nminus 1
k0f
k+(12)
2
A
(31)
Theorem 1 For scheme (24) we have the following stableconclusion
Q Um
( 1113857le (α minus 1) δxϕ1113868111386811138681113868
111386811138681113868111386821 + 1113944
mminus 1
k0Mkψ
2A
+(α minus 1)Δt 1113944mminus 1
k0f
k+(12)
2
A forall0lemleN
(32)
322 Optimal Error Estimates Combining equations (21)and (23) with (24) we get an error equation as follows
(α minus 1)ΔtAδtek+1i + M0Aδte
k+1i minus
(α minus 1)Δt2
δ2xek+1
(α minus 1)Δt
2δ2xe
k+ 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857Aδte
ni
+(α minus 1)ΔtRk+(12)i
(33)
where Rk+(12)i O(Δt2 + h4) and ek
i uki minus uk
i forallkge 0Multiplying hδte
k+1i on both sides of equation (33) and
summing up with respect to i from 1 to M minus 1 we get
(α minus 1)Δt δtek+1
2
A+ M0 δte
k+1
2
A+α minus 12
δxek+1
11138681113868111386811138681113868
111386811138681113868111386811138682
1
α minus 12
δxek
11138681113868111386811138681113868
111386811138681113868111386811138682
1+ 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 Aδte
n δte
k+11113872 1113873
+(α minus 1)Δt Rk+(12)
δtek+1
1113872 1113873
(34)
By the triangle inequality and Lemma 2 we obtain
1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 Aδte
n δte
k+11113872 1113873
le 1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δten
2A
+ δtek+1
2
A1113874 1113875
1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δten
2A
+12
M0 minus Mk( 1113857 δtek+1
2
A
le 1113944k
n1
12Mkminus n δte
n
2A
minus 1113944k
n1
12Mkminus n+1 δte
n
2A
+12M0 δte
k+1
2
A
(35)
Combining equation (34) with (35) we have
1113944
k+1
n1Mkminus n+1 δte
n
2A
+(α minus 1) δxek+1
11138681113868111386811138681113868
111386811138681113868111386811138682
1le 1113944
k
n1Mkminus n δte
n
2A
+(α minus 1) δxek
11138681113868111386811138681113868
111386811138681113868111386811138682
1+(α minus 1)Δt R
k+(12)
2
(36)
By the definition of Q in stability analysis the inequality(36) can be rearranged as
Mathematical Problems in Engineering 5
Q ek+1
1113872 1113873leQ ek
1113872 1113873 +(α minus 1)Δt Rk+(12)
2 (37)
Summing up with respect to k from 0 to N minus 1 we get
Q eN
1113872 1113873leQ e0
1113872 1113873 + C Δt2 + h4
1113872 11138732 (38)
Observing that the initial error e0 0 implies Q(e0) 0)en we have
(α minus 1) δxeN
11138681113868111386811138681113868
111386811138681113868111386811138682
1leQ e
N1113872 1113873leC Δt2 + h
41113872 1113873
2
1113944
N
n1MNminus n δte
n
2AleQ e
N1113872 1113873leC Δt2 + h
41113872 1113873
2
(39)
Theorem 2 Suppose that the exact solution of the fractionalCattaneo equation is smooth sufficiently then there exists apositive constant C independent of h k and Δt such that
ek
11138681113868111386811138681113868
111386811138681113868111386811138681leC Δt2 + h
41113872 1113873 forall0le kleN (40)
where eki uk
i minus uki
4 Compact Finite Difference Scheme inTwo Dimensions
In this section the following fractional Cattaneo equation intwo dimensions will be considered
zu(x y t)
zt+
zαu(x y t)
ztα
z2u(x y t)
zx2 +z2u(x y t)
zy2 + f(x y t) (x y t) isin (a b) times(c d) times(0 T]
u(x y 0) ϕ(x y)zu
zt
1113868111386811138681113868111386811138681113868 t0 ψ(x y) ale xle b cleyled
u(a y t) u(b y t) u(x c t) u(x d t) 0 tgt 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(41)
where 1lt αlt 2 f(x y t) is the source termϕ(x y) andψ(x y) are the given functions and zαuztα isdefined by the new Caputo fractional derivative withoutsingular kernel
In order to construct the finite difference schemes therectangle [a b] times [c d] is discretized with xi
a + ihx (0le ileMx) and yj c + jhy (0le jleMy) andthe time interval [0 T] is discretized with tk kΔt (0le kleN)where hx (b minus a)Mx hy (d minus c) My and Δt TN arethe spatial grid and temporal step sizes respectively DenoteΩh
(xi yj) 0le ileMx 0le1113966 jleMy and ΩΔt tk 0le kleN1113864 1113865then Ωh timesΩΔt is a discretization of the physical computationaldomain [a b] times [c d] times [0 T] uk
ij u(xi yj tk) denotes thevalues of function u at the grid points and uk
ij denotes the valuesof the numerical solution at the point (xi yj tk)
Denote Vh v| v (v00 v01 vMxMy)1113882 1113883 and V0
h
v| v isin Vh v0j vMxj vi0 viMy 0 0 le i leMx 0le j1113882
leMy For any mesh function v isin Vh we use the followingnotations
δxviminus (12)j vij minus viminus 1j
hx
δ2xvij vi+1j minus 2vij + viminus 1j
h2x
δyvijminus (12) vij minus vijminus 1
hy
δ2yvij vij+1 minus 2vij + vijminus 1
h2y
1le ileMx minus 1 1le jleMy minus 1
(42)
and define the average operator
Axvij
112
vi+1j + 10vij + viminus 1j1113872 1113873 1le ileMx minus 1 0le jleMy
vij i 0 Mx 0le jleMy
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
Ayvij
112
vij+1 + 10vij + vijminus 11113872 1113873 1le jleMy minus 1 0le ileMx
vij j 0 My 0le ileMx
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(43)
6 Mathematical Problems in Engineering
It is clear that
Axvij I +h2
x
12δ2x1113888 1113889vij
Ayvij I +h2
y
12δ2y⎛⎝ ⎞⎠vij
(44)
We also denote AxAy(u v) (AxAyu v) It is easy to seethat AxAy AyAx
For any gird function u v isin V0h the discrete inner
product and norms are defined as follows
(u v) hxhy 1113944
Mxminus 1
i11113944
My minus 1
j1uijvij
u22 (u u)
(45)
For the average operator AxAy define
AxAy(v v)≜ AxAyv v1113872 1113873 v2A (46)
41 Compact Finite Difference Scheme At the node(xi yj tk+(12)) the differential equation is rewritten as
zu
ztxi yj tk+(12)1113872 1113873 +
zαu
ztαxi yj tk+(12)1113872 1113873
z2u
zx2 xi yj tk+(12)1113872 1113873 +z2u
zy2 xi yj tk+(12)1113872 1113873
+ f xi yj tk+(12)1113872 1113873
1le ileMx minus 1 1le jleMy minus 1 1le kleN minus 1
(47)
For the approximation of the time-fractional derivativewe have the following approximation [45]
CF0 D
αt u xi yj tk+(12)1113872 1113873
1(α minus 1)Δt
M0δtuk+1ij minus 1113944
k
n1Mkminus n(⎡⎣
minus Mkminus n+11113857δtunij minus Mkψij
⎤⎦
+ Rk+(12)ij
(48)
where the truncation error Rk+(12)ij O(Δt2) and
Mn exp1 minus α2 minus αΔtn1113876 1113877 minus exp
1 minus α2 minus αΔt(n + 1)1113876 1113877 (49)
Furthermore we also have
z2u
zx2 xi yj tk+(12)1113872 1113873 12
δ2xuk+1ij + δ2xuk
ij1113872 1113873
Ax
+ O Δt2 + h4x1113872 1113873
(50)
z2u
zy2 xi yj tk+(12)1113872 1113873 12
δ2yuk+1ij + δ2yuk
ij1113872 1113873
Ay
+ O Δt2 + h4y1113872 1113873
(51)
zu
ztxi yj tk+(12)1113872 1113873 δtu
k+1ij + O Δt21113872 1113873 (52)
Substituting (48) and (50)sim(52) into (47) leads to
AxAyδtuk+1ij +
1(α minus 1)τ
AxAy M0δtuk+1ij1113890
minus 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857δtu
nij minus Mkψij
⎤⎦
12
δ2xuk+1ij + δ2xu
kij1113872 1113873 +
12
δ2yuk+1ij + δ2yu
kij1113872 1113873 + AxAyf
k+(12)ij
+ Rk+(12)ij
1le ileMx minus 1 1le jleMy minus 1 0le kleN minus 1
(53)
and there exists a constant C depending on the function uand its derivatives such that
Rk+(12)ij leC Δt2 + h
4x + h
4y1113872 1113873 (54)
By the initial and boundary conditions we have
u0ij ϕij 1le ileMx minus 1 1le jleMy minus 1
uk0j u
kMxj u
ki0 u
kiMy
0 0le kleN(55)
Omitting the truncation error Rk+(12)ij and replacing the
true solution ukij with numerical solution uk
ij a compactfinite difference scheme can be obtained as follows
(α minus 1)ΔtAxAyδtUk+1ij + M0AxAyδtU
k+1ij
minus Ay
(α minus 1)Δt2
δ2xUk+1ij minus Ax
(α minus 1)Δt2
δ2yUk+1ij
Ay
(α minus 1)Δt2
δ2xukij + Ax
(α minus 1)Δt2
δ2yukij
+ 1113944k
n1Mkminus n minus Mkminus n+1( 1113857AxAyδtu
nij + MkAxAyψij
+ (α minus 1)ΔtAxAyfk+(12)ij
1le ileMx minus 1 1le jleMy minus 1 1le kleN minus 1
u0ij ϕij 1le ileMx minus 1 1le jleMy minus 1
uk0j u
kMxj u
ki0 u
kiMy
0 0le kleN
(56)
Mathematical Problems in Engineering 7
42 Stability Analysis and Optimal Error Estimates
421 Stability Analysis
Definition 4 (see [46]) For any gird function u isin V0h define
the norm
⦀nablahu⦀2A δxu
2
minush2
y
12δyδxu
2
⎛⎝ ⎞⎠ + δyu
2
minush2
x
12δxδyu
2
1113888 1113889
(57)
)e lemmas below is useful in the subsequent analysis ofstability
Lemma 3 (see [46]) For any gird function u isin V0h the
following equation is held13u
2 le u2A le u
2 (58)
Lemma 4 (see [46]) For any gird function u isin V0h the
following equation is held23nablahu
2 le⦀nablahu⦀2A le nablahu
2
Ayδ2xu
nminus (12)+ Axδ
2yu
nminus (12) δtu
nminus (12)1113872 1113873
minus12Δt⦀nablahu
n⦀2A minus ⦀nablahunminus 1⦀
2A1113874 1113875
(59)
where nablahu2 δxu22 + δyu22
Multiplying hxhyδtUk+1ij on both sides of equation (56)
and summing up wrt i j from 1 to (Mx minus 1) and from 1 to(My minus 1) respectively the following equation is obtained
(α minus 1)Δt δtUk+1
2
A+ M0 δtU
k+1
2
A
(α minus 1)Δt Ayδ2xU
k+(12)+ Axδ
2yU
k+(12) δtU
k+11113872 1113873 + 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 AyAxδtU
n δtU
k+11113872 1113873
+ Mk AyAxψ δtUk+1
1113872 1113873 +(α minus 1)Δt AyAxfk+(12)
δtUk+1
1113872 1113873
(60)
Observing Lemma 4 we have
Ayδ2xUk+(12) + Axδ
2yUk+(12) δtU
k+11113872 1113873 minus12Δt⦀nablahu
k+1⦀2A minus ⦀nablahu
k⦀2A1113874 1113875 (61)
(α minus 1)Δt AyAxfk+(12) δtUk+11113872 1113873le
(α minus 1)Δt2
fk+(12)
2
A+ δtu
k+1
2
A1113874 1113875 (62)
By the triangle inequality and Lemma 2 we obtain
1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 AyAxδtU
n δtU
k+11113872 1113873 + Mk AyAxψ δtU
k+11113872 1113873
le 1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δtUn
2A
+ δtUk+1
2
A1113874 1113875 +
12Mk ψ
2A + δtU
k+1
2
A1113874 1113875
1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δtUn
2A
+12
M0 minus Mk( 1113857 δtUk+1
2
A+12Mk ψ
2A + δtU
k+1
2
A1113874 1113875
1113944k
n1
12Mkminus n δtU
n
2A
minus 1113944k
n1
12Mkminus n+1 δtU
n
2A
+12M0 δtU
k+1
2
A+12Mkψ
2A
(63)
Combining equation (60) with (61)sim(63)we get
1113944
k+1
n1Mkminus n+1 δtU
n
2A
+(α minus 1)⦀nablahUk+1⦀
2A
le 1113944k
n1Mkminus n δtU
n
2A
+(α minus 1) nablahUk
2
A+ Mkψ
2A +(α minus 1)Δt⦀fk+(12)⦀
2A
(64)
8 Mathematical Problems in Engineering
Let
Q Uk
1113872 1113873 1113944k
n1Mkminus n δtU
n
2A
+(α minus 1)⦀nablahUk⦀
2A (65)
Summing up with respect to k from 0 to N minus 1 we get
Q UN
1113872 1113873leQ U0
1113872 1113873 + 1113944Nminus 1
k0Mkψ
2A +(α minus 1)Δt 1113944
Nminus 1
k0f
k+(12)
2
A
(66)
Noting that U0 ϕ we have Q(U0) (α minus 1)nablahϕ2A Itfollows that
Q UN
1113872 1113873le (α minus 1)⦀nablahϕ⦀2A + 1113944
Nminus 1
k0Mkψ
2A
+(α minus 1)Δt 1113944Nminus 1
k0f
k+(12)
2
A
(67)
Theorem 3 For the compact finite difference scheme (56)the following stability inequality holds
Q Um
( 1113857le (α minus 1) nablahϕ1113868111386811138681113868
1113868111386811138681113868
2A
+ 1113944
mminus 1
k0Mkψ
2A
+(α minus 1)Δt 1113944
mminus 1
k0f
k+(12)
2
A forall0lemleN
(68)
Similar to the stability the convergence can also beanalyzed
Theorem 4 Suppose that the exact solution of the fractionalCattaneo equation is sufficiently smooth then there exists apositive constant C independent of h k and Δt such that
ek
11138681113868111386811138681113868
111386811138681113868111386811138681leC Δt2 + h
41113872 1113873 forall0le kleN (69)
where ekij uk
ij minus ukij and h max hx hy1113966 1113967
5 Efficient Storage and Fast Evaluation of theCaputondashFabrizio Fractional Derivative
Since time-fractional derivative operator is nonlocal thetraditional direct method for numerically solving the frac-tional Cattaneo equations generally requires total O(MN)
memory units and O(MN2) computational complexitywhere N andM are the total number of time steps and spacesteps respectively
In this section we develop a fast solution method for thefinite difference scheme of the time-fractional Cattaneoequation
Let
Nkn Mkminus n minus Mkminus n+1
exp1 minus α2 minus α
(k minus n)Δt1113874 1113875
middot 1 minus 2exp1 minus α2 minus αΔt1113874 1113875 + exp
1 minus α2 minus α
2Δt1113874 11138751113876 1113877
1le nle k
(70)
then
Nk+1n Mk+1minus n minus Mk+1minus n+1
exp1 minus α2 minus α
(k minus n)Δt1113874 1113875
middot 1 minus 2exp1 minus α2 minus αΔt1113874 1113875 + exp
1 minus α2 minus α
2Δt1113874 11138751113876 1113877exp1 minus α2 minus αΔt1113874 1113875
exp1 minus α2 minus αΔt1113874 1113875N
kn
Nk+1k+1 M0 minus M1 1 minus 2exp
1 minus α2 minus αΔt1113874 1113875 + exp
1 minus α2 minus α
2Δt1113874 1113875
(71)
So we have
1113944
k+1
n1N
k+1n Aδtu
ni exp
1 minus α2 minus αΔt1113874 1113875 1113944
k
n1N
knAδtu
ni + N
k+1k+1Aδtu
k+1i
1113944
k+1
n1N
k+1n AyAxδtu
nij exp
1 minus α2 minus αΔt1113874 1113875 1113944
k
n1N
knAyAxδtu
nij
+ Nk+1k+1AyAxδtU
k+1ij
(72)
Remark We find that at the k-th level only O(1) operationsare needed to compute the k-th level since the (k minus 1)-thlevel is known at that point )us the total operations arereduced from O(N2) to O(N) and the memory require-ment decreases from O(N) to O(1) We conclude that thatthis fast method significantly reduces the total computa-tional cost from O(MN2) to O(MN) and the memoryrequirement from O(MN) to O(M)
6 Numerical Experiments
In this section we carry out several numerical experiments tocheck the effectiveness of the proposed scheme )e conver-gence rate and CPU consumption are all compared in thesimulations We take the space-time domainΩ [0 1] T 1for one-dimensional case and Ω [0 1] times [0 1] T 1 fortwo-dimensional case )ese simulations are implemented inMatlab and the numerical experiments are run on a computerwith 4GB memory )e time-fractional Cattaneo equation ofthe following forms is considered
Example 1 We provide the exact solution u(x t)
etsin(πx) and for different α we have different f(x t)
accordingly
Mathematical Problems in Engineering 9
Table 1 Considering Δt h2 the discrete linfin error and convergence rates of u with different α for Example 1
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 15320e minus 04 mdash 15941e minus 04 mdash 18076e minus 04 mdash2minus 4 95109e minus 06 40097 98981e minus 06 40094 11230e minus 05 400862minus 5 59343e minus 07 40024 61762e minus 07 40024 70079e minus 07 400222minus 6 37074e minus 08 40006 38585e minus 08 40006 43783e minus 08 40005
Table 2 Considering h Δt
radic the discrete l2 error and convergence rates of u with different α for Example 1
Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
2minus 4 00017 mdash 00018 mdash 00021 mdash2minus 6 10832e minus 04 40382 11271e minus 04 40373 12781e minus 04 403442minus 8 67252e minus 06 40097 69990e minus 06 40094 79405e minus 06 400862minus 10 41961e minus 07 40024 43672e minus 07 40023 49553e minus 07 400212minus 12 26215e minus 08 40006 27284e minus 08 40005 30959e minus 08 40005
Table 3 )e CPU time consumption of the fast compact difference scheme and direct difference scheme for Example 1
Δtα 125 α 15 α 175
DCD FCD DCD FCD DCD FCD11000 34008 18876 34632 19032 33540 1918812500 142740 46800 141960 47580 138840 4742415000 470967 93600 463010 94692 461450 9375617500 994194 139464 956754 140088 967206 138684110 000 1699318 186733 1641130 186421 1658290 184393125 000 9816362 466910 9509820 469251 9591409 460046150 000 38269853 934133 36923096 940062 37547412 919625
N times104
0
500
1000
1500
2000
2500
3000
3500
4000
CPU
tim
e
Fast evaluationDirect evaluation
0 05 1 15 2 25 3 35 4 45 5
(a)
Fast evaluationDirect evaluation
log10 (N)
ndash05
0
05
1
15
2
25
3
35
4
log1
0 (C
PU ti
me)
25 3 35 4 45 5
(b)
Figure 1 )e CPU time (a) and the log-log CPU time (b) versus the total number of time steps N for Example 1
10 Mathematical Problems in Engineering
Table 4 Considering Δt 2minus 13 the discrete linfin error and convergence rates of u with different α for Example 1
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 22486e minus 04 mdash 22733e minus 04 mdash 23705e minus 04 mdash2minus 4 13985e minus 05 40071 14140e minus 05 40069 14745e minus 05 400692minus 5 86897e minus 07 40084 87882e minus 07 40081 91726e minus 07 400682minus 6 50194e minus 08 41137 51024e minus 08 41063 54092e minus 08 40838
Table 5 Considering h 0001 the discrete linfin error and convergence rates of u with different α for Example 1
Δtα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 00046 mdash 00043 mdash 00036 mdash2minus 4 00011 20641 00011 19668 90063e minus 04 199902minus 5 28662e minus 04 19403 27171e minus 04 20174 22518e minus 04 199992minus 6 71658e minus 05 19999 67928e minus 05 20000 56295e minus 05 200002minus 7 17915e minus 05 20000 16982e minus 05 20000 14074e minus 05 20000
x
ndash02
0
02
04
06
08
1
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(a)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(b)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(c)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(d)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(e)
ndash02
0
02
04
06
08
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(f)
Figure 2 Considering c 0001 α 15 M N 100 real and numerical solutions of u at T 1 for Example 2 with the value of x0 equalto (a) 0 (b) 02 (c) 04 (d) 06 (e) 08 and (f) 1
Mathematical Problems in Engineering 11
zu(x t)
zt+
zαu(x t)
ztα
z2u(x t)
zx2 + f(x t)
u(x 0) sin(πx)zu
zt
1113868111386811138681113868111386811138681113868 t0 sin(πx)
u(0 t) u(1 t) 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(73)
In Tables 1 and 2 we take Δt h2 and h Δt
radicto ex-
amine the discrete linfin-norm (l2-norm) errors and
corresponding spatial and temporal convergence rates re-spectively We list the errors and convergence rates (order) ofthe proposed compact finite difference (CD) scheme which isalmost O(Δt2 + h4) for different α Additionally Table 3shows the CPU time (CPU) consumed by direct compact(DCD) scheme and fast compact difference (FCD) schemerespectively It is obvious that the FCD scheme has a sig-nificantly reduced CPU time over the DCD scheme Forinstance when α 15 we choose h 01 and Δt 150 000and observe that the FCD scheme consumes only 94 secondswhile the DCD scheme consumes 3692 seconds We can find
x
075
08
085
09
095
1
105
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(a)
0
01
02
03
04
05
06
07
08
09
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(b)
0
01
02
03
04
05
06
07
08
09
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(c)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(d)
Figure 3 Considering x0 05 α 15 M N 100 real and numerical solutions of u at T 1 for Example 2 with the value of c equal to(a) 1 (b) 01 (c) 001 and (d) 0001
12 Mathematical Problems in Engineering
that the performance of the FCD scheme will be moreconspicuous as the time step size Δt decreases
In Figure 1 we set h 01 and α 175 and change thetotal number of time steps N to plot out the CPU time (inseconds) of the FCD scheme and DCD scheme We canobserve that the CPU time increases almost linearly withrespect to N for the FCD scheme while the DCD schemescales like O(N2)
Tables 4 and 5 show the discrete linfin errors and con-vergence rates of the compact finite difference scheme forExample 1 )e space rates are almost O(h4) for fixedΔt 2minus 13 and the time convergence rates are always O(Δt2)for fixed h 0001 We can conclude that the numericalconvergence rates of our scheme approach almost toO(Δt2 + h4)
Example 2 )e example is described by
zu(x t)
zt+
zαu(x t)
ztα
z2u(x t)
zx2 + f(x t)
u(x 0) 0zu
zt
1113868111386811138681113868111386811138681113868t0 0
u(0 t) t2exp minusx20
c1113888 1113889 u(1 t) t2exp minus
1 minus x0( 11138572
c1113888 1113889
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(74)
Note that the exact solution of the above problem isu(x t) t2exp(minus (x minus x0)
2c) where x0 isin [0 1] and c is a
N times104
0
1000
2000
3000
4000
5000
6000
7000
8000CP
U ti
me
Fast evaluationDirect evaluation
0 05 1 15 2 25 3 35 4 45 5
(a)
Fast evaluationDirect evaluation
log10 (N)
0
05
1
15
2
25
3
35
4
log1
0 (C
PU ti
me)
25 3 35 4 45 5
(b)
Figure 4 Considering x0 05 c 001 α 15 andM 10 the CPU time (a) and the log-log CPU time (b) versus the total number oftime steps N for Example 2
Table 6 Considering Δt h2 andx0 05 the discrete linfin error and convergence rates of u with different α and c for Example 2
c hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
1
2minus 3 73224e minus 05 mdash 55643e minus 05 mdash 38478e minus 05 mdash2minus 4 45707e-06 40018 34695e minus 06 40034 23900e minus 06 400902minus 5 28558e minus 07 40004 21671e minus 07 40009 14913e minus 07 400242minus 6 17847e minus 08 40001 13543e minus 08 40001 93169e minus 09 40006
01
2minus 3 00012 mdash 00012 mdash 00012 mdash2minus 4 70462e minus 05 40900 69731e minus 05 41051 69007e minus 05 412012minus 5 43571e minus 06 40154 43110e minus 06 40157 42650e minus 06 401612minus 6 27160e minus 07 40038 26871e minus 07 40039 26583e minus 07 40040
001
2minus 4 00090 mdash 00090 mdash 00090 mdash2minus 5 49257e minus 04 41915 49235e minus 04 41922 49232e minus 04 419232minus 6 29868e minus 05 40437 29855e minus 05 40436 29854e minus 05 404362minus 7 18531e minus 06 40106 18523e minus 06 40106 18522e minus 06 40106
Mathematical Problems in Engineering 13
Table 7 Considering h Δt
radicand c 001 the discrete l2 error and convergence rates of u with different α andx0 for Example 2
x0 Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
02minus 8 00044 mdash 00044 mdash 00042 mdash2minus 10 24205e minus 04 41841 23721e minus 04 42133 23098e minus 04 418452minus 12 14684e minus 05 40430 14390e minus 05 40430 14013e minus 05 40429
0252minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14902e minus 04 41794 14894e minus 04 41802 14892e minus 04 418032minus 12 90680e minus 06 40386 90633e minus 06 40386 90616e minus 06 40386
052minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14848e minus 04 41846 14842e minus 04 41852 14842e minus 04 418522minus 12 90323e minus 06 40390 90288e minus 06 40390 90286e minus 06 40390
0752minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14902e minus 04 41794 14894e minus 04 41802 14892e minus 04 418032minus 12 90680e minus 06 40386 90633e minus 06 40386 90616e minus 06 40386
12minus 8 00044 mdash 00044 mdash 00042 mdash2minus 10 24205e minus 04 41841 23721e minus 04 42133 23098e minus 04 418452minus 12 14684e minus 05 40430 14390e minus 05 40430 14013e minus 05 40429
0 2 4 6 8 10times104
0
2000
4000
6000
8000
10000
12000
14000
N
CPU
tim
e
Fast evaluationDirect evaluation
(a)
Fast evaluationDirect evaluation
36 38 4 42 44 46 48 515
2
25
3
35
4
45
log10 (N)
log1
0 (C
PU ti
me)
(b)
Figure 5 )e CPU time (a) and the log-log CPU time (b) versus the total number of time steps N for Example 3
Table 8 Considering Δt h2 the discrete linfin error and convergence rates of u with different α for Example 3
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 2 00028 mdash 00028 mdash 00030 mdash2minus 3 16948e minus 04 40462 17328e minus 04 40142 18411e minus 04 402632minus 4 10522e minus 05 40096 10759e minus 05 40095 11434e minus 05 400922minus 5 65652e minus 07 40024 67133e minus 07 40024 71346e minus 07 40024
14 Mathematical Problems in Engineering
constant and for different α we have different f(x t)
accordinglyWe apply the fast compact difference scheme to dis-
cretize the equation In Figure 2 we setc 0001 α 15 andM N 100 and plot exact andnumerical solutions at time T 1 for Example 2 with dif-ferent x0 For x0 05 α 15 andM N 100 we alsoplot exact and numerical solutions with the different c inFigure 3 In Figure 4 for h 01 and α 15 we vary thetotal number of time steps N to plot out the CPU time (inseconds) of the FCD scheme and DCD scheme )e nu-merical experiments verified our theoretical results In Ta-ble 6 by equating Δt h2 and fixing x0 05 we compute
the discrete linfin error and convergence rates with differentfractional derivative orders α and different c It shows thatthe compact finite difference scheme has space accuracy offourth order and temporal accuracy of second order We seth
Δt
radicand fix c 001 and the discrete l2 error and
convergence rates with different α and x0 are displayed inTable 7 )ese numerical convergence rates are almostapproaching O(Δt2 + h4) as Example 1
Example 3 If the exact solution is given byu(x t) etsin(πx)sin(πy) we have different f(x y t) fordifferent α accordingly
zu(x y t)
zt+
zαu(x y t)
ztα
z2u(x y t)
zx2 +z2u(x y t)
zy2 + f(x y t)
u(x 0) sin(πx)sin(πy)
zu
zt | t0 sin(πx)sin(πy)
u(0 y t) u(1 y t) u(x 0 t) u(x 1 t) 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(75)
In Figure 5 h 01 and α 175 are fixed and the totalnumber of time steps N vary to plot out the CPU time (inseconds) of the FCD procedure and DCD procedure and itpresents an approximately linear computation complexityfor FCD procedure We set Δt h2 in Table 8 and h
Δt
radic
in Table 9 the discrete linfin error discrete l2 error andconvergence rates with different derivative orders α arepresented )e fourth-order space accuracy and second-order temporal accuracy can be observed clearly
7 Conclusion
In this paper we develop and analyze a fast compact finitedifference procedure for the Cattaneo equation equipped withtime-fractional derivative without singular kernel )e time-fractional derivative is of CaputondashFabrizio type with the orderof α(1lt αlt 2) Compact difference discretization is appliedto obtain a high-order approximation for spatial derivatives ofinteger order in the partial differential equation and theCaputondashFabrizio fractional derivative is discretized by means
of CrankndashNicolson approximation It has been proved thatthe proposed compact finite difference scheme has spatialaccuracy of fourth order and temporal accuracy of secondorder Since the fractional derivatives are history dependentand nonlocal huge memory for storage and computationalcost are required )is means extreme difficulty especially fora long-time simulation Enlightened by the treatment forCaputo fractional derivative [32] we develop an effective fastevaluation procedure for the new CaputondashFabrizio fractionalderivative for the compact finite difference scheme Severalnumerical experiments have been carried out to show theconvergence orders and applicability of the scheme
Inspired by the work [43] the topic about modelling andnumerical solutions of porous media flow equipped withfractional derivatives is very interesting and challenging andwill be our main research direction in the future
Data Availability
All data generated or analyzed during this study are includedin this article
Table 9 Considering h Δt
radic the discrete l2 error and convergence rates of u with different α for Example 3
Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
2minus 4 00014 mdash 00014 mdash 00015 mdash2minus 6 84738e minus 05 40463 86640e minus 05 40142 92056e minus 05 402632minus 8 52610e minus 06 40096 53795e minus 06 40095 57168e minus 06 400922minus 10 32826e minus 07 40024 33566e minus 07 40024 35673e minus 07 40023
Mathematical Problems in Engineering 15
Conflicts of Interest
)e authors declare that they have no conflicts of interest
Acknowledgments
)is work was supported in part by the National NaturalScience Foundation of China under Grant nos 91630207and 11971272
References
[1] A Chaves ldquoA fractional diffusion equation to describe Levyflightsrdquo Physics Letters A vol 239 no 1-2 pp 13ndash16 1998
[2] M Giona and H E Roman ldquoFractional diffusion equation fortransport phenomena in random mediardquo Physica A Statis-tical Mechanics and its Applications vol 185 no 1ndash4pp 87ndash97 1992
[3] R Metzler and J Klafter ldquo)e random walkrsquos guide toanomalous diffusion a fractional dynamics approachrdquoPhysics Reports vol 339 no 1 pp 1ndash77 2000
[4] R L Magin Fractional Calculus in Bioengineering BegellHouse Redding Redding CT USA 2006
[5] R L Magin O Abdullah D Baleanu and X J ZhouldquoAnomalous diffusion expressed through fractional orderdifferential operators in the Bloch-Torrey equationrdquo Journalof Magnetic Resonance vol 190 no 2 pp 255ndash270 2008
[6] I Podlubny Fractional Differential Equations An Introduc-tion to Fractional Derivatives Fractional Differential Equa-tions to Methods of Beir Solution and Some of BeirApplications Vol 198 Elsevier Amsterdam Netherlands1998
[7] M Ran and C Zhang ldquoCompact difference scheme for a classof fractional-in-space nonlinear damped wave equations intwo space dimensionsrdquo Computers amp Mathematics withApplications vol 71 no 5 pp 1151ndash1162 2016
[8] A H Bhrawy and M A Zaky ldquoHighly accurate numericalschemes for multi-dimensional space variable-order frac-tional Schrodinger equationsrdquo Computers amp Mathematicswith Applications vol 73 no 6 pp 1100ndash1117 2017
[9] S B Yuste ldquoWeighted average finite difference methods forfractional diffusion equationsrdquo Journal of ComputationalPhysics vol 216 no 1 pp 264ndash274 2006
[10] T A M Langlands and B I Henry ldquo)e accuracy andstability of an implicit solution method for the fractionaldiffusion equationrdquo Journal of Computational Physicsvol 205 no 2 pp 719ndash736 2005
[11] C-M Chen F Liu I Turner and V Anh ldquoA Fourier method forthe fractional diffusion equation describing sub-diffusionrdquo Journalof Computational Physics vol 227 no 2 pp 886ndash897 2007
[12] Y Lin and C Xu ldquoFinite differencespectral approximationsfor the time-fractional diffusion equationrdquo Journal of Com-putational Physics vol 225 no 2 pp 1533ndash1552 2007
[13] G-h Gao and Z-z Sun ldquoA compact finite difference schemefor the fractional sub-diffusion equationsrdquo Journal of Com-putational Physics vol 230 no 3 pp 586ndash595 2011
[14] F Zeng C Li F Liu and I Turner ldquoNumerical algorithms fortime-fractional subdiffusion equation with second-order ac-curacyrdquo SIAM Journal on Scientific Computing vol 37 no 1pp A55ndashA78 2015
[15] G-H Gao Z-Z Sun and H-W Zhang ldquoA new fractionalnumerical differentiation formula to approximate the Caputofractional derivative and its applicationsrdquo Journal of Com-putational Physics vol 259 no 2 pp 33ndash50 2014
[16] H Li J Cao and C Li ldquoHigh-order approximation to Caputoderivatives and Caputo-type advection-diffusion equations(III)rdquo Journal of Computational and Applied Mathematicsvol 299 pp 159ndash175 2016
[17] A A Alikhanov ldquoA new difference scheme for the timefractional diffusion equationrdquo Journal of ComputationalPhysics vol 280 pp 424ndash438 2015
[18] Z Wang and S Vong ldquoCompact difference schemes for themodified anomalous fractional sub-diffusion equation and thefractional diffusion-wave equationrdquo Journal of ComputationalPhysics vol 277 pp 1ndash15 2014
[19] H-K Pang and H-W Sun ldquoFourth order finite differenceschemes for time-space fractional sub-diffusion equationsrdquoComputers amp Mathematics with Applications vol 71 no 6pp 1287ndash1302 2016
[20] M Dehghan M Abbaszadeh and W Deng ldquoFourth-ordernumerical method for the space-time tempered fractionaldiffusion-wave equationrdquo Applied Mathematics Lettersvol 73 no 11 pp 120ndash127 2017
[21] H Sun Z-Z Sun and G-H Gao ldquoSome high order dif-ference schemes for the space and time fractional Bloch-Torrey equationsrdquo Applied Mathematics and Computationvol 281 pp 356ndash380 2016
[22] S Arshad J Huang A Q M Khaliq and Y Tang ldquoTrape-zoidal scheme for time-space fractional diffusion equationwith Riesz derivativerdquo Journal of Computational Physicsvol 350 pp 1ndash15 2017
[23] H Wang K Wang and T Sircar ldquoA direct O(Nlog2N) finitedifferencemethod for fractional diffusion equationsrdquo Journal ofComputational Physics vol 229 no 21 pp 8095ndash8104 2010
[24] H Wang and T S Basu ldquoA fast finite difference method fortwo-dimensional space-fractional diffusion equationsrdquo SIAMJournal on Scientific Computing vol 34 no 5 pp A2444ndashA2458 2012
[25] H Wang and N Du ldquoA superfast-preconditioned iterativemethod for steady-state space-fractional diffusion equationsrdquoJournal of Computational Physics vol 240 pp 49ndash57 2013
[26] H Wang and N Du ldquoA fast finite difference method forthree-dimensional time-dependent space-fractional diffusionequations and its efficient implementationrdquo Journal ofComputational Physics vol 253 pp 50ndash63 2013
[27] H Wang and N Du ldquoFast alternating-direction finite dif-ference methods for three-dimensional space-fractional dif-fusion equationsrdquo Journal of Computational Physics vol 258pp 305ndash318 2014
[28] K Wang and H Wang ldquoA fast characteristic finite differencemethod for fractional advection-diffusion equationsrdquo Ad-vances in Water Resources vol 34 no 7 pp 810ndash816 2011
[29] H Wang and H Tian ldquoA fast Galerkin method with efficientmatrix assembly and storage for a peridynamic modelrdquoJournal of Computational Physics vol 231 no 23pp 7730ndash7738 2012
[30] H Tian H Wang and W Wang ldquoAn efficient collocationmethod for a non-local diffusion modelrdquo InternationalJournal of Numerical Analysis amp Modeling vol 10 no 4pp 815ndash825 2013
[31] H Wang and H Tian ldquoA fast and faithful collocation methodwith efficient matrix assembly for a two-dimensional nonlocaldiffusion modelrdquo Computer Methods in Applied Mechanicsand Engineering vol 273 no 5 pp 19ndash36 2014
[32] S Jiang J Zhang Q Zhang and Z Zhang ldquoFast evaluation ofthe Caputo fractional derivative and its applications tofractional diffusion equationsrdquo Communications in Compu-tational Physics vol 21 no 3 pp 650ndash678 2017
16 Mathematical Problems in Engineering
[33] Y Yan Z-Z Sun and J Zhang ldquoFast evaluation of the Caputofractional derivative and its applications to fractional diffusionequations a second-order schemerdquo Communications inComputational Physics vol 22 no 4 pp 1028ndash1048 2017
[34] X Lu H-K Pang and H-W Sun ldquoFast approximate in-version of a block triangular Toeplitz matrix with applicationsto fractional sub-diffusion equationsrdquo Numerical Linear Al-gebra with Applications vol 22 no 5 pp 866ndash882 2015
[35] M Caputo and M Fabrizio ldquoA new definition of fractionalderivative without singular kernelrdquo Progress in FractionalDifferentiation and Applications vol 1 no 2 pp 73ndash85 2015
[36] A Atangana ldquoOn the new fractional derivative and appli-cation to nonlinear Fisherrsquos reaction-diffusion equationrdquoAppliedMathematics and Computation vol 273 pp 948ndash9562016
[37] M D Ortigueira and J Tenreiro Machado ldquoA critical analysisof the Caputo-Fabrizio operatorrdquo Communications in Non-linear Science and Numerical Simulation vol 59 pp 608ndash6112018
[38] V E Tarasov ldquoNo nonlocality no fractional derivativerdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 62 pp 157ndash163 2018
[39] V Tarasov ldquoCaputo-Fabrizio operator in terms of integerderivatives memory or distributed lagrdquo Computational andApplied Mathematics vol 38 p 113 2019
[40] A Giusti ldquoA comment on some new definitions of fractionalderivativerdquoNonlinear Dynamics vol 93 no 3 pp 1757ndash17632018
[41] A Atangana and J F Gomez-Aguilar ldquoFractional derivativeswith no-index law property application to chaos and sta-tisticsrdquo Chaos Solitons amp Fractals vol 114 pp 516ndash535 2018
[42] J Hristov ldquoResponse functions in linear viscoelastic consti-tutive equations and related fractional operatorsrdquo Mathe-matical Modelling of Natural Phenomena vol 14 no 3pp 1ndash34 2019
[43] L X Vivas-Cruz A Gonzalez-Calderon M A Taneco-Hernandez and D P Luis ldquo)eoretical analysis of a model offluid flow in a reservoir with the Caputo-Fabrizio operatorrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 84 Article ID 105186 2020
[44] H-L Liao and Z-Z Sun ldquoMaximum norm error bounds ofADI and compact ADI methods for solving parabolicequationsrdquo Numerical Methods for Partial DifferentialEquations vol 26 no 1 pp 37ndash60 2010
[45] Z Liu A Cheng and X Li ldquoA second order Crank-Nicolsonscheme for fractional Cattaneo equation based on newfractional derivativerdquoAppliedMathematics and Computationvol 311 pp 361ndash374 2017
[46] Z Sun and G Gao Finite Difference Method for FractionalDifferential Equations Science Press Beijing China 2015
Mathematical Problems in Engineering 17
Q ek+1
1113872 1113873leQ ek
1113872 1113873 +(α minus 1)Δt Rk+(12)
2 (37)
Summing up with respect to k from 0 to N minus 1 we get
Q eN
1113872 1113873leQ e0
1113872 1113873 + C Δt2 + h4
1113872 11138732 (38)
Observing that the initial error e0 0 implies Q(e0) 0)en we have
(α minus 1) δxeN
11138681113868111386811138681113868
111386811138681113868111386811138682
1leQ e
N1113872 1113873leC Δt2 + h
41113872 1113873
2
1113944
N
n1MNminus n δte
n
2AleQ e
N1113872 1113873leC Δt2 + h
41113872 1113873
2
(39)
Theorem 2 Suppose that the exact solution of the fractionalCattaneo equation is smooth sufficiently then there exists apositive constant C independent of h k and Δt such that
ek
11138681113868111386811138681113868
111386811138681113868111386811138681leC Δt2 + h
41113872 1113873 forall0le kleN (40)
where eki uk
i minus uki
4 Compact Finite Difference Scheme inTwo Dimensions
In this section the following fractional Cattaneo equation intwo dimensions will be considered
zu(x y t)
zt+
zαu(x y t)
ztα
z2u(x y t)
zx2 +z2u(x y t)
zy2 + f(x y t) (x y t) isin (a b) times(c d) times(0 T]
u(x y 0) ϕ(x y)zu
zt
1113868111386811138681113868111386811138681113868 t0 ψ(x y) ale xle b cleyled
u(a y t) u(b y t) u(x c t) u(x d t) 0 tgt 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(41)
where 1lt αlt 2 f(x y t) is the source termϕ(x y) andψ(x y) are the given functions and zαuztα isdefined by the new Caputo fractional derivative withoutsingular kernel
In order to construct the finite difference schemes therectangle [a b] times [c d] is discretized with xi
a + ihx (0le ileMx) and yj c + jhy (0le jleMy) andthe time interval [0 T] is discretized with tk kΔt (0le kleN)where hx (b minus a)Mx hy (d minus c) My and Δt TN arethe spatial grid and temporal step sizes respectively DenoteΩh
(xi yj) 0le ileMx 0le1113966 jleMy and ΩΔt tk 0le kleN1113864 1113865then Ωh timesΩΔt is a discretization of the physical computationaldomain [a b] times [c d] times [0 T] uk
ij u(xi yj tk) denotes thevalues of function u at the grid points and uk
ij denotes the valuesof the numerical solution at the point (xi yj tk)
Denote Vh v| v (v00 v01 vMxMy)1113882 1113883 and V0
h
v| v isin Vh v0j vMxj vi0 viMy 0 0 le i leMx 0le j1113882
leMy For any mesh function v isin Vh we use the followingnotations
δxviminus (12)j vij minus viminus 1j
hx
δ2xvij vi+1j minus 2vij + viminus 1j
h2x
δyvijminus (12) vij minus vijminus 1
hy
δ2yvij vij+1 minus 2vij + vijminus 1
h2y
1le ileMx minus 1 1le jleMy minus 1
(42)
and define the average operator
Axvij
112
vi+1j + 10vij + viminus 1j1113872 1113873 1le ileMx minus 1 0le jleMy
vij i 0 Mx 0le jleMy
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
Ayvij
112
vij+1 + 10vij + vijminus 11113872 1113873 1le jleMy minus 1 0le ileMx
vij j 0 My 0le ileMx
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(43)
6 Mathematical Problems in Engineering
It is clear that
Axvij I +h2
x
12δ2x1113888 1113889vij
Ayvij I +h2
y
12δ2y⎛⎝ ⎞⎠vij
(44)
We also denote AxAy(u v) (AxAyu v) It is easy to seethat AxAy AyAx
For any gird function u v isin V0h the discrete inner
product and norms are defined as follows
(u v) hxhy 1113944
Mxminus 1
i11113944
My minus 1
j1uijvij
u22 (u u)
(45)
For the average operator AxAy define
AxAy(v v)≜ AxAyv v1113872 1113873 v2A (46)
41 Compact Finite Difference Scheme At the node(xi yj tk+(12)) the differential equation is rewritten as
zu
ztxi yj tk+(12)1113872 1113873 +
zαu
ztαxi yj tk+(12)1113872 1113873
z2u
zx2 xi yj tk+(12)1113872 1113873 +z2u
zy2 xi yj tk+(12)1113872 1113873
+ f xi yj tk+(12)1113872 1113873
1le ileMx minus 1 1le jleMy minus 1 1le kleN minus 1
(47)
For the approximation of the time-fractional derivativewe have the following approximation [45]
CF0 D
αt u xi yj tk+(12)1113872 1113873
1(α minus 1)Δt
M0δtuk+1ij minus 1113944
k
n1Mkminus n(⎡⎣
minus Mkminus n+11113857δtunij minus Mkψij
⎤⎦
+ Rk+(12)ij
(48)
where the truncation error Rk+(12)ij O(Δt2) and
Mn exp1 minus α2 minus αΔtn1113876 1113877 minus exp
1 minus α2 minus αΔt(n + 1)1113876 1113877 (49)
Furthermore we also have
z2u
zx2 xi yj tk+(12)1113872 1113873 12
δ2xuk+1ij + δ2xuk
ij1113872 1113873
Ax
+ O Δt2 + h4x1113872 1113873
(50)
z2u
zy2 xi yj tk+(12)1113872 1113873 12
δ2yuk+1ij + δ2yuk
ij1113872 1113873
Ay
+ O Δt2 + h4y1113872 1113873
(51)
zu
ztxi yj tk+(12)1113872 1113873 δtu
k+1ij + O Δt21113872 1113873 (52)
Substituting (48) and (50)sim(52) into (47) leads to
AxAyδtuk+1ij +
1(α minus 1)τ
AxAy M0δtuk+1ij1113890
minus 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857δtu
nij minus Mkψij
⎤⎦
12
δ2xuk+1ij + δ2xu
kij1113872 1113873 +
12
δ2yuk+1ij + δ2yu
kij1113872 1113873 + AxAyf
k+(12)ij
+ Rk+(12)ij
1le ileMx minus 1 1le jleMy minus 1 0le kleN minus 1
(53)
and there exists a constant C depending on the function uand its derivatives such that
Rk+(12)ij leC Δt2 + h
4x + h
4y1113872 1113873 (54)
By the initial and boundary conditions we have
u0ij ϕij 1le ileMx minus 1 1le jleMy minus 1
uk0j u
kMxj u
ki0 u
kiMy
0 0le kleN(55)
Omitting the truncation error Rk+(12)ij and replacing the
true solution ukij with numerical solution uk
ij a compactfinite difference scheme can be obtained as follows
(α minus 1)ΔtAxAyδtUk+1ij + M0AxAyδtU
k+1ij
minus Ay
(α minus 1)Δt2
δ2xUk+1ij minus Ax
(α minus 1)Δt2
δ2yUk+1ij
Ay
(α minus 1)Δt2
δ2xukij + Ax
(α minus 1)Δt2
δ2yukij
+ 1113944k
n1Mkminus n minus Mkminus n+1( 1113857AxAyδtu
nij + MkAxAyψij
+ (α minus 1)ΔtAxAyfk+(12)ij
1le ileMx minus 1 1le jleMy minus 1 1le kleN minus 1
u0ij ϕij 1le ileMx minus 1 1le jleMy minus 1
uk0j u
kMxj u
ki0 u
kiMy
0 0le kleN
(56)
Mathematical Problems in Engineering 7
42 Stability Analysis and Optimal Error Estimates
421 Stability Analysis
Definition 4 (see [46]) For any gird function u isin V0h define
the norm
⦀nablahu⦀2A δxu
2
minush2
y
12δyδxu
2
⎛⎝ ⎞⎠ + δyu
2
minush2
x
12δxδyu
2
1113888 1113889
(57)
)e lemmas below is useful in the subsequent analysis ofstability
Lemma 3 (see [46]) For any gird function u isin V0h the
following equation is held13u
2 le u2A le u
2 (58)
Lemma 4 (see [46]) For any gird function u isin V0h the
following equation is held23nablahu
2 le⦀nablahu⦀2A le nablahu
2
Ayδ2xu
nminus (12)+ Axδ
2yu
nminus (12) δtu
nminus (12)1113872 1113873
minus12Δt⦀nablahu
n⦀2A minus ⦀nablahunminus 1⦀
2A1113874 1113875
(59)
where nablahu2 δxu22 + δyu22
Multiplying hxhyδtUk+1ij on both sides of equation (56)
and summing up wrt i j from 1 to (Mx minus 1) and from 1 to(My minus 1) respectively the following equation is obtained
(α minus 1)Δt δtUk+1
2
A+ M0 δtU
k+1
2
A
(α minus 1)Δt Ayδ2xU
k+(12)+ Axδ
2yU
k+(12) δtU
k+11113872 1113873 + 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 AyAxδtU
n δtU
k+11113872 1113873
+ Mk AyAxψ δtUk+1
1113872 1113873 +(α minus 1)Δt AyAxfk+(12)
δtUk+1
1113872 1113873
(60)
Observing Lemma 4 we have
Ayδ2xUk+(12) + Axδ
2yUk+(12) δtU
k+11113872 1113873 minus12Δt⦀nablahu
k+1⦀2A minus ⦀nablahu
k⦀2A1113874 1113875 (61)
(α minus 1)Δt AyAxfk+(12) δtUk+11113872 1113873le
(α minus 1)Δt2
fk+(12)
2
A+ δtu
k+1
2
A1113874 1113875 (62)
By the triangle inequality and Lemma 2 we obtain
1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 AyAxδtU
n δtU
k+11113872 1113873 + Mk AyAxψ δtU
k+11113872 1113873
le 1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δtUn
2A
+ δtUk+1
2
A1113874 1113875 +
12Mk ψ
2A + δtU
k+1
2
A1113874 1113875
1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δtUn
2A
+12
M0 minus Mk( 1113857 δtUk+1
2
A+12Mk ψ
2A + δtU
k+1
2
A1113874 1113875
1113944k
n1
12Mkminus n δtU
n
2A
minus 1113944k
n1
12Mkminus n+1 δtU
n
2A
+12M0 δtU
k+1
2
A+12Mkψ
2A
(63)
Combining equation (60) with (61)sim(63)we get
1113944
k+1
n1Mkminus n+1 δtU
n
2A
+(α minus 1)⦀nablahUk+1⦀
2A
le 1113944k
n1Mkminus n δtU
n
2A
+(α minus 1) nablahUk
2
A+ Mkψ
2A +(α minus 1)Δt⦀fk+(12)⦀
2A
(64)
8 Mathematical Problems in Engineering
Let
Q Uk
1113872 1113873 1113944k
n1Mkminus n δtU
n
2A
+(α minus 1)⦀nablahUk⦀
2A (65)
Summing up with respect to k from 0 to N minus 1 we get
Q UN
1113872 1113873leQ U0
1113872 1113873 + 1113944Nminus 1
k0Mkψ
2A +(α minus 1)Δt 1113944
Nminus 1
k0f
k+(12)
2
A
(66)
Noting that U0 ϕ we have Q(U0) (α minus 1)nablahϕ2A Itfollows that
Q UN
1113872 1113873le (α minus 1)⦀nablahϕ⦀2A + 1113944
Nminus 1
k0Mkψ
2A
+(α minus 1)Δt 1113944Nminus 1
k0f
k+(12)
2
A
(67)
Theorem 3 For the compact finite difference scheme (56)the following stability inequality holds
Q Um
( 1113857le (α minus 1) nablahϕ1113868111386811138681113868
1113868111386811138681113868
2A
+ 1113944
mminus 1
k0Mkψ
2A
+(α minus 1)Δt 1113944
mminus 1
k0f
k+(12)
2
A forall0lemleN
(68)
Similar to the stability the convergence can also beanalyzed
Theorem 4 Suppose that the exact solution of the fractionalCattaneo equation is sufficiently smooth then there exists apositive constant C independent of h k and Δt such that
ek
11138681113868111386811138681113868
111386811138681113868111386811138681leC Δt2 + h
41113872 1113873 forall0le kleN (69)
where ekij uk
ij minus ukij and h max hx hy1113966 1113967
5 Efficient Storage and Fast Evaluation of theCaputondashFabrizio Fractional Derivative
Since time-fractional derivative operator is nonlocal thetraditional direct method for numerically solving the frac-tional Cattaneo equations generally requires total O(MN)
memory units and O(MN2) computational complexitywhere N andM are the total number of time steps and spacesteps respectively
In this section we develop a fast solution method for thefinite difference scheme of the time-fractional Cattaneoequation
Let
Nkn Mkminus n minus Mkminus n+1
exp1 minus α2 minus α
(k minus n)Δt1113874 1113875
middot 1 minus 2exp1 minus α2 minus αΔt1113874 1113875 + exp
1 minus α2 minus α
2Δt1113874 11138751113876 1113877
1le nle k
(70)
then
Nk+1n Mk+1minus n minus Mk+1minus n+1
exp1 minus α2 minus α
(k minus n)Δt1113874 1113875
middot 1 minus 2exp1 minus α2 minus αΔt1113874 1113875 + exp
1 minus α2 minus α
2Δt1113874 11138751113876 1113877exp1 minus α2 minus αΔt1113874 1113875
exp1 minus α2 minus αΔt1113874 1113875N
kn
Nk+1k+1 M0 minus M1 1 minus 2exp
1 minus α2 minus αΔt1113874 1113875 + exp
1 minus α2 minus α
2Δt1113874 1113875
(71)
So we have
1113944
k+1
n1N
k+1n Aδtu
ni exp
1 minus α2 minus αΔt1113874 1113875 1113944
k
n1N
knAδtu
ni + N
k+1k+1Aδtu
k+1i
1113944
k+1
n1N
k+1n AyAxδtu
nij exp
1 minus α2 minus αΔt1113874 1113875 1113944
k
n1N
knAyAxδtu
nij
+ Nk+1k+1AyAxδtU
k+1ij
(72)
Remark We find that at the k-th level only O(1) operationsare needed to compute the k-th level since the (k minus 1)-thlevel is known at that point )us the total operations arereduced from O(N2) to O(N) and the memory require-ment decreases from O(N) to O(1) We conclude that thatthis fast method significantly reduces the total computa-tional cost from O(MN2) to O(MN) and the memoryrequirement from O(MN) to O(M)
6 Numerical Experiments
In this section we carry out several numerical experiments tocheck the effectiveness of the proposed scheme )e conver-gence rate and CPU consumption are all compared in thesimulations We take the space-time domainΩ [0 1] T 1for one-dimensional case and Ω [0 1] times [0 1] T 1 fortwo-dimensional case )ese simulations are implemented inMatlab and the numerical experiments are run on a computerwith 4GB memory )e time-fractional Cattaneo equation ofthe following forms is considered
Example 1 We provide the exact solution u(x t)
etsin(πx) and for different α we have different f(x t)
accordingly
Mathematical Problems in Engineering 9
Table 1 Considering Δt h2 the discrete linfin error and convergence rates of u with different α for Example 1
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 15320e minus 04 mdash 15941e minus 04 mdash 18076e minus 04 mdash2minus 4 95109e minus 06 40097 98981e minus 06 40094 11230e minus 05 400862minus 5 59343e minus 07 40024 61762e minus 07 40024 70079e minus 07 400222minus 6 37074e minus 08 40006 38585e minus 08 40006 43783e minus 08 40005
Table 2 Considering h Δt
radic the discrete l2 error and convergence rates of u with different α for Example 1
Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
2minus 4 00017 mdash 00018 mdash 00021 mdash2minus 6 10832e minus 04 40382 11271e minus 04 40373 12781e minus 04 403442minus 8 67252e minus 06 40097 69990e minus 06 40094 79405e minus 06 400862minus 10 41961e minus 07 40024 43672e minus 07 40023 49553e minus 07 400212minus 12 26215e minus 08 40006 27284e minus 08 40005 30959e minus 08 40005
Table 3 )e CPU time consumption of the fast compact difference scheme and direct difference scheme for Example 1
Δtα 125 α 15 α 175
DCD FCD DCD FCD DCD FCD11000 34008 18876 34632 19032 33540 1918812500 142740 46800 141960 47580 138840 4742415000 470967 93600 463010 94692 461450 9375617500 994194 139464 956754 140088 967206 138684110 000 1699318 186733 1641130 186421 1658290 184393125 000 9816362 466910 9509820 469251 9591409 460046150 000 38269853 934133 36923096 940062 37547412 919625
N times104
0
500
1000
1500
2000
2500
3000
3500
4000
CPU
tim
e
Fast evaluationDirect evaluation
0 05 1 15 2 25 3 35 4 45 5
(a)
Fast evaluationDirect evaluation
log10 (N)
ndash05
0
05
1
15
2
25
3
35
4
log1
0 (C
PU ti
me)
25 3 35 4 45 5
(b)
Figure 1 )e CPU time (a) and the log-log CPU time (b) versus the total number of time steps N for Example 1
10 Mathematical Problems in Engineering
Table 4 Considering Δt 2minus 13 the discrete linfin error and convergence rates of u with different α for Example 1
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 22486e minus 04 mdash 22733e minus 04 mdash 23705e minus 04 mdash2minus 4 13985e minus 05 40071 14140e minus 05 40069 14745e minus 05 400692minus 5 86897e minus 07 40084 87882e minus 07 40081 91726e minus 07 400682minus 6 50194e minus 08 41137 51024e minus 08 41063 54092e minus 08 40838
Table 5 Considering h 0001 the discrete linfin error and convergence rates of u with different α for Example 1
Δtα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 00046 mdash 00043 mdash 00036 mdash2minus 4 00011 20641 00011 19668 90063e minus 04 199902minus 5 28662e minus 04 19403 27171e minus 04 20174 22518e minus 04 199992minus 6 71658e minus 05 19999 67928e minus 05 20000 56295e minus 05 200002minus 7 17915e minus 05 20000 16982e minus 05 20000 14074e minus 05 20000
x
ndash02
0
02
04
06
08
1
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(a)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(b)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(c)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(d)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(e)
ndash02
0
02
04
06
08
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(f)
Figure 2 Considering c 0001 α 15 M N 100 real and numerical solutions of u at T 1 for Example 2 with the value of x0 equalto (a) 0 (b) 02 (c) 04 (d) 06 (e) 08 and (f) 1
Mathematical Problems in Engineering 11
zu(x t)
zt+
zαu(x t)
ztα
z2u(x t)
zx2 + f(x t)
u(x 0) sin(πx)zu
zt
1113868111386811138681113868111386811138681113868 t0 sin(πx)
u(0 t) u(1 t) 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(73)
In Tables 1 and 2 we take Δt h2 and h Δt
radicto ex-
amine the discrete linfin-norm (l2-norm) errors and
corresponding spatial and temporal convergence rates re-spectively We list the errors and convergence rates (order) ofthe proposed compact finite difference (CD) scheme which isalmost O(Δt2 + h4) for different α Additionally Table 3shows the CPU time (CPU) consumed by direct compact(DCD) scheme and fast compact difference (FCD) schemerespectively It is obvious that the FCD scheme has a sig-nificantly reduced CPU time over the DCD scheme Forinstance when α 15 we choose h 01 and Δt 150 000and observe that the FCD scheme consumes only 94 secondswhile the DCD scheme consumes 3692 seconds We can find
x
075
08
085
09
095
1
105
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(a)
0
01
02
03
04
05
06
07
08
09
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(b)
0
01
02
03
04
05
06
07
08
09
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(c)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(d)
Figure 3 Considering x0 05 α 15 M N 100 real and numerical solutions of u at T 1 for Example 2 with the value of c equal to(a) 1 (b) 01 (c) 001 and (d) 0001
12 Mathematical Problems in Engineering
that the performance of the FCD scheme will be moreconspicuous as the time step size Δt decreases
In Figure 1 we set h 01 and α 175 and change thetotal number of time steps N to plot out the CPU time (inseconds) of the FCD scheme and DCD scheme We canobserve that the CPU time increases almost linearly withrespect to N for the FCD scheme while the DCD schemescales like O(N2)
Tables 4 and 5 show the discrete linfin errors and con-vergence rates of the compact finite difference scheme forExample 1 )e space rates are almost O(h4) for fixedΔt 2minus 13 and the time convergence rates are always O(Δt2)for fixed h 0001 We can conclude that the numericalconvergence rates of our scheme approach almost toO(Δt2 + h4)
Example 2 )e example is described by
zu(x t)
zt+
zαu(x t)
ztα
z2u(x t)
zx2 + f(x t)
u(x 0) 0zu
zt
1113868111386811138681113868111386811138681113868t0 0
u(0 t) t2exp minusx20
c1113888 1113889 u(1 t) t2exp minus
1 minus x0( 11138572
c1113888 1113889
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(74)
Note that the exact solution of the above problem isu(x t) t2exp(minus (x minus x0)
2c) where x0 isin [0 1] and c is a
N times104
0
1000
2000
3000
4000
5000
6000
7000
8000CP
U ti
me
Fast evaluationDirect evaluation
0 05 1 15 2 25 3 35 4 45 5
(a)
Fast evaluationDirect evaluation
log10 (N)
0
05
1
15
2
25
3
35
4
log1
0 (C
PU ti
me)
25 3 35 4 45 5
(b)
Figure 4 Considering x0 05 c 001 α 15 andM 10 the CPU time (a) and the log-log CPU time (b) versus the total number oftime steps N for Example 2
Table 6 Considering Δt h2 andx0 05 the discrete linfin error and convergence rates of u with different α and c for Example 2
c hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
1
2minus 3 73224e minus 05 mdash 55643e minus 05 mdash 38478e minus 05 mdash2minus 4 45707e-06 40018 34695e minus 06 40034 23900e minus 06 400902minus 5 28558e minus 07 40004 21671e minus 07 40009 14913e minus 07 400242minus 6 17847e minus 08 40001 13543e minus 08 40001 93169e minus 09 40006
01
2minus 3 00012 mdash 00012 mdash 00012 mdash2minus 4 70462e minus 05 40900 69731e minus 05 41051 69007e minus 05 412012minus 5 43571e minus 06 40154 43110e minus 06 40157 42650e minus 06 401612minus 6 27160e minus 07 40038 26871e minus 07 40039 26583e minus 07 40040
001
2minus 4 00090 mdash 00090 mdash 00090 mdash2minus 5 49257e minus 04 41915 49235e minus 04 41922 49232e minus 04 419232minus 6 29868e minus 05 40437 29855e minus 05 40436 29854e minus 05 404362minus 7 18531e minus 06 40106 18523e minus 06 40106 18522e minus 06 40106
Mathematical Problems in Engineering 13
Table 7 Considering h Δt
radicand c 001 the discrete l2 error and convergence rates of u with different α andx0 for Example 2
x0 Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
02minus 8 00044 mdash 00044 mdash 00042 mdash2minus 10 24205e minus 04 41841 23721e minus 04 42133 23098e minus 04 418452minus 12 14684e minus 05 40430 14390e minus 05 40430 14013e minus 05 40429
0252minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14902e minus 04 41794 14894e minus 04 41802 14892e minus 04 418032minus 12 90680e minus 06 40386 90633e minus 06 40386 90616e minus 06 40386
052minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14848e minus 04 41846 14842e minus 04 41852 14842e minus 04 418522minus 12 90323e minus 06 40390 90288e minus 06 40390 90286e minus 06 40390
0752minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14902e minus 04 41794 14894e minus 04 41802 14892e minus 04 418032minus 12 90680e minus 06 40386 90633e minus 06 40386 90616e minus 06 40386
12minus 8 00044 mdash 00044 mdash 00042 mdash2minus 10 24205e minus 04 41841 23721e minus 04 42133 23098e minus 04 418452minus 12 14684e minus 05 40430 14390e minus 05 40430 14013e minus 05 40429
0 2 4 6 8 10times104
0
2000
4000
6000
8000
10000
12000
14000
N
CPU
tim
e
Fast evaluationDirect evaluation
(a)
Fast evaluationDirect evaluation
36 38 4 42 44 46 48 515
2
25
3
35
4
45
log10 (N)
log1
0 (C
PU ti
me)
(b)
Figure 5 )e CPU time (a) and the log-log CPU time (b) versus the total number of time steps N for Example 3
Table 8 Considering Δt h2 the discrete linfin error and convergence rates of u with different α for Example 3
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 2 00028 mdash 00028 mdash 00030 mdash2minus 3 16948e minus 04 40462 17328e minus 04 40142 18411e minus 04 402632minus 4 10522e minus 05 40096 10759e minus 05 40095 11434e minus 05 400922minus 5 65652e minus 07 40024 67133e minus 07 40024 71346e minus 07 40024
14 Mathematical Problems in Engineering
constant and for different α we have different f(x t)
accordinglyWe apply the fast compact difference scheme to dis-
cretize the equation In Figure 2 we setc 0001 α 15 andM N 100 and plot exact andnumerical solutions at time T 1 for Example 2 with dif-ferent x0 For x0 05 α 15 andM N 100 we alsoplot exact and numerical solutions with the different c inFigure 3 In Figure 4 for h 01 and α 15 we vary thetotal number of time steps N to plot out the CPU time (inseconds) of the FCD scheme and DCD scheme )e nu-merical experiments verified our theoretical results In Ta-ble 6 by equating Δt h2 and fixing x0 05 we compute
the discrete linfin error and convergence rates with differentfractional derivative orders α and different c It shows thatthe compact finite difference scheme has space accuracy offourth order and temporal accuracy of second order We seth
Δt
radicand fix c 001 and the discrete l2 error and
convergence rates with different α and x0 are displayed inTable 7 )ese numerical convergence rates are almostapproaching O(Δt2 + h4) as Example 1
Example 3 If the exact solution is given byu(x t) etsin(πx)sin(πy) we have different f(x y t) fordifferent α accordingly
zu(x y t)
zt+
zαu(x y t)
ztα
z2u(x y t)
zx2 +z2u(x y t)
zy2 + f(x y t)
u(x 0) sin(πx)sin(πy)
zu
zt | t0 sin(πx)sin(πy)
u(0 y t) u(1 y t) u(x 0 t) u(x 1 t) 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(75)
In Figure 5 h 01 and α 175 are fixed and the totalnumber of time steps N vary to plot out the CPU time (inseconds) of the FCD procedure and DCD procedure and itpresents an approximately linear computation complexityfor FCD procedure We set Δt h2 in Table 8 and h
Δt
radic
in Table 9 the discrete linfin error discrete l2 error andconvergence rates with different derivative orders α arepresented )e fourth-order space accuracy and second-order temporal accuracy can be observed clearly
7 Conclusion
In this paper we develop and analyze a fast compact finitedifference procedure for the Cattaneo equation equipped withtime-fractional derivative without singular kernel )e time-fractional derivative is of CaputondashFabrizio type with the orderof α(1lt αlt 2) Compact difference discretization is appliedto obtain a high-order approximation for spatial derivatives ofinteger order in the partial differential equation and theCaputondashFabrizio fractional derivative is discretized by means
of CrankndashNicolson approximation It has been proved thatthe proposed compact finite difference scheme has spatialaccuracy of fourth order and temporal accuracy of secondorder Since the fractional derivatives are history dependentand nonlocal huge memory for storage and computationalcost are required )is means extreme difficulty especially fora long-time simulation Enlightened by the treatment forCaputo fractional derivative [32] we develop an effective fastevaluation procedure for the new CaputondashFabrizio fractionalderivative for the compact finite difference scheme Severalnumerical experiments have been carried out to show theconvergence orders and applicability of the scheme
Inspired by the work [43] the topic about modelling andnumerical solutions of porous media flow equipped withfractional derivatives is very interesting and challenging andwill be our main research direction in the future
Data Availability
All data generated or analyzed during this study are includedin this article
Table 9 Considering h Δt
radic the discrete l2 error and convergence rates of u with different α for Example 3
Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
2minus 4 00014 mdash 00014 mdash 00015 mdash2minus 6 84738e minus 05 40463 86640e minus 05 40142 92056e minus 05 402632minus 8 52610e minus 06 40096 53795e minus 06 40095 57168e minus 06 400922minus 10 32826e minus 07 40024 33566e minus 07 40024 35673e minus 07 40023
Mathematical Problems in Engineering 15
Conflicts of Interest
)e authors declare that they have no conflicts of interest
Acknowledgments
)is work was supported in part by the National NaturalScience Foundation of China under Grant nos 91630207and 11971272
References
[1] A Chaves ldquoA fractional diffusion equation to describe Levyflightsrdquo Physics Letters A vol 239 no 1-2 pp 13ndash16 1998
[2] M Giona and H E Roman ldquoFractional diffusion equation fortransport phenomena in random mediardquo Physica A Statis-tical Mechanics and its Applications vol 185 no 1ndash4pp 87ndash97 1992
[3] R Metzler and J Klafter ldquo)e random walkrsquos guide toanomalous diffusion a fractional dynamics approachrdquoPhysics Reports vol 339 no 1 pp 1ndash77 2000
[4] R L Magin Fractional Calculus in Bioengineering BegellHouse Redding Redding CT USA 2006
[5] R L Magin O Abdullah D Baleanu and X J ZhouldquoAnomalous diffusion expressed through fractional orderdifferential operators in the Bloch-Torrey equationrdquo Journalof Magnetic Resonance vol 190 no 2 pp 255ndash270 2008
[6] I Podlubny Fractional Differential Equations An Introduc-tion to Fractional Derivatives Fractional Differential Equa-tions to Methods of Beir Solution and Some of BeirApplications Vol 198 Elsevier Amsterdam Netherlands1998
[7] M Ran and C Zhang ldquoCompact difference scheme for a classof fractional-in-space nonlinear damped wave equations intwo space dimensionsrdquo Computers amp Mathematics withApplications vol 71 no 5 pp 1151ndash1162 2016
[8] A H Bhrawy and M A Zaky ldquoHighly accurate numericalschemes for multi-dimensional space variable-order frac-tional Schrodinger equationsrdquo Computers amp Mathematicswith Applications vol 73 no 6 pp 1100ndash1117 2017
[9] S B Yuste ldquoWeighted average finite difference methods forfractional diffusion equationsrdquo Journal of ComputationalPhysics vol 216 no 1 pp 264ndash274 2006
[10] T A M Langlands and B I Henry ldquo)e accuracy andstability of an implicit solution method for the fractionaldiffusion equationrdquo Journal of Computational Physicsvol 205 no 2 pp 719ndash736 2005
[11] C-M Chen F Liu I Turner and V Anh ldquoA Fourier method forthe fractional diffusion equation describing sub-diffusionrdquo Journalof Computational Physics vol 227 no 2 pp 886ndash897 2007
[12] Y Lin and C Xu ldquoFinite differencespectral approximationsfor the time-fractional diffusion equationrdquo Journal of Com-putational Physics vol 225 no 2 pp 1533ndash1552 2007
[13] G-h Gao and Z-z Sun ldquoA compact finite difference schemefor the fractional sub-diffusion equationsrdquo Journal of Com-putational Physics vol 230 no 3 pp 586ndash595 2011
[14] F Zeng C Li F Liu and I Turner ldquoNumerical algorithms fortime-fractional subdiffusion equation with second-order ac-curacyrdquo SIAM Journal on Scientific Computing vol 37 no 1pp A55ndashA78 2015
[15] G-H Gao Z-Z Sun and H-W Zhang ldquoA new fractionalnumerical differentiation formula to approximate the Caputofractional derivative and its applicationsrdquo Journal of Com-putational Physics vol 259 no 2 pp 33ndash50 2014
[16] H Li J Cao and C Li ldquoHigh-order approximation to Caputoderivatives and Caputo-type advection-diffusion equations(III)rdquo Journal of Computational and Applied Mathematicsvol 299 pp 159ndash175 2016
[17] A A Alikhanov ldquoA new difference scheme for the timefractional diffusion equationrdquo Journal of ComputationalPhysics vol 280 pp 424ndash438 2015
[18] Z Wang and S Vong ldquoCompact difference schemes for themodified anomalous fractional sub-diffusion equation and thefractional diffusion-wave equationrdquo Journal of ComputationalPhysics vol 277 pp 1ndash15 2014
[19] H-K Pang and H-W Sun ldquoFourth order finite differenceschemes for time-space fractional sub-diffusion equationsrdquoComputers amp Mathematics with Applications vol 71 no 6pp 1287ndash1302 2016
[20] M Dehghan M Abbaszadeh and W Deng ldquoFourth-ordernumerical method for the space-time tempered fractionaldiffusion-wave equationrdquo Applied Mathematics Lettersvol 73 no 11 pp 120ndash127 2017
[21] H Sun Z-Z Sun and G-H Gao ldquoSome high order dif-ference schemes for the space and time fractional Bloch-Torrey equationsrdquo Applied Mathematics and Computationvol 281 pp 356ndash380 2016
[22] S Arshad J Huang A Q M Khaliq and Y Tang ldquoTrape-zoidal scheme for time-space fractional diffusion equationwith Riesz derivativerdquo Journal of Computational Physicsvol 350 pp 1ndash15 2017
[23] H Wang K Wang and T Sircar ldquoA direct O(Nlog2N) finitedifferencemethod for fractional diffusion equationsrdquo Journal ofComputational Physics vol 229 no 21 pp 8095ndash8104 2010
[24] H Wang and T S Basu ldquoA fast finite difference method fortwo-dimensional space-fractional diffusion equationsrdquo SIAMJournal on Scientific Computing vol 34 no 5 pp A2444ndashA2458 2012
[25] H Wang and N Du ldquoA superfast-preconditioned iterativemethod for steady-state space-fractional diffusion equationsrdquoJournal of Computational Physics vol 240 pp 49ndash57 2013
[26] H Wang and N Du ldquoA fast finite difference method forthree-dimensional time-dependent space-fractional diffusionequations and its efficient implementationrdquo Journal ofComputational Physics vol 253 pp 50ndash63 2013
[27] H Wang and N Du ldquoFast alternating-direction finite dif-ference methods for three-dimensional space-fractional dif-fusion equationsrdquo Journal of Computational Physics vol 258pp 305ndash318 2014
[28] K Wang and H Wang ldquoA fast characteristic finite differencemethod for fractional advection-diffusion equationsrdquo Ad-vances in Water Resources vol 34 no 7 pp 810ndash816 2011
[29] H Wang and H Tian ldquoA fast Galerkin method with efficientmatrix assembly and storage for a peridynamic modelrdquoJournal of Computational Physics vol 231 no 23pp 7730ndash7738 2012
[30] H Tian H Wang and W Wang ldquoAn efficient collocationmethod for a non-local diffusion modelrdquo InternationalJournal of Numerical Analysis amp Modeling vol 10 no 4pp 815ndash825 2013
[31] H Wang and H Tian ldquoA fast and faithful collocation methodwith efficient matrix assembly for a two-dimensional nonlocaldiffusion modelrdquo Computer Methods in Applied Mechanicsand Engineering vol 273 no 5 pp 19ndash36 2014
[32] S Jiang J Zhang Q Zhang and Z Zhang ldquoFast evaluation ofthe Caputo fractional derivative and its applications tofractional diffusion equationsrdquo Communications in Compu-tational Physics vol 21 no 3 pp 650ndash678 2017
16 Mathematical Problems in Engineering
[33] Y Yan Z-Z Sun and J Zhang ldquoFast evaluation of the Caputofractional derivative and its applications to fractional diffusionequations a second-order schemerdquo Communications inComputational Physics vol 22 no 4 pp 1028ndash1048 2017
[34] X Lu H-K Pang and H-W Sun ldquoFast approximate in-version of a block triangular Toeplitz matrix with applicationsto fractional sub-diffusion equationsrdquo Numerical Linear Al-gebra with Applications vol 22 no 5 pp 866ndash882 2015
[35] M Caputo and M Fabrizio ldquoA new definition of fractionalderivative without singular kernelrdquo Progress in FractionalDifferentiation and Applications vol 1 no 2 pp 73ndash85 2015
[36] A Atangana ldquoOn the new fractional derivative and appli-cation to nonlinear Fisherrsquos reaction-diffusion equationrdquoAppliedMathematics and Computation vol 273 pp 948ndash9562016
[37] M D Ortigueira and J Tenreiro Machado ldquoA critical analysisof the Caputo-Fabrizio operatorrdquo Communications in Non-linear Science and Numerical Simulation vol 59 pp 608ndash6112018
[38] V E Tarasov ldquoNo nonlocality no fractional derivativerdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 62 pp 157ndash163 2018
[39] V Tarasov ldquoCaputo-Fabrizio operator in terms of integerderivatives memory or distributed lagrdquo Computational andApplied Mathematics vol 38 p 113 2019
[40] A Giusti ldquoA comment on some new definitions of fractionalderivativerdquoNonlinear Dynamics vol 93 no 3 pp 1757ndash17632018
[41] A Atangana and J F Gomez-Aguilar ldquoFractional derivativeswith no-index law property application to chaos and sta-tisticsrdquo Chaos Solitons amp Fractals vol 114 pp 516ndash535 2018
[42] J Hristov ldquoResponse functions in linear viscoelastic consti-tutive equations and related fractional operatorsrdquo Mathe-matical Modelling of Natural Phenomena vol 14 no 3pp 1ndash34 2019
[43] L X Vivas-Cruz A Gonzalez-Calderon M A Taneco-Hernandez and D P Luis ldquo)eoretical analysis of a model offluid flow in a reservoir with the Caputo-Fabrizio operatorrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 84 Article ID 105186 2020
[44] H-L Liao and Z-Z Sun ldquoMaximum norm error bounds ofADI and compact ADI methods for solving parabolicequationsrdquo Numerical Methods for Partial DifferentialEquations vol 26 no 1 pp 37ndash60 2010
[45] Z Liu A Cheng and X Li ldquoA second order Crank-Nicolsonscheme for fractional Cattaneo equation based on newfractional derivativerdquoAppliedMathematics and Computationvol 311 pp 361ndash374 2017
[46] Z Sun and G Gao Finite Difference Method for FractionalDifferential Equations Science Press Beijing China 2015
Mathematical Problems in Engineering 17
It is clear that
Axvij I +h2
x
12δ2x1113888 1113889vij
Ayvij I +h2
y
12δ2y⎛⎝ ⎞⎠vij
(44)
We also denote AxAy(u v) (AxAyu v) It is easy to seethat AxAy AyAx
For any gird function u v isin V0h the discrete inner
product and norms are defined as follows
(u v) hxhy 1113944
Mxminus 1
i11113944
My minus 1
j1uijvij
u22 (u u)
(45)
For the average operator AxAy define
AxAy(v v)≜ AxAyv v1113872 1113873 v2A (46)
41 Compact Finite Difference Scheme At the node(xi yj tk+(12)) the differential equation is rewritten as
zu
ztxi yj tk+(12)1113872 1113873 +
zαu
ztαxi yj tk+(12)1113872 1113873
z2u
zx2 xi yj tk+(12)1113872 1113873 +z2u
zy2 xi yj tk+(12)1113872 1113873
+ f xi yj tk+(12)1113872 1113873
1le ileMx minus 1 1le jleMy minus 1 1le kleN minus 1
(47)
For the approximation of the time-fractional derivativewe have the following approximation [45]
CF0 D
αt u xi yj tk+(12)1113872 1113873
1(α minus 1)Δt
M0δtuk+1ij minus 1113944
k
n1Mkminus n(⎡⎣
minus Mkminus n+11113857δtunij minus Mkψij
⎤⎦
+ Rk+(12)ij
(48)
where the truncation error Rk+(12)ij O(Δt2) and
Mn exp1 minus α2 minus αΔtn1113876 1113877 minus exp
1 minus α2 minus αΔt(n + 1)1113876 1113877 (49)
Furthermore we also have
z2u
zx2 xi yj tk+(12)1113872 1113873 12
δ2xuk+1ij + δ2xuk
ij1113872 1113873
Ax
+ O Δt2 + h4x1113872 1113873
(50)
z2u
zy2 xi yj tk+(12)1113872 1113873 12
δ2yuk+1ij + δ2yuk
ij1113872 1113873
Ay
+ O Δt2 + h4y1113872 1113873
(51)
zu
ztxi yj tk+(12)1113872 1113873 δtu
k+1ij + O Δt21113872 1113873 (52)
Substituting (48) and (50)sim(52) into (47) leads to
AxAyδtuk+1ij +
1(α minus 1)τ
AxAy M0δtuk+1ij1113890
minus 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857δtu
nij minus Mkψij
⎤⎦
12
δ2xuk+1ij + δ2xu
kij1113872 1113873 +
12
δ2yuk+1ij + δ2yu
kij1113872 1113873 + AxAyf
k+(12)ij
+ Rk+(12)ij
1le ileMx minus 1 1le jleMy minus 1 0le kleN minus 1
(53)
and there exists a constant C depending on the function uand its derivatives such that
Rk+(12)ij leC Δt2 + h
4x + h
4y1113872 1113873 (54)
By the initial and boundary conditions we have
u0ij ϕij 1le ileMx minus 1 1le jleMy minus 1
uk0j u
kMxj u
ki0 u
kiMy
0 0le kleN(55)
Omitting the truncation error Rk+(12)ij and replacing the
true solution ukij with numerical solution uk
ij a compactfinite difference scheme can be obtained as follows
(α minus 1)ΔtAxAyδtUk+1ij + M0AxAyδtU
k+1ij
minus Ay
(α minus 1)Δt2
δ2xUk+1ij minus Ax
(α minus 1)Δt2
δ2yUk+1ij
Ay
(α minus 1)Δt2
δ2xukij + Ax
(α minus 1)Δt2
δ2yukij
+ 1113944k
n1Mkminus n minus Mkminus n+1( 1113857AxAyδtu
nij + MkAxAyψij
+ (α minus 1)ΔtAxAyfk+(12)ij
1le ileMx minus 1 1le jleMy minus 1 1le kleN minus 1
u0ij ϕij 1le ileMx minus 1 1le jleMy minus 1
uk0j u
kMxj u
ki0 u
kiMy
0 0le kleN
(56)
Mathematical Problems in Engineering 7
42 Stability Analysis and Optimal Error Estimates
421 Stability Analysis
Definition 4 (see [46]) For any gird function u isin V0h define
the norm
⦀nablahu⦀2A δxu
2
minush2
y
12δyδxu
2
⎛⎝ ⎞⎠ + δyu
2
minush2
x
12δxδyu
2
1113888 1113889
(57)
)e lemmas below is useful in the subsequent analysis ofstability
Lemma 3 (see [46]) For any gird function u isin V0h the
following equation is held13u
2 le u2A le u
2 (58)
Lemma 4 (see [46]) For any gird function u isin V0h the
following equation is held23nablahu
2 le⦀nablahu⦀2A le nablahu
2
Ayδ2xu
nminus (12)+ Axδ
2yu
nminus (12) δtu
nminus (12)1113872 1113873
minus12Δt⦀nablahu
n⦀2A minus ⦀nablahunminus 1⦀
2A1113874 1113875
(59)
where nablahu2 δxu22 + δyu22
Multiplying hxhyδtUk+1ij on both sides of equation (56)
and summing up wrt i j from 1 to (Mx minus 1) and from 1 to(My minus 1) respectively the following equation is obtained
(α minus 1)Δt δtUk+1
2
A+ M0 δtU
k+1
2
A
(α minus 1)Δt Ayδ2xU
k+(12)+ Axδ
2yU
k+(12) δtU
k+11113872 1113873 + 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 AyAxδtU
n δtU
k+11113872 1113873
+ Mk AyAxψ δtUk+1
1113872 1113873 +(α minus 1)Δt AyAxfk+(12)
δtUk+1
1113872 1113873
(60)
Observing Lemma 4 we have
Ayδ2xUk+(12) + Axδ
2yUk+(12) δtU
k+11113872 1113873 minus12Δt⦀nablahu
k+1⦀2A minus ⦀nablahu
k⦀2A1113874 1113875 (61)
(α minus 1)Δt AyAxfk+(12) δtUk+11113872 1113873le
(α minus 1)Δt2
fk+(12)
2
A+ δtu
k+1
2
A1113874 1113875 (62)
By the triangle inequality and Lemma 2 we obtain
1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 AyAxδtU
n δtU
k+11113872 1113873 + Mk AyAxψ δtU
k+11113872 1113873
le 1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δtUn
2A
+ δtUk+1
2
A1113874 1113875 +
12Mk ψ
2A + δtU
k+1
2
A1113874 1113875
1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δtUn
2A
+12
M0 minus Mk( 1113857 δtUk+1
2
A+12Mk ψ
2A + δtU
k+1
2
A1113874 1113875
1113944k
n1
12Mkminus n δtU
n
2A
minus 1113944k
n1
12Mkminus n+1 δtU
n
2A
+12M0 δtU
k+1
2
A+12Mkψ
2A
(63)
Combining equation (60) with (61)sim(63)we get
1113944
k+1
n1Mkminus n+1 δtU
n
2A
+(α minus 1)⦀nablahUk+1⦀
2A
le 1113944k
n1Mkminus n δtU
n
2A
+(α minus 1) nablahUk
2
A+ Mkψ
2A +(α minus 1)Δt⦀fk+(12)⦀
2A
(64)
8 Mathematical Problems in Engineering
Let
Q Uk
1113872 1113873 1113944k
n1Mkminus n δtU
n
2A
+(α minus 1)⦀nablahUk⦀
2A (65)
Summing up with respect to k from 0 to N minus 1 we get
Q UN
1113872 1113873leQ U0
1113872 1113873 + 1113944Nminus 1
k0Mkψ
2A +(α minus 1)Δt 1113944
Nminus 1
k0f
k+(12)
2
A
(66)
Noting that U0 ϕ we have Q(U0) (α minus 1)nablahϕ2A Itfollows that
Q UN
1113872 1113873le (α minus 1)⦀nablahϕ⦀2A + 1113944
Nminus 1
k0Mkψ
2A
+(α minus 1)Δt 1113944Nminus 1
k0f
k+(12)
2
A
(67)
Theorem 3 For the compact finite difference scheme (56)the following stability inequality holds
Q Um
( 1113857le (α minus 1) nablahϕ1113868111386811138681113868
1113868111386811138681113868
2A
+ 1113944
mminus 1
k0Mkψ
2A
+(α minus 1)Δt 1113944
mminus 1
k0f
k+(12)
2
A forall0lemleN
(68)
Similar to the stability the convergence can also beanalyzed
Theorem 4 Suppose that the exact solution of the fractionalCattaneo equation is sufficiently smooth then there exists apositive constant C independent of h k and Δt such that
ek
11138681113868111386811138681113868
111386811138681113868111386811138681leC Δt2 + h
41113872 1113873 forall0le kleN (69)
where ekij uk
ij minus ukij and h max hx hy1113966 1113967
5 Efficient Storage and Fast Evaluation of theCaputondashFabrizio Fractional Derivative
Since time-fractional derivative operator is nonlocal thetraditional direct method for numerically solving the frac-tional Cattaneo equations generally requires total O(MN)
memory units and O(MN2) computational complexitywhere N andM are the total number of time steps and spacesteps respectively
In this section we develop a fast solution method for thefinite difference scheme of the time-fractional Cattaneoequation
Let
Nkn Mkminus n minus Mkminus n+1
exp1 minus α2 minus α
(k minus n)Δt1113874 1113875
middot 1 minus 2exp1 minus α2 minus αΔt1113874 1113875 + exp
1 minus α2 minus α
2Δt1113874 11138751113876 1113877
1le nle k
(70)
then
Nk+1n Mk+1minus n minus Mk+1minus n+1
exp1 minus α2 minus α
(k minus n)Δt1113874 1113875
middot 1 minus 2exp1 minus α2 minus αΔt1113874 1113875 + exp
1 minus α2 minus α
2Δt1113874 11138751113876 1113877exp1 minus α2 minus αΔt1113874 1113875
exp1 minus α2 minus αΔt1113874 1113875N
kn
Nk+1k+1 M0 minus M1 1 minus 2exp
1 minus α2 minus αΔt1113874 1113875 + exp
1 minus α2 minus α
2Δt1113874 1113875
(71)
So we have
1113944
k+1
n1N
k+1n Aδtu
ni exp
1 minus α2 minus αΔt1113874 1113875 1113944
k
n1N
knAδtu
ni + N
k+1k+1Aδtu
k+1i
1113944
k+1
n1N
k+1n AyAxδtu
nij exp
1 minus α2 minus αΔt1113874 1113875 1113944
k
n1N
knAyAxδtu
nij
+ Nk+1k+1AyAxδtU
k+1ij
(72)
Remark We find that at the k-th level only O(1) operationsare needed to compute the k-th level since the (k minus 1)-thlevel is known at that point )us the total operations arereduced from O(N2) to O(N) and the memory require-ment decreases from O(N) to O(1) We conclude that thatthis fast method significantly reduces the total computa-tional cost from O(MN2) to O(MN) and the memoryrequirement from O(MN) to O(M)
6 Numerical Experiments
In this section we carry out several numerical experiments tocheck the effectiveness of the proposed scheme )e conver-gence rate and CPU consumption are all compared in thesimulations We take the space-time domainΩ [0 1] T 1for one-dimensional case and Ω [0 1] times [0 1] T 1 fortwo-dimensional case )ese simulations are implemented inMatlab and the numerical experiments are run on a computerwith 4GB memory )e time-fractional Cattaneo equation ofthe following forms is considered
Example 1 We provide the exact solution u(x t)
etsin(πx) and for different α we have different f(x t)
accordingly
Mathematical Problems in Engineering 9
Table 1 Considering Δt h2 the discrete linfin error and convergence rates of u with different α for Example 1
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 15320e minus 04 mdash 15941e minus 04 mdash 18076e minus 04 mdash2minus 4 95109e minus 06 40097 98981e minus 06 40094 11230e minus 05 400862minus 5 59343e minus 07 40024 61762e minus 07 40024 70079e minus 07 400222minus 6 37074e minus 08 40006 38585e minus 08 40006 43783e minus 08 40005
Table 2 Considering h Δt
radic the discrete l2 error and convergence rates of u with different α for Example 1
Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
2minus 4 00017 mdash 00018 mdash 00021 mdash2minus 6 10832e minus 04 40382 11271e minus 04 40373 12781e minus 04 403442minus 8 67252e minus 06 40097 69990e minus 06 40094 79405e minus 06 400862minus 10 41961e minus 07 40024 43672e minus 07 40023 49553e minus 07 400212minus 12 26215e minus 08 40006 27284e minus 08 40005 30959e minus 08 40005
Table 3 )e CPU time consumption of the fast compact difference scheme and direct difference scheme for Example 1
Δtα 125 α 15 α 175
DCD FCD DCD FCD DCD FCD11000 34008 18876 34632 19032 33540 1918812500 142740 46800 141960 47580 138840 4742415000 470967 93600 463010 94692 461450 9375617500 994194 139464 956754 140088 967206 138684110 000 1699318 186733 1641130 186421 1658290 184393125 000 9816362 466910 9509820 469251 9591409 460046150 000 38269853 934133 36923096 940062 37547412 919625
N times104
0
500
1000
1500
2000
2500
3000
3500
4000
CPU
tim
e
Fast evaluationDirect evaluation
0 05 1 15 2 25 3 35 4 45 5
(a)
Fast evaluationDirect evaluation
log10 (N)
ndash05
0
05
1
15
2
25
3
35
4
log1
0 (C
PU ti
me)
25 3 35 4 45 5
(b)
Figure 1 )e CPU time (a) and the log-log CPU time (b) versus the total number of time steps N for Example 1
10 Mathematical Problems in Engineering
Table 4 Considering Δt 2minus 13 the discrete linfin error and convergence rates of u with different α for Example 1
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 22486e minus 04 mdash 22733e minus 04 mdash 23705e minus 04 mdash2minus 4 13985e minus 05 40071 14140e minus 05 40069 14745e minus 05 400692minus 5 86897e minus 07 40084 87882e minus 07 40081 91726e minus 07 400682minus 6 50194e minus 08 41137 51024e minus 08 41063 54092e minus 08 40838
Table 5 Considering h 0001 the discrete linfin error and convergence rates of u with different α for Example 1
Δtα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 00046 mdash 00043 mdash 00036 mdash2minus 4 00011 20641 00011 19668 90063e minus 04 199902minus 5 28662e minus 04 19403 27171e minus 04 20174 22518e minus 04 199992minus 6 71658e minus 05 19999 67928e minus 05 20000 56295e minus 05 200002minus 7 17915e minus 05 20000 16982e minus 05 20000 14074e minus 05 20000
x
ndash02
0
02
04
06
08
1
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(a)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(b)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(c)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(d)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(e)
ndash02
0
02
04
06
08
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(f)
Figure 2 Considering c 0001 α 15 M N 100 real and numerical solutions of u at T 1 for Example 2 with the value of x0 equalto (a) 0 (b) 02 (c) 04 (d) 06 (e) 08 and (f) 1
Mathematical Problems in Engineering 11
zu(x t)
zt+
zαu(x t)
ztα
z2u(x t)
zx2 + f(x t)
u(x 0) sin(πx)zu
zt
1113868111386811138681113868111386811138681113868 t0 sin(πx)
u(0 t) u(1 t) 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(73)
In Tables 1 and 2 we take Δt h2 and h Δt
radicto ex-
amine the discrete linfin-norm (l2-norm) errors and
corresponding spatial and temporal convergence rates re-spectively We list the errors and convergence rates (order) ofthe proposed compact finite difference (CD) scheme which isalmost O(Δt2 + h4) for different α Additionally Table 3shows the CPU time (CPU) consumed by direct compact(DCD) scheme and fast compact difference (FCD) schemerespectively It is obvious that the FCD scheme has a sig-nificantly reduced CPU time over the DCD scheme Forinstance when α 15 we choose h 01 and Δt 150 000and observe that the FCD scheme consumes only 94 secondswhile the DCD scheme consumes 3692 seconds We can find
x
075
08
085
09
095
1
105
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(a)
0
01
02
03
04
05
06
07
08
09
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(b)
0
01
02
03
04
05
06
07
08
09
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(c)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(d)
Figure 3 Considering x0 05 α 15 M N 100 real and numerical solutions of u at T 1 for Example 2 with the value of c equal to(a) 1 (b) 01 (c) 001 and (d) 0001
12 Mathematical Problems in Engineering
that the performance of the FCD scheme will be moreconspicuous as the time step size Δt decreases
In Figure 1 we set h 01 and α 175 and change thetotal number of time steps N to plot out the CPU time (inseconds) of the FCD scheme and DCD scheme We canobserve that the CPU time increases almost linearly withrespect to N for the FCD scheme while the DCD schemescales like O(N2)
Tables 4 and 5 show the discrete linfin errors and con-vergence rates of the compact finite difference scheme forExample 1 )e space rates are almost O(h4) for fixedΔt 2minus 13 and the time convergence rates are always O(Δt2)for fixed h 0001 We can conclude that the numericalconvergence rates of our scheme approach almost toO(Δt2 + h4)
Example 2 )e example is described by
zu(x t)
zt+
zαu(x t)
ztα
z2u(x t)
zx2 + f(x t)
u(x 0) 0zu
zt
1113868111386811138681113868111386811138681113868t0 0
u(0 t) t2exp minusx20
c1113888 1113889 u(1 t) t2exp minus
1 minus x0( 11138572
c1113888 1113889
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(74)
Note that the exact solution of the above problem isu(x t) t2exp(minus (x minus x0)
2c) where x0 isin [0 1] and c is a
N times104
0
1000
2000
3000
4000
5000
6000
7000
8000CP
U ti
me
Fast evaluationDirect evaluation
0 05 1 15 2 25 3 35 4 45 5
(a)
Fast evaluationDirect evaluation
log10 (N)
0
05
1
15
2
25
3
35
4
log1
0 (C
PU ti
me)
25 3 35 4 45 5
(b)
Figure 4 Considering x0 05 c 001 α 15 andM 10 the CPU time (a) and the log-log CPU time (b) versus the total number oftime steps N for Example 2
Table 6 Considering Δt h2 andx0 05 the discrete linfin error and convergence rates of u with different α and c for Example 2
c hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
1
2minus 3 73224e minus 05 mdash 55643e minus 05 mdash 38478e minus 05 mdash2minus 4 45707e-06 40018 34695e minus 06 40034 23900e minus 06 400902minus 5 28558e minus 07 40004 21671e minus 07 40009 14913e minus 07 400242minus 6 17847e minus 08 40001 13543e minus 08 40001 93169e minus 09 40006
01
2minus 3 00012 mdash 00012 mdash 00012 mdash2minus 4 70462e minus 05 40900 69731e minus 05 41051 69007e minus 05 412012minus 5 43571e minus 06 40154 43110e minus 06 40157 42650e minus 06 401612minus 6 27160e minus 07 40038 26871e minus 07 40039 26583e minus 07 40040
001
2minus 4 00090 mdash 00090 mdash 00090 mdash2minus 5 49257e minus 04 41915 49235e minus 04 41922 49232e minus 04 419232minus 6 29868e minus 05 40437 29855e minus 05 40436 29854e minus 05 404362minus 7 18531e minus 06 40106 18523e minus 06 40106 18522e minus 06 40106
Mathematical Problems in Engineering 13
Table 7 Considering h Δt
radicand c 001 the discrete l2 error and convergence rates of u with different α andx0 for Example 2
x0 Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
02minus 8 00044 mdash 00044 mdash 00042 mdash2minus 10 24205e minus 04 41841 23721e minus 04 42133 23098e minus 04 418452minus 12 14684e minus 05 40430 14390e minus 05 40430 14013e minus 05 40429
0252minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14902e minus 04 41794 14894e minus 04 41802 14892e minus 04 418032minus 12 90680e minus 06 40386 90633e minus 06 40386 90616e minus 06 40386
052minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14848e minus 04 41846 14842e minus 04 41852 14842e minus 04 418522minus 12 90323e minus 06 40390 90288e minus 06 40390 90286e minus 06 40390
0752minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14902e minus 04 41794 14894e minus 04 41802 14892e minus 04 418032minus 12 90680e minus 06 40386 90633e minus 06 40386 90616e minus 06 40386
12minus 8 00044 mdash 00044 mdash 00042 mdash2minus 10 24205e minus 04 41841 23721e minus 04 42133 23098e minus 04 418452minus 12 14684e minus 05 40430 14390e minus 05 40430 14013e minus 05 40429
0 2 4 6 8 10times104
0
2000
4000
6000
8000
10000
12000
14000
N
CPU
tim
e
Fast evaluationDirect evaluation
(a)
Fast evaluationDirect evaluation
36 38 4 42 44 46 48 515
2
25
3
35
4
45
log10 (N)
log1
0 (C
PU ti
me)
(b)
Figure 5 )e CPU time (a) and the log-log CPU time (b) versus the total number of time steps N for Example 3
Table 8 Considering Δt h2 the discrete linfin error and convergence rates of u with different α for Example 3
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 2 00028 mdash 00028 mdash 00030 mdash2minus 3 16948e minus 04 40462 17328e minus 04 40142 18411e minus 04 402632minus 4 10522e minus 05 40096 10759e minus 05 40095 11434e minus 05 400922minus 5 65652e minus 07 40024 67133e minus 07 40024 71346e minus 07 40024
14 Mathematical Problems in Engineering
constant and for different α we have different f(x t)
accordinglyWe apply the fast compact difference scheme to dis-
cretize the equation In Figure 2 we setc 0001 α 15 andM N 100 and plot exact andnumerical solutions at time T 1 for Example 2 with dif-ferent x0 For x0 05 α 15 andM N 100 we alsoplot exact and numerical solutions with the different c inFigure 3 In Figure 4 for h 01 and α 15 we vary thetotal number of time steps N to plot out the CPU time (inseconds) of the FCD scheme and DCD scheme )e nu-merical experiments verified our theoretical results In Ta-ble 6 by equating Δt h2 and fixing x0 05 we compute
the discrete linfin error and convergence rates with differentfractional derivative orders α and different c It shows thatthe compact finite difference scheme has space accuracy offourth order and temporal accuracy of second order We seth
Δt
radicand fix c 001 and the discrete l2 error and
convergence rates with different α and x0 are displayed inTable 7 )ese numerical convergence rates are almostapproaching O(Δt2 + h4) as Example 1
Example 3 If the exact solution is given byu(x t) etsin(πx)sin(πy) we have different f(x y t) fordifferent α accordingly
zu(x y t)
zt+
zαu(x y t)
ztα
z2u(x y t)
zx2 +z2u(x y t)
zy2 + f(x y t)
u(x 0) sin(πx)sin(πy)
zu
zt | t0 sin(πx)sin(πy)
u(0 y t) u(1 y t) u(x 0 t) u(x 1 t) 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(75)
In Figure 5 h 01 and α 175 are fixed and the totalnumber of time steps N vary to plot out the CPU time (inseconds) of the FCD procedure and DCD procedure and itpresents an approximately linear computation complexityfor FCD procedure We set Δt h2 in Table 8 and h
Δt
radic
in Table 9 the discrete linfin error discrete l2 error andconvergence rates with different derivative orders α arepresented )e fourth-order space accuracy and second-order temporal accuracy can be observed clearly
7 Conclusion
In this paper we develop and analyze a fast compact finitedifference procedure for the Cattaneo equation equipped withtime-fractional derivative without singular kernel )e time-fractional derivative is of CaputondashFabrizio type with the orderof α(1lt αlt 2) Compact difference discretization is appliedto obtain a high-order approximation for spatial derivatives ofinteger order in the partial differential equation and theCaputondashFabrizio fractional derivative is discretized by means
of CrankndashNicolson approximation It has been proved thatthe proposed compact finite difference scheme has spatialaccuracy of fourth order and temporal accuracy of secondorder Since the fractional derivatives are history dependentand nonlocal huge memory for storage and computationalcost are required )is means extreme difficulty especially fora long-time simulation Enlightened by the treatment forCaputo fractional derivative [32] we develop an effective fastevaluation procedure for the new CaputondashFabrizio fractionalderivative for the compact finite difference scheme Severalnumerical experiments have been carried out to show theconvergence orders and applicability of the scheme
Inspired by the work [43] the topic about modelling andnumerical solutions of porous media flow equipped withfractional derivatives is very interesting and challenging andwill be our main research direction in the future
Data Availability
All data generated or analyzed during this study are includedin this article
Table 9 Considering h Δt
radic the discrete l2 error and convergence rates of u with different α for Example 3
Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
2minus 4 00014 mdash 00014 mdash 00015 mdash2minus 6 84738e minus 05 40463 86640e minus 05 40142 92056e minus 05 402632minus 8 52610e minus 06 40096 53795e minus 06 40095 57168e minus 06 400922minus 10 32826e minus 07 40024 33566e minus 07 40024 35673e minus 07 40023
Mathematical Problems in Engineering 15
Conflicts of Interest
)e authors declare that they have no conflicts of interest
Acknowledgments
)is work was supported in part by the National NaturalScience Foundation of China under Grant nos 91630207and 11971272
References
[1] A Chaves ldquoA fractional diffusion equation to describe Levyflightsrdquo Physics Letters A vol 239 no 1-2 pp 13ndash16 1998
[2] M Giona and H E Roman ldquoFractional diffusion equation fortransport phenomena in random mediardquo Physica A Statis-tical Mechanics and its Applications vol 185 no 1ndash4pp 87ndash97 1992
[3] R Metzler and J Klafter ldquo)e random walkrsquos guide toanomalous diffusion a fractional dynamics approachrdquoPhysics Reports vol 339 no 1 pp 1ndash77 2000
[4] R L Magin Fractional Calculus in Bioengineering BegellHouse Redding Redding CT USA 2006
[5] R L Magin O Abdullah D Baleanu and X J ZhouldquoAnomalous diffusion expressed through fractional orderdifferential operators in the Bloch-Torrey equationrdquo Journalof Magnetic Resonance vol 190 no 2 pp 255ndash270 2008
[6] I Podlubny Fractional Differential Equations An Introduc-tion to Fractional Derivatives Fractional Differential Equa-tions to Methods of Beir Solution and Some of BeirApplications Vol 198 Elsevier Amsterdam Netherlands1998
[7] M Ran and C Zhang ldquoCompact difference scheme for a classof fractional-in-space nonlinear damped wave equations intwo space dimensionsrdquo Computers amp Mathematics withApplications vol 71 no 5 pp 1151ndash1162 2016
[8] A H Bhrawy and M A Zaky ldquoHighly accurate numericalschemes for multi-dimensional space variable-order frac-tional Schrodinger equationsrdquo Computers amp Mathematicswith Applications vol 73 no 6 pp 1100ndash1117 2017
[9] S B Yuste ldquoWeighted average finite difference methods forfractional diffusion equationsrdquo Journal of ComputationalPhysics vol 216 no 1 pp 264ndash274 2006
[10] T A M Langlands and B I Henry ldquo)e accuracy andstability of an implicit solution method for the fractionaldiffusion equationrdquo Journal of Computational Physicsvol 205 no 2 pp 719ndash736 2005
[11] C-M Chen F Liu I Turner and V Anh ldquoA Fourier method forthe fractional diffusion equation describing sub-diffusionrdquo Journalof Computational Physics vol 227 no 2 pp 886ndash897 2007
[12] Y Lin and C Xu ldquoFinite differencespectral approximationsfor the time-fractional diffusion equationrdquo Journal of Com-putational Physics vol 225 no 2 pp 1533ndash1552 2007
[13] G-h Gao and Z-z Sun ldquoA compact finite difference schemefor the fractional sub-diffusion equationsrdquo Journal of Com-putational Physics vol 230 no 3 pp 586ndash595 2011
[14] F Zeng C Li F Liu and I Turner ldquoNumerical algorithms fortime-fractional subdiffusion equation with second-order ac-curacyrdquo SIAM Journal on Scientific Computing vol 37 no 1pp A55ndashA78 2015
[15] G-H Gao Z-Z Sun and H-W Zhang ldquoA new fractionalnumerical differentiation formula to approximate the Caputofractional derivative and its applicationsrdquo Journal of Com-putational Physics vol 259 no 2 pp 33ndash50 2014
[16] H Li J Cao and C Li ldquoHigh-order approximation to Caputoderivatives and Caputo-type advection-diffusion equations(III)rdquo Journal of Computational and Applied Mathematicsvol 299 pp 159ndash175 2016
[17] A A Alikhanov ldquoA new difference scheme for the timefractional diffusion equationrdquo Journal of ComputationalPhysics vol 280 pp 424ndash438 2015
[18] Z Wang and S Vong ldquoCompact difference schemes for themodified anomalous fractional sub-diffusion equation and thefractional diffusion-wave equationrdquo Journal of ComputationalPhysics vol 277 pp 1ndash15 2014
[19] H-K Pang and H-W Sun ldquoFourth order finite differenceschemes for time-space fractional sub-diffusion equationsrdquoComputers amp Mathematics with Applications vol 71 no 6pp 1287ndash1302 2016
[20] M Dehghan M Abbaszadeh and W Deng ldquoFourth-ordernumerical method for the space-time tempered fractionaldiffusion-wave equationrdquo Applied Mathematics Lettersvol 73 no 11 pp 120ndash127 2017
[21] H Sun Z-Z Sun and G-H Gao ldquoSome high order dif-ference schemes for the space and time fractional Bloch-Torrey equationsrdquo Applied Mathematics and Computationvol 281 pp 356ndash380 2016
[22] S Arshad J Huang A Q M Khaliq and Y Tang ldquoTrape-zoidal scheme for time-space fractional diffusion equationwith Riesz derivativerdquo Journal of Computational Physicsvol 350 pp 1ndash15 2017
[23] H Wang K Wang and T Sircar ldquoA direct O(Nlog2N) finitedifferencemethod for fractional diffusion equationsrdquo Journal ofComputational Physics vol 229 no 21 pp 8095ndash8104 2010
[24] H Wang and T S Basu ldquoA fast finite difference method fortwo-dimensional space-fractional diffusion equationsrdquo SIAMJournal on Scientific Computing vol 34 no 5 pp A2444ndashA2458 2012
[25] H Wang and N Du ldquoA superfast-preconditioned iterativemethod for steady-state space-fractional diffusion equationsrdquoJournal of Computational Physics vol 240 pp 49ndash57 2013
[26] H Wang and N Du ldquoA fast finite difference method forthree-dimensional time-dependent space-fractional diffusionequations and its efficient implementationrdquo Journal ofComputational Physics vol 253 pp 50ndash63 2013
[27] H Wang and N Du ldquoFast alternating-direction finite dif-ference methods for three-dimensional space-fractional dif-fusion equationsrdquo Journal of Computational Physics vol 258pp 305ndash318 2014
[28] K Wang and H Wang ldquoA fast characteristic finite differencemethod for fractional advection-diffusion equationsrdquo Ad-vances in Water Resources vol 34 no 7 pp 810ndash816 2011
[29] H Wang and H Tian ldquoA fast Galerkin method with efficientmatrix assembly and storage for a peridynamic modelrdquoJournal of Computational Physics vol 231 no 23pp 7730ndash7738 2012
[30] H Tian H Wang and W Wang ldquoAn efficient collocationmethod for a non-local diffusion modelrdquo InternationalJournal of Numerical Analysis amp Modeling vol 10 no 4pp 815ndash825 2013
[31] H Wang and H Tian ldquoA fast and faithful collocation methodwith efficient matrix assembly for a two-dimensional nonlocaldiffusion modelrdquo Computer Methods in Applied Mechanicsand Engineering vol 273 no 5 pp 19ndash36 2014
[32] S Jiang J Zhang Q Zhang and Z Zhang ldquoFast evaluation ofthe Caputo fractional derivative and its applications tofractional diffusion equationsrdquo Communications in Compu-tational Physics vol 21 no 3 pp 650ndash678 2017
16 Mathematical Problems in Engineering
[33] Y Yan Z-Z Sun and J Zhang ldquoFast evaluation of the Caputofractional derivative and its applications to fractional diffusionequations a second-order schemerdquo Communications inComputational Physics vol 22 no 4 pp 1028ndash1048 2017
[34] X Lu H-K Pang and H-W Sun ldquoFast approximate in-version of a block triangular Toeplitz matrix with applicationsto fractional sub-diffusion equationsrdquo Numerical Linear Al-gebra with Applications vol 22 no 5 pp 866ndash882 2015
[35] M Caputo and M Fabrizio ldquoA new definition of fractionalderivative without singular kernelrdquo Progress in FractionalDifferentiation and Applications vol 1 no 2 pp 73ndash85 2015
[36] A Atangana ldquoOn the new fractional derivative and appli-cation to nonlinear Fisherrsquos reaction-diffusion equationrdquoAppliedMathematics and Computation vol 273 pp 948ndash9562016
[37] M D Ortigueira and J Tenreiro Machado ldquoA critical analysisof the Caputo-Fabrizio operatorrdquo Communications in Non-linear Science and Numerical Simulation vol 59 pp 608ndash6112018
[38] V E Tarasov ldquoNo nonlocality no fractional derivativerdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 62 pp 157ndash163 2018
[39] V Tarasov ldquoCaputo-Fabrizio operator in terms of integerderivatives memory or distributed lagrdquo Computational andApplied Mathematics vol 38 p 113 2019
[40] A Giusti ldquoA comment on some new definitions of fractionalderivativerdquoNonlinear Dynamics vol 93 no 3 pp 1757ndash17632018
[41] A Atangana and J F Gomez-Aguilar ldquoFractional derivativeswith no-index law property application to chaos and sta-tisticsrdquo Chaos Solitons amp Fractals vol 114 pp 516ndash535 2018
[42] J Hristov ldquoResponse functions in linear viscoelastic consti-tutive equations and related fractional operatorsrdquo Mathe-matical Modelling of Natural Phenomena vol 14 no 3pp 1ndash34 2019
[43] L X Vivas-Cruz A Gonzalez-Calderon M A Taneco-Hernandez and D P Luis ldquo)eoretical analysis of a model offluid flow in a reservoir with the Caputo-Fabrizio operatorrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 84 Article ID 105186 2020
[44] H-L Liao and Z-Z Sun ldquoMaximum norm error bounds ofADI and compact ADI methods for solving parabolicequationsrdquo Numerical Methods for Partial DifferentialEquations vol 26 no 1 pp 37ndash60 2010
[45] Z Liu A Cheng and X Li ldquoA second order Crank-Nicolsonscheme for fractional Cattaneo equation based on newfractional derivativerdquoAppliedMathematics and Computationvol 311 pp 361ndash374 2017
[46] Z Sun and G Gao Finite Difference Method for FractionalDifferential Equations Science Press Beijing China 2015
Mathematical Problems in Engineering 17
42 Stability Analysis and Optimal Error Estimates
421 Stability Analysis
Definition 4 (see [46]) For any gird function u isin V0h define
the norm
⦀nablahu⦀2A δxu
2
minush2
y
12δyδxu
2
⎛⎝ ⎞⎠ + δyu
2
minush2
x
12δxδyu
2
1113888 1113889
(57)
)e lemmas below is useful in the subsequent analysis ofstability
Lemma 3 (see [46]) For any gird function u isin V0h the
following equation is held13u
2 le u2A le u
2 (58)
Lemma 4 (see [46]) For any gird function u isin V0h the
following equation is held23nablahu
2 le⦀nablahu⦀2A le nablahu
2
Ayδ2xu
nminus (12)+ Axδ
2yu
nminus (12) δtu
nminus (12)1113872 1113873
minus12Δt⦀nablahu
n⦀2A minus ⦀nablahunminus 1⦀
2A1113874 1113875
(59)
where nablahu2 δxu22 + δyu22
Multiplying hxhyδtUk+1ij on both sides of equation (56)
and summing up wrt i j from 1 to (Mx minus 1) and from 1 to(My minus 1) respectively the following equation is obtained
(α minus 1)Δt δtUk+1
2
A+ M0 δtU
k+1
2
A
(α minus 1)Δt Ayδ2xU
k+(12)+ Axδ
2yU
k+(12) δtU
k+11113872 1113873 + 1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 AyAxδtU
n δtU
k+11113872 1113873
+ Mk AyAxψ δtUk+1
1113872 1113873 +(α minus 1)Δt AyAxfk+(12)
δtUk+1
1113872 1113873
(60)
Observing Lemma 4 we have
Ayδ2xUk+(12) + Axδ
2yUk+(12) δtU
k+11113872 1113873 minus12Δt⦀nablahu
k+1⦀2A minus ⦀nablahu
k⦀2A1113874 1113875 (61)
(α minus 1)Δt AyAxfk+(12) δtUk+11113872 1113873le
(α minus 1)Δt2
fk+(12)
2
A+ δtu
k+1
2
A1113874 1113875 (62)
By the triangle inequality and Lemma 2 we obtain
1113944
k
n1Mkminus n minus Mkminus n+1( 1113857 AyAxδtU
n δtU
k+11113872 1113873 + Mk AyAxψ δtU
k+11113872 1113873
le 1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δtUn
2A
+ δtUk+1
2
A1113874 1113875 +
12Mk ψ
2A + δtU
k+1
2
A1113874 1113875
1113944k
n1
12
Mkminus n minus Mkminus n+1( 1113857 δtUn
2A
+12
M0 minus Mk( 1113857 δtUk+1
2
A+12Mk ψ
2A + δtU
k+1
2
A1113874 1113875
1113944k
n1
12Mkminus n δtU
n
2A
minus 1113944k
n1
12Mkminus n+1 δtU
n
2A
+12M0 δtU
k+1
2
A+12Mkψ
2A
(63)
Combining equation (60) with (61)sim(63)we get
1113944
k+1
n1Mkminus n+1 δtU
n
2A
+(α minus 1)⦀nablahUk+1⦀
2A
le 1113944k
n1Mkminus n δtU
n
2A
+(α minus 1) nablahUk
2
A+ Mkψ
2A +(α minus 1)Δt⦀fk+(12)⦀
2A
(64)
8 Mathematical Problems in Engineering
Let
Q Uk
1113872 1113873 1113944k
n1Mkminus n δtU
n
2A
+(α minus 1)⦀nablahUk⦀
2A (65)
Summing up with respect to k from 0 to N minus 1 we get
Q UN
1113872 1113873leQ U0
1113872 1113873 + 1113944Nminus 1
k0Mkψ
2A +(α minus 1)Δt 1113944
Nminus 1
k0f
k+(12)
2
A
(66)
Noting that U0 ϕ we have Q(U0) (α minus 1)nablahϕ2A Itfollows that
Q UN
1113872 1113873le (α minus 1)⦀nablahϕ⦀2A + 1113944
Nminus 1
k0Mkψ
2A
+(α minus 1)Δt 1113944Nminus 1
k0f
k+(12)
2
A
(67)
Theorem 3 For the compact finite difference scheme (56)the following stability inequality holds
Q Um
( 1113857le (α minus 1) nablahϕ1113868111386811138681113868
1113868111386811138681113868
2A
+ 1113944
mminus 1
k0Mkψ
2A
+(α minus 1)Δt 1113944
mminus 1
k0f
k+(12)
2
A forall0lemleN
(68)
Similar to the stability the convergence can also beanalyzed
Theorem 4 Suppose that the exact solution of the fractionalCattaneo equation is sufficiently smooth then there exists apositive constant C independent of h k and Δt such that
ek
11138681113868111386811138681113868
111386811138681113868111386811138681leC Δt2 + h
41113872 1113873 forall0le kleN (69)
where ekij uk
ij minus ukij and h max hx hy1113966 1113967
5 Efficient Storage and Fast Evaluation of theCaputondashFabrizio Fractional Derivative
Since time-fractional derivative operator is nonlocal thetraditional direct method for numerically solving the frac-tional Cattaneo equations generally requires total O(MN)
memory units and O(MN2) computational complexitywhere N andM are the total number of time steps and spacesteps respectively
In this section we develop a fast solution method for thefinite difference scheme of the time-fractional Cattaneoequation
Let
Nkn Mkminus n minus Mkminus n+1
exp1 minus α2 minus α
(k minus n)Δt1113874 1113875
middot 1 minus 2exp1 minus α2 minus αΔt1113874 1113875 + exp
1 minus α2 minus α
2Δt1113874 11138751113876 1113877
1le nle k
(70)
then
Nk+1n Mk+1minus n minus Mk+1minus n+1
exp1 minus α2 minus α
(k minus n)Δt1113874 1113875
middot 1 minus 2exp1 minus α2 minus αΔt1113874 1113875 + exp
1 minus α2 minus α
2Δt1113874 11138751113876 1113877exp1 minus α2 minus αΔt1113874 1113875
exp1 minus α2 minus αΔt1113874 1113875N
kn
Nk+1k+1 M0 minus M1 1 minus 2exp
1 minus α2 minus αΔt1113874 1113875 + exp
1 minus α2 minus α
2Δt1113874 1113875
(71)
So we have
1113944
k+1
n1N
k+1n Aδtu
ni exp
1 minus α2 minus αΔt1113874 1113875 1113944
k
n1N
knAδtu
ni + N
k+1k+1Aδtu
k+1i
1113944
k+1
n1N
k+1n AyAxδtu
nij exp
1 minus α2 minus αΔt1113874 1113875 1113944
k
n1N
knAyAxδtu
nij
+ Nk+1k+1AyAxδtU
k+1ij
(72)
Remark We find that at the k-th level only O(1) operationsare needed to compute the k-th level since the (k minus 1)-thlevel is known at that point )us the total operations arereduced from O(N2) to O(N) and the memory require-ment decreases from O(N) to O(1) We conclude that thatthis fast method significantly reduces the total computa-tional cost from O(MN2) to O(MN) and the memoryrequirement from O(MN) to O(M)
6 Numerical Experiments
In this section we carry out several numerical experiments tocheck the effectiveness of the proposed scheme )e conver-gence rate and CPU consumption are all compared in thesimulations We take the space-time domainΩ [0 1] T 1for one-dimensional case and Ω [0 1] times [0 1] T 1 fortwo-dimensional case )ese simulations are implemented inMatlab and the numerical experiments are run on a computerwith 4GB memory )e time-fractional Cattaneo equation ofthe following forms is considered
Example 1 We provide the exact solution u(x t)
etsin(πx) and for different α we have different f(x t)
accordingly
Mathematical Problems in Engineering 9
Table 1 Considering Δt h2 the discrete linfin error and convergence rates of u with different α for Example 1
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 15320e minus 04 mdash 15941e minus 04 mdash 18076e minus 04 mdash2minus 4 95109e minus 06 40097 98981e minus 06 40094 11230e minus 05 400862minus 5 59343e minus 07 40024 61762e minus 07 40024 70079e minus 07 400222minus 6 37074e minus 08 40006 38585e minus 08 40006 43783e minus 08 40005
Table 2 Considering h Δt
radic the discrete l2 error and convergence rates of u with different α for Example 1
Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
2minus 4 00017 mdash 00018 mdash 00021 mdash2minus 6 10832e minus 04 40382 11271e minus 04 40373 12781e minus 04 403442minus 8 67252e minus 06 40097 69990e minus 06 40094 79405e minus 06 400862minus 10 41961e minus 07 40024 43672e minus 07 40023 49553e minus 07 400212minus 12 26215e minus 08 40006 27284e minus 08 40005 30959e minus 08 40005
Table 3 )e CPU time consumption of the fast compact difference scheme and direct difference scheme for Example 1
Δtα 125 α 15 α 175
DCD FCD DCD FCD DCD FCD11000 34008 18876 34632 19032 33540 1918812500 142740 46800 141960 47580 138840 4742415000 470967 93600 463010 94692 461450 9375617500 994194 139464 956754 140088 967206 138684110 000 1699318 186733 1641130 186421 1658290 184393125 000 9816362 466910 9509820 469251 9591409 460046150 000 38269853 934133 36923096 940062 37547412 919625
N times104
0
500
1000
1500
2000
2500
3000
3500
4000
CPU
tim
e
Fast evaluationDirect evaluation
0 05 1 15 2 25 3 35 4 45 5
(a)
Fast evaluationDirect evaluation
log10 (N)
ndash05
0
05
1
15
2
25
3
35
4
log1
0 (C
PU ti
me)
25 3 35 4 45 5
(b)
Figure 1 )e CPU time (a) and the log-log CPU time (b) versus the total number of time steps N for Example 1
10 Mathematical Problems in Engineering
Table 4 Considering Δt 2minus 13 the discrete linfin error and convergence rates of u with different α for Example 1
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 22486e minus 04 mdash 22733e minus 04 mdash 23705e minus 04 mdash2minus 4 13985e minus 05 40071 14140e minus 05 40069 14745e minus 05 400692minus 5 86897e minus 07 40084 87882e minus 07 40081 91726e minus 07 400682minus 6 50194e minus 08 41137 51024e minus 08 41063 54092e minus 08 40838
Table 5 Considering h 0001 the discrete linfin error and convergence rates of u with different α for Example 1
Δtα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 00046 mdash 00043 mdash 00036 mdash2minus 4 00011 20641 00011 19668 90063e minus 04 199902minus 5 28662e minus 04 19403 27171e minus 04 20174 22518e minus 04 199992minus 6 71658e minus 05 19999 67928e minus 05 20000 56295e minus 05 200002minus 7 17915e minus 05 20000 16982e minus 05 20000 14074e minus 05 20000
x
ndash02
0
02
04
06
08
1
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(a)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(b)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(c)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(d)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(e)
ndash02
0
02
04
06
08
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(f)
Figure 2 Considering c 0001 α 15 M N 100 real and numerical solutions of u at T 1 for Example 2 with the value of x0 equalto (a) 0 (b) 02 (c) 04 (d) 06 (e) 08 and (f) 1
Mathematical Problems in Engineering 11
zu(x t)
zt+
zαu(x t)
ztα
z2u(x t)
zx2 + f(x t)
u(x 0) sin(πx)zu
zt
1113868111386811138681113868111386811138681113868 t0 sin(πx)
u(0 t) u(1 t) 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(73)
In Tables 1 and 2 we take Δt h2 and h Δt
radicto ex-
amine the discrete linfin-norm (l2-norm) errors and
corresponding spatial and temporal convergence rates re-spectively We list the errors and convergence rates (order) ofthe proposed compact finite difference (CD) scheme which isalmost O(Δt2 + h4) for different α Additionally Table 3shows the CPU time (CPU) consumed by direct compact(DCD) scheme and fast compact difference (FCD) schemerespectively It is obvious that the FCD scheme has a sig-nificantly reduced CPU time over the DCD scheme Forinstance when α 15 we choose h 01 and Δt 150 000and observe that the FCD scheme consumes only 94 secondswhile the DCD scheme consumes 3692 seconds We can find
x
075
08
085
09
095
1
105
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(a)
0
01
02
03
04
05
06
07
08
09
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(b)
0
01
02
03
04
05
06
07
08
09
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(c)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(d)
Figure 3 Considering x0 05 α 15 M N 100 real and numerical solutions of u at T 1 for Example 2 with the value of c equal to(a) 1 (b) 01 (c) 001 and (d) 0001
12 Mathematical Problems in Engineering
that the performance of the FCD scheme will be moreconspicuous as the time step size Δt decreases
In Figure 1 we set h 01 and α 175 and change thetotal number of time steps N to plot out the CPU time (inseconds) of the FCD scheme and DCD scheme We canobserve that the CPU time increases almost linearly withrespect to N for the FCD scheme while the DCD schemescales like O(N2)
Tables 4 and 5 show the discrete linfin errors and con-vergence rates of the compact finite difference scheme forExample 1 )e space rates are almost O(h4) for fixedΔt 2minus 13 and the time convergence rates are always O(Δt2)for fixed h 0001 We can conclude that the numericalconvergence rates of our scheme approach almost toO(Δt2 + h4)
Example 2 )e example is described by
zu(x t)
zt+
zαu(x t)
ztα
z2u(x t)
zx2 + f(x t)
u(x 0) 0zu
zt
1113868111386811138681113868111386811138681113868t0 0
u(0 t) t2exp minusx20
c1113888 1113889 u(1 t) t2exp minus
1 minus x0( 11138572
c1113888 1113889
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(74)
Note that the exact solution of the above problem isu(x t) t2exp(minus (x minus x0)
2c) where x0 isin [0 1] and c is a
N times104
0
1000
2000
3000
4000
5000
6000
7000
8000CP
U ti
me
Fast evaluationDirect evaluation
0 05 1 15 2 25 3 35 4 45 5
(a)
Fast evaluationDirect evaluation
log10 (N)
0
05
1
15
2
25
3
35
4
log1
0 (C
PU ti
me)
25 3 35 4 45 5
(b)
Figure 4 Considering x0 05 c 001 α 15 andM 10 the CPU time (a) and the log-log CPU time (b) versus the total number oftime steps N for Example 2
Table 6 Considering Δt h2 andx0 05 the discrete linfin error and convergence rates of u with different α and c for Example 2
c hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
1
2minus 3 73224e minus 05 mdash 55643e minus 05 mdash 38478e minus 05 mdash2minus 4 45707e-06 40018 34695e minus 06 40034 23900e minus 06 400902minus 5 28558e minus 07 40004 21671e minus 07 40009 14913e minus 07 400242minus 6 17847e minus 08 40001 13543e minus 08 40001 93169e minus 09 40006
01
2minus 3 00012 mdash 00012 mdash 00012 mdash2minus 4 70462e minus 05 40900 69731e minus 05 41051 69007e minus 05 412012minus 5 43571e minus 06 40154 43110e minus 06 40157 42650e minus 06 401612minus 6 27160e minus 07 40038 26871e minus 07 40039 26583e minus 07 40040
001
2minus 4 00090 mdash 00090 mdash 00090 mdash2minus 5 49257e minus 04 41915 49235e minus 04 41922 49232e minus 04 419232minus 6 29868e minus 05 40437 29855e minus 05 40436 29854e minus 05 404362minus 7 18531e minus 06 40106 18523e minus 06 40106 18522e minus 06 40106
Mathematical Problems in Engineering 13
Table 7 Considering h Δt
radicand c 001 the discrete l2 error and convergence rates of u with different α andx0 for Example 2
x0 Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
02minus 8 00044 mdash 00044 mdash 00042 mdash2minus 10 24205e minus 04 41841 23721e minus 04 42133 23098e minus 04 418452minus 12 14684e minus 05 40430 14390e minus 05 40430 14013e minus 05 40429
0252minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14902e minus 04 41794 14894e minus 04 41802 14892e minus 04 418032minus 12 90680e minus 06 40386 90633e minus 06 40386 90616e minus 06 40386
052minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14848e minus 04 41846 14842e minus 04 41852 14842e minus 04 418522minus 12 90323e minus 06 40390 90288e minus 06 40390 90286e minus 06 40390
0752minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14902e minus 04 41794 14894e minus 04 41802 14892e minus 04 418032minus 12 90680e minus 06 40386 90633e minus 06 40386 90616e minus 06 40386
12minus 8 00044 mdash 00044 mdash 00042 mdash2minus 10 24205e minus 04 41841 23721e minus 04 42133 23098e minus 04 418452minus 12 14684e minus 05 40430 14390e minus 05 40430 14013e minus 05 40429
0 2 4 6 8 10times104
0
2000
4000
6000
8000
10000
12000
14000
N
CPU
tim
e
Fast evaluationDirect evaluation
(a)
Fast evaluationDirect evaluation
36 38 4 42 44 46 48 515
2
25
3
35
4
45
log10 (N)
log1
0 (C
PU ti
me)
(b)
Figure 5 )e CPU time (a) and the log-log CPU time (b) versus the total number of time steps N for Example 3
Table 8 Considering Δt h2 the discrete linfin error and convergence rates of u with different α for Example 3
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 2 00028 mdash 00028 mdash 00030 mdash2minus 3 16948e minus 04 40462 17328e minus 04 40142 18411e minus 04 402632minus 4 10522e minus 05 40096 10759e minus 05 40095 11434e minus 05 400922minus 5 65652e minus 07 40024 67133e minus 07 40024 71346e minus 07 40024
14 Mathematical Problems in Engineering
constant and for different α we have different f(x t)
accordinglyWe apply the fast compact difference scheme to dis-
cretize the equation In Figure 2 we setc 0001 α 15 andM N 100 and plot exact andnumerical solutions at time T 1 for Example 2 with dif-ferent x0 For x0 05 α 15 andM N 100 we alsoplot exact and numerical solutions with the different c inFigure 3 In Figure 4 for h 01 and α 15 we vary thetotal number of time steps N to plot out the CPU time (inseconds) of the FCD scheme and DCD scheme )e nu-merical experiments verified our theoretical results In Ta-ble 6 by equating Δt h2 and fixing x0 05 we compute
the discrete linfin error and convergence rates with differentfractional derivative orders α and different c It shows thatthe compact finite difference scheme has space accuracy offourth order and temporal accuracy of second order We seth
Δt
radicand fix c 001 and the discrete l2 error and
convergence rates with different α and x0 are displayed inTable 7 )ese numerical convergence rates are almostapproaching O(Δt2 + h4) as Example 1
Example 3 If the exact solution is given byu(x t) etsin(πx)sin(πy) we have different f(x y t) fordifferent α accordingly
zu(x y t)
zt+
zαu(x y t)
ztα
z2u(x y t)
zx2 +z2u(x y t)
zy2 + f(x y t)
u(x 0) sin(πx)sin(πy)
zu
zt | t0 sin(πx)sin(πy)
u(0 y t) u(1 y t) u(x 0 t) u(x 1 t) 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(75)
In Figure 5 h 01 and α 175 are fixed and the totalnumber of time steps N vary to plot out the CPU time (inseconds) of the FCD procedure and DCD procedure and itpresents an approximately linear computation complexityfor FCD procedure We set Δt h2 in Table 8 and h
Δt
radic
in Table 9 the discrete linfin error discrete l2 error andconvergence rates with different derivative orders α arepresented )e fourth-order space accuracy and second-order temporal accuracy can be observed clearly
7 Conclusion
In this paper we develop and analyze a fast compact finitedifference procedure for the Cattaneo equation equipped withtime-fractional derivative without singular kernel )e time-fractional derivative is of CaputondashFabrizio type with the orderof α(1lt αlt 2) Compact difference discretization is appliedto obtain a high-order approximation for spatial derivatives ofinteger order in the partial differential equation and theCaputondashFabrizio fractional derivative is discretized by means
of CrankndashNicolson approximation It has been proved thatthe proposed compact finite difference scheme has spatialaccuracy of fourth order and temporal accuracy of secondorder Since the fractional derivatives are history dependentand nonlocal huge memory for storage and computationalcost are required )is means extreme difficulty especially fora long-time simulation Enlightened by the treatment forCaputo fractional derivative [32] we develop an effective fastevaluation procedure for the new CaputondashFabrizio fractionalderivative for the compact finite difference scheme Severalnumerical experiments have been carried out to show theconvergence orders and applicability of the scheme
Inspired by the work [43] the topic about modelling andnumerical solutions of porous media flow equipped withfractional derivatives is very interesting and challenging andwill be our main research direction in the future
Data Availability
All data generated or analyzed during this study are includedin this article
Table 9 Considering h Δt
radic the discrete l2 error and convergence rates of u with different α for Example 3
Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
2minus 4 00014 mdash 00014 mdash 00015 mdash2minus 6 84738e minus 05 40463 86640e minus 05 40142 92056e minus 05 402632minus 8 52610e minus 06 40096 53795e minus 06 40095 57168e minus 06 400922minus 10 32826e minus 07 40024 33566e minus 07 40024 35673e minus 07 40023
Mathematical Problems in Engineering 15
Conflicts of Interest
)e authors declare that they have no conflicts of interest
Acknowledgments
)is work was supported in part by the National NaturalScience Foundation of China under Grant nos 91630207and 11971272
References
[1] A Chaves ldquoA fractional diffusion equation to describe Levyflightsrdquo Physics Letters A vol 239 no 1-2 pp 13ndash16 1998
[2] M Giona and H E Roman ldquoFractional diffusion equation fortransport phenomena in random mediardquo Physica A Statis-tical Mechanics and its Applications vol 185 no 1ndash4pp 87ndash97 1992
[3] R Metzler and J Klafter ldquo)e random walkrsquos guide toanomalous diffusion a fractional dynamics approachrdquoPhysics Reports vol 339 no 1 pp 1ndash77 2000
[4] R L Magin Fractional Calculus in Bioengineering BegellHouse Redding Redding CT USA 2006
[5] R L Magin O Abdullah D Baleanu and X J ZhouldquoAnomalous diffusion expressed through fractional orderdifferential operators in the Bloch-Torrey equationrdquo Journalof Magnetic Resonance vol 190 no 2 pp 255ndash270 2008
[6] I Podlubny Fractional Differential Equations An Introduc-tion to Fractional Derivatives Fractional Differential Equa-tions to Methods of Beir Solution and Some of BeirApplications Vol 198 Elsevier Amsterdam Netherlands1998
[7] M Ran and C Zhang ldquoCompact difference scheme for a classof fractional-in-space nonlinear damped wave equations intwo space dimensionsrdquo Computers amp Mathematics withApplications vol 71 no 5 pp 1151ndash1162 2016
[8] A H Bhrawy and M A Zaky ldquoHighly accurate numericalschemes for multi-dimensional space variable-order frac-tional Schrodinger equationsrdquo Computers amp Mathematicswith Applications vol 73 no 6 pp 1100ndash1117 2017
[9] S B Yuste ldquoWeighted average finite difference methods forfractional diffusion equationsrdquo Journal of ComputationalPhysics vol 216 no 1 pp 264ndash274 2006
[10] T A M Langlands and B I Henry ldquo)e accuracy andstability of an implicit solution method for the fractionaldiffusion equationrdquo Journal of Computational Physicsvol 205 no 2 pp 719ndash736 2005
[11] C-M Chen F Liu I Turner and V Anh ldquoA Fourier method forthe fractional diffusion equation describing sub-diffusionrdquo Journalof Computational Physics vol 227 no 2 pp 886ndash897 2007
[12] Y Lin and C Xu ldquoFinite differencespectral approximationsfor the time-fractional diffusion equationrdquo Journal of Com-putational Physics vol 225 no 2 pp 1533ndash1552 2007
[13] G-h Gao and Z-z Sun ldquoA compact finite difference schemefor the fractional sub-diffusion equationsrdquo Journal of Com-putational Physics vol 230 no 3 pp 586ndash595 2011
[14] F Zeng C Li F Liu and I Turner ldquoNumerical algorithms fortime-fractional subdiffusion equation with second-order ac-curacyrdquo SIAM Journal on Scientific Computing vol 37 no 1pp A55ndashA78 2015
[15] G-H Gao Z-Z Sun and H-W Zhang ldquoA new fractionalnumerical differentiation formula to approximate the Caputofractional derivative and its applicationsrdquo Journal of Com-putational Physics vol 259 no 2 pp 33ndash50 2014
[16] H Li J Cao and C Li ldquoHigh-order approximation to Caputoderivatives and Caputo-type advection-diffusion equations(III)rdquo Journal of Computational and Applied Mathematicsvol 299 pp 159ndash175 2016
[17] A A Alikhanov ldquoA new difference scheme for the timefractional diffusion equationrdquo Journal of ComputationalPhysics vol 280 pp 424ndash438 2015
[18] Z Wang and S Vong ldquoCompact difference schemes for themodified anomalous fractional sub-diffusion equation and thefractional diffusion-wave equationrdquo Journal of ComputationalPhysics vol 277 pp 1ndash15 2014
[19] H-K Pang and H-W Sun ldquoFourth order finite differenceschemes for time-space fractional sub-diffusion equationsrdquoComputers amp Mathematics with Applications vol 71 no 6pp 1287ndash1302 2016
[20] M Dehghan M Abbaszadeh and W Deng ldquoFourth-ordernumerical method for the space-time tempered fractionaldiffusion-wave equationrdquo Applied Mathematics Lettersvol 73 no 11 pp 120ndash127 2017
[21] H Sun Z-Z Sun and G-H Gao ldquoSome high order dif-ference schemes for the space and time fractional Bloch-Torrey equationsrdquo Applied Mathematics and Computationvol 281 pp 356ndash380 2016
[22] S Arshad J Huang A Q M Khaliq and Y Tang ldquoTrape-zoidal scheme for time-space fractional diffusion equationwith Riesz derivativerdquo Journal of Computational Physicsvol 350 pp 1ndash15 2017
[23] H Wang K Wang and T Sircar ldquoA direct O(Nlog2N) finitedifferencemethod for fractional diffusion equationsrdquo Journal ofComputational Physics vol 229 no 21 pp 8095ndash8104 2010
[24] H Wang and T S Basu ldquoA fast finite difference method fortwo-dimensional space-fractional diffusion equationsrdquo SIAMJournal on Scientific Computing vol 34 no 5 pp A2444ndashA2458 2012
[25] H Wang and N Du ldquoA superfast-preconditioned iterativemethod for steady-state space-fractional diffusion equationsrdquoJournal of Computational Physics vol 240 pp 49ndash57 2013
[26] H Wang and N Du ldquoA fast finite difference method forthree-dimensional time-dependent space-fractional diffusionequations and its efficient implementationrdquo Journal ofComputational Physics vol 253 pp 50ndash63 2013
[27] H Wang and N Du ldquoFast alternating-direction finite dif-ference methods for three-dimensional space-fractional dif-fusion equationsrdquo Journal of Computational Physics vol 258pp 305ndash318 2014
[28] K Wang and H Wang ldquoA fast characteristic finite differencemethod for fractional advection-diffusion equationsrdquo Ad-vances in Water Resources vol 34 no 7 pp 810ndash816 2011
[29] H Wang and H Tian ldquoA fast Galerkin method with efficientmatrix assembly and storage for a peridynamic modelrdquoJournal of Computational Physics vol 231 no 23pp 7730ndash7738 2012
[30] H Tian H Wang and W Wang ldquoAn efficient collocationmethod for a non-local diffusion modelrdquo InternationalJournal of Numerical Analysis amp Modeling vol 10 no 4pp 815ndash825 2013
[31] H Wang and H Tian ldquoA fast and faithful collocation methodwith efficient matrix assembly for a two-dimensional nonlocaldiffusion modelrdquo Computer Methods in Applied Mechanicsand Engineering vol 273 no 5 pp 19ndash36 2014
[32] S Jiang J Zhang Q Zhang and Z Zhang ldquoFast evaluation ofthe Caputo fractional derivative and its applications tofractional diffusion equationsrdquo Communications in Compu-tational Physics vol 21 no 3 pp 650ndash678 2017
16 Mathematical Problems in Engineering
[33] Y Yan Z-Z Sun and J Zhang ldquoFast evaluation of the Caputofractional derivative and its applications to fractional diffusionequations a second-order schemerdquo Communications inComputational Physics vol 22 no 4 pp 1028ndash1048 2017
[34] X Lu H-K Pang and H-W Sun ldquoFast approximate in-version of a block triangular Toeplitz matrix with applicationsto fractional sub-diffusion equationsrdquo Numerical Linear Al-gebra with Applications vol 22 no 5 pp 866ndash882 2015
[35] M Caputo and M Fabrizio ldquoA new definition of fractionalderivative without singular kernelrdquo Progress in FractionalDifferentiation and Applications vol 1 no 2 pp 73ndash85 2015
[36] A Atangana ldquoOn the new fractional derivative and appli-cation to nonlinear Fisherrsquos reaction-diffusion equationrdquoAppliedMathematics and Computation vol 273 pp 948ndash9562016
[37] M D Ortigueira and J Tenreiro Machado ldquoA critical analysisof the Caputo-Fabrizio operatorrdquo Communications in Non-linear Science and Numerical Simulation vol 59 pp 608ndash6112018
[38] V E Tarasov ldquoNo nonlocality no fractional derivativerdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 62 pp 157ndash163 2018
[39] V Tarasov ldquoCaputo-Fabrizio operator in terms of integerderivatives memory or distributed lagrdquo Computational andApplied Mathematics vol 38 p 113 2019
[40] A Giusti ldquoA comment on some new definitions of fractionalderivativerdquoNonlinear Dynamics vol 93 no 3 pp 1757ndash17632018
[41] A Atangana and J F Gomez-Aguilar ldquoFractional derivativeswith no-index law property application to chaos and sta-tisticsrdquo Chaos Solitons amp Fractals vol 114 pp 516ndash535 2018
[42] J Hristov ldquoResponse functions in linear viscoelastic consti-tutive equations and related fractional operatorsrdquo Mathe-matical Modelling of Natural Phenomena vol 14 no 3pp 1ndash34 2019
[43] L X Vivas-Cruz A Gonzalez-Calderon M A Taneco-Hernandez and D P Luis ldquo)eoretical analysis of a model offluid flow in a reservoir with the Caputo-Fabrizio operatorrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 84 Article ID 105186 2020
[44] H-L Liao and Z-Z Sun ldquoMaximum norm error bounds ofADI and compact ADI methods for solving parabolicequationsrdquo Numerical Methods for Partial DifferentialEquations vol 26 no 1 pp 37ndash60 2010
[45] Z Liu A Cheng and X Li ldquoA second order Crank-Nicolsonscheme for fractional Cattaneo equation based on newfractional derivativerdquoAppliedMathematics and Computationvol 311 pp 361ndash374 2017
[46] Z Sun and G Gao Finite Difference Method for FractionalDifferential Equations Science Press Beijing China 2015
Mathematical Problems in Engineering 17
Let
Q Uk
1113872 1113873 1113944k
n1Mkminus n δtU
n
2A
+(α minus 1)⦀nablahUk⦀
2A (65)
Summing up with respect to k from 0 to N minus 1 we get
Q UN
1113872 1113873leQ U0
1113872 1113873 + 1113944Nminus 1
k0Mkψ
2A +(α minus 1)Δt 1113944
Nminus 1
k0f
k+(12)
2
A
(66)
Noting that U0 ϕ we have Q(U0) (α minus 1)nablahϕ2A Itfollows that
Q UN
1113872 1113873le (α minus 1)⦀nablahϕ⦀2A + 1113944
Nminus 1
k0Mkψ
2A
+(α minus 1)Δt 1113944Nminus 1
k0f
k+(12)
2
A
(67)
Theorem 3 For the compact finite difference scheme (56)the following stability inequality holds
Q Um
( 1113857le (α minus 1) nablahϕ1113868111386811138681113868
1113868111386811138681113868
2A
+ 1113944
mminus 1
k0Mkψ
2A
+(α minus 1)Δt 1113944
mminus 1
k0f
k+(12)
2
A forall0lemleN
(68)
Similar to the stability the convergence can also beanalyzed
Theorem 4 Suppose that the exact solution of the fractionalCattaneo equation is sufficiently smooth then there exists apositive constant C independent of h k and Δt such that
ek
11138681113868111386811138681113868
111386811138681113868111386811138681leC Δt2 + h
41113872 1113873 forall0le kleN (69)
where ekij uk
ij minus ukij and h max hx hy1113966 1113967
5 Efficient Storage and Fast Evaluation of theCaputondashFabrizio Fractional Derivative
Since time-fractional derivative operator is nonlocal thetraditional direct method for numerically solving the frac-tional Cattaneo equations generally requires total O(MN)
memory units and O(MN2) computational complexitywhere N andM are the total number of time steps and spacesteps respectively
In this section we develop a fast solution method for thefinite difference scheme of the time-fractional Cattaneoequation
Let
Nkn Mkminus n minus Mkminus n+1
exp1 minus α2 minus α
(k minus n)Δt1113874 1113875
middot 1 minus 2exp1 minus α2 minus αΔt1113874 1113875 + exp
1 minus α2 minus α
2Δt1113874 11138751113876 1113877
1le nle k
(70)
then
Nk+1n Mk+1minus n minus Mk+1minus n+1
exp1 minus α2 minus α
(k minus n)Δt1113874 1113875
middot 1 minus 2exp1 minus α2 minus αΔt1113874 1113875 + exp
1 minus α2 minus α
2Δt1113874 11138751113876 1113877exp1 minus α2 minus αΔt1113874 1113875
exp1 minus α2 minus αΔt1113874 1113875N
kn
Nk+1k+1 M0 minus M1 1 minus 2exp
1 minus α2 minus αΔt1113874 1113875 + exp
1 minus α2 minus α
2Δt1113874 1113875
(71)
So we have
1113944
k+1
n1N
k+1n Aδtu
ni exp
1 minus α2 minus αΔt1113874 1113875 1113944
k
n1N
knAδtu
ni + N
k+1k+1Aδtu
k+1i
1113944
k+1
n1N
k+1n AyAxδtu
nij exp
1 minus α2 minus αΔt1113874 1113875 1113944
k
n1N
knAyAxδtu
nij
+ Nk+1k+1AyAxδtU
k+1ij
(72)
Remark We find that at the k-th level only O(1) operationsare needed to compute the k-th level since the (k minus 1)-thlevel is known at that point )us the total operations arereduced from O(N2) to O(N) and the memory require-ment decreases from O(N) to O(1) We conclude that thatthis fast method significantly reduces the total computa-tional cost from O(MN2) to O(MN) and the memoryrequirement from O(MN) to O(M)
6 Numerical Experiments
In this section we carry out several numerical experiments tocheck the effectiveness of the proposed scheme )e conver-gence rate and CPU consumption are all compared in thesimulations We take the space-time domainΩ [0 1] T 1for one-dimensional case and Ω [0 1] times [0 1] T 1 fortwo-dimensional case )ese simulations are implemented inMatlab and the numerical experiments are run on a computerwith 4GB memory )e time-fractional Cattaneo equation ofthe following forms is considered
Example 1 We provide the exact solution u(x t)
etsin(πx) and for different α we have different f(x t)
accordingly
Mathematical Problems in Engineering 9
Table 1 Considering Δt h2 the discrete linfin error and convergence rates of u with different α for Example 1
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 15320e minus 04 mdash 15941e minus 04 mdash 18076e minus 04 mdash2minus 4 95109e minus 06 40097 98981e minus 06 40094 11230e minus 05 400862minus 5 59343e minus 07 40024 61762e minus 07 40024 70079e minus 07 400222minus 6 37074e minus 08 40006 38585e minus 08 40006 43783e minus 08 40005
Table 2 Considering h Δt
radic the discrete l2 error and convergence rates of u with different α for Example 1
Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
2minus 4 00017 mdash 00018 mdash 00021 mdash2minus 6 10832e minus 04 40382 11271e minus 04 40373 12781e minus 04 403442minus 8 67252e minus 06 40097 69990e minus 06 40094 79405e minus 06 400862minus 10 41961e minus 07 40024 43672e minus 07 40023 49553e minus 07 400212minus 12 26215e minus 08 40006 27284e minus 08 40005 30959e minus 08 40005
Table 3 )e CPU time consumption of the fast compact difference scheme and direct difference scheme for Example 1
Δtα 125 α 15 α 175
DCD FCD DCD FCD DCD FCD11000 34008 18876 34632 19032 33540 1918812500 142740 46800 141960 47580 138840 4742415000 470967 93600 463010 94692 461450 9375617500 994194 139464 956754 140088 967206 138684110 000 1699318 186733 1641130 186421 1658290 184393125 000 9816362 466910 9509820 469251 9591409 460046150 000 38269853 934133 36923096 940062 37547412 919625
N times104
0
500
1000
1500
2000
2500
3000
3500
4000
CPU
tim
e
Fast evaluationDirect evaluation
0 05 1 15 2 25 3 35 4 45 5
(a)
Fast evaluationDirect evaluation
log10 (N)
ndash05
0
05
1
15
2
25
3
35
4
log1
0 (C
PU ti
me)
25 3 35 4 45 5
(b)
Figure 1 )e CPU time (a) and the log-log CPU time (b) versus the total number of time steps N for Example 1
10 Mathematical Problems in Engineering
Table 4 Considering Δt 2minus 13 the discrete linfin error and convergence rates of u with different α for Example 1
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 22486e minus 04 mdash 22733e minus 04 mdash 23705e minus 04 mdash2minus 4 13985e minus 05 40071 14140e minus 05 40069 14745e minus 05 400692minus 5 86897e minus 07 40084 87882e minus 07 40081 91726e minus 07 400682minus 6 50194e minus 08 41137 51024e minus 08 41063 54092e minus 08 40838
Table 5 Considering h 0001 the discrete linfin error and convergence rates of u with different α for Example 1
Δtα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 00046 mdash 00043 mdash 00036 mdash2minus 4 00011 20641 00011 19668 90063e minus 04 199902minus 5 28662e minus 04 19403 27171e minus 04 20174 22518e minus 04 199992minus 6 71658e minus 05 19999 67928e minus 05 20000 56295e minus 05 200002minus 7 17915e minus 05 20000 16982e minus 05 20000 14074e minus 05 20000
x
ndash02
0
02
04
06
08
1
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(a)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(b)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(c)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(d)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(e)
ndash02
0
02
04
06
08
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(f)
Figure 2 Considering c 0001 α 15 M N 100 real and numerical solutions of u at T 1 for Example 2 with the value of x0 equalto (a) 0 (b) 02 (c) 04 (d) 06 (e) 08 and (f) 1
Mathematical Problems in Engineering 11
zu(x t)
zt+
zαu(x t)
ztα
z2u(x t)
zx2 + f(x t)
u(x 0) sin(πx)zu
zt
1113868111386811138681113868111386811138681113868 t0 sin(πx)
u(0 t) u(1 t) 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(73)
In Tables 1 and 2 we take Δt h2 and h Δt
radicto ex-
amine the discrete linfin-norm (l2-norm) errors and
corresponding spatial and temporal convergence rates re-spectively We list the errors and convergence rates (order) ofthe proposed compact finite difference (CD) scheme which isalmost O(Δt2 + h4) for different α Additionally Table 3shows the CPU time (CPU) consumed by direct compact(DCD) scheme and fast compact difference (FCD) schemerespectively It is obvious that the FCD scheme has a sig-nificantly reduced CPU time over the DCD scheme Forinstance when α 15 we choose h 01 and Δt 150 000and observe that the FCD scheme consumes only 94 secondswhile the DCD scheme consumes 3692 seconds We can find
x
075
08
085
09
095
1
105
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(a)
0
01
02
03
04
05
06
07
08
09
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(b)
0
01
02
03
04
05
06
07
08
09
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(c)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(d)
Figure 3 Considering x0 05 α 15 M N 100 real and numerical solutions of u at T 1 for Example 2 with the value of c equal to(a) 1 (b) 01 (c) 001 and (d) 0001
12 Mathematical Problems in Engineering
that the performance of the FCD scheme will be moreconspicuous as the time step size Δt decreases
In Figure 1 we set h 01 and α 175 and change thetotal number of time steps N to plot out the CPU time (inseconds) of the FCD scheme and DCD scheme We canobserve that the CPU time increases almost linearly withrespect to N for the FCD scheme while the DCD schemescales like O(N2)
Tables 4 and 5 show the discrete linfin errors and con-vergence rates of the compact finite difference scheme forExample 1 )e space rates are almost O(h4) for fixedΔt 2minus 13 and the time convergence rates are always O(Δt2)for fixed h 0001 We can conclude that the numericalconvergence rates of our scheme approach almost toO(Δt2 + h4)
Example 2 )e example is described by
zu(x t)
zt+
zαu(x t)
ztα
z2u(x t)
zx2 + f(x t)
u(x 0) 0zu
zt
1113868111386811138681113868111386811138681113868t0 0
u(0 t) t2exp minusx20
c1113888 1113889 u(1 t) t2exp minus
1 minus x0( 11138572
c1113888 1113889
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(74)
Note that the exact solution of the above problem isu(x t) t2exp(minus (x minus x0)
2c) where x0 isin [0 1] and c is a
N times104
0
1000
2000
3000
4000
5000
6000
7000
8000CP
U ti
me
Fast evaluationDirect evaluation
0 05 1 15 2 25 3 35 4 45 5
(a)
Fast evaluationDirect evaluation
log10 (N)
0
05
1
15
2
25
3
35
4
log1
0 (C
PU ti
me)
25 3 35 4 45 5
(b)
Figure 4 Considering x0 05 c 001 α 15 andM 10 the CPU time (a) and the log-log CPU time (b) versus the total number oftime steps N for Example 2
Table 6 Considering Δt h2 andx0 05 the discrete linfin error and convergence rates of u with different α and c for Example 2
c hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
1
2minus 3 73224e minus 05 mdash 55643e minus 05 mdash 38478e minus 05 mdash2minus 4 45707e-06 40018 34695e minus 06 40034 23900e minus 06 400902minus 5 28558e minus 07 40004 21671e minus 07 40009 14913e minus 07 400242minus 6 17847e minus 08 40001 13543e minus 08 40001 93169e minus 09 40006
01
2minus 3 00012 mdash 00012 mdash 00012 mdash2minus 4 70462e minus 05 40900 69731e minus 05 41051 69007e minus 05 412012minus 5 43571e minus 06 40154 43110e minus 06 40157 42650e minus 06 401612minus 6 27160e minus 07 40038 26871e minus 07 40039 26583e minus 07 40040
001
2minus 4 00090 mdash 00090 mdash 00090 mdash2minus 5 49257e minus 04 41915 49235e minus 04 41922 49232e minus 04 419232minus 6 29868e minus 05 40437 29855e minus 05 40436 29854e minus 05 404362minus 7 18531e minus 06 40106 18523e minus 06 40106 18522e minus 06 40106
Mathematical Problems in Engineering 13
Table 7 Considering h Δt
radicand c 001 the discrete l2 error and convergence rates of u with different α andx0 for Example 2
x0 Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
02minus 8 00044 mdash 00044 mdash 00042 mdash2minus 10 24205e minus 04 41841 23721e minus 04 42133 23098e minus 04 418452minus 12 14684e minus 05 40430 14390e minus 05 40430 14013e minus 05 40429
0252minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14902e minus 04 41794 14894e minus 04 41802 14892e minus 04 418032minus 12 90680e minus 06 40386 90633e minus 06 40386 90616e minus 06 40386
052minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14848e minus 04 41846 14842e minus 04 41852 14842e minus 04 418522minus 12 90323e minus 06 40390 90288e minus 06 40390 90286e minus 06 40390
0752minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14902e minus 04 41794 14894e minus 04 41802 14892e minus 04 418032minus 12 90680e minus 06 40386 90633e minus 06 40386 90616e minus 06 40386
12minus 8 00044 mdash 00044 mdash 00042 mdash2minus 10 24205e minus 04 41841 23721e minus 04 42133 23098e minus 04 418452minus 12 14684e minus 05 40430 14390e minus 05 40430 14013e minus 05 40429
0 2 4 6 8 10times104
0
2000
4000
6000
8000
10000
12000
14000
N
CPU
tim
e
Fast evaluationDirect evaluation
(a)
Fast evaluationDirect evaluation
36 38 4 42 44 46 48 515
2
25
3
35
4
45
log10 (N)
log1
0 (C
PU ti
me)
(b)
Figure 5 )e CPU time (a) and the log-log CPU time (b) versus the total number of time steps N for Example 3
Table 8 Considering Δt h2 the discrete linfin error and convergence rates of u with different α for Example 3
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 2 00028 mdash 00028 mdash 00030 mdash2minus 3 16948e minus 04 40462 17328e minus 04 40142 18411e minus 04 402632minus 4 10522e minus 05 40096 10759e minus 05 40095 11434e minus 05 400922minus 5 65652e minus 07 40024 67133e minus 07 40024 71346e minus 07 40024
14 Mathematical Problems in Engineering
constant and for different α we have different f(x t)
accordinglyWe apply the fast compact difference scheme to dis-
cretize the equation In Figure 2 we setc 0001 α 15 andM N 100 and plot exact andnumerical solutions at time T 1 for Example 2 with dif-ferent x0 For x0 05 α 15 andM N 100 we alsoplot exact and numerical solutions with the different c inFigure 3 In Figure 4 for h 01 and α 15 we vary thetotal number of time steps N to plot out the CPU time (inseconds) of the FCD scheme and DCD scheme )e nu-merical experiments verified our theoretical results In Ta-ble 6 by equating Δt h2 and fixing x0 05 we compute
the discrete linfin error and convergence rates with differentfractional derivative orders α and different c It shows thatthe compact finite difference scheme has space accuracy offourth order and temporal accuracy of second order We seth
Δt
radicand fix c 001 and the discrete l2 error and
convergence rates with different α and x0 are displayed inTable 7 )ese numerical convergence rates are almostapproaching O(Δt2 + h4) as Example 1
Example 3 If the exact solution is given byu(x t) etsin(πx)sin(πy) we have different f(x y t) fordifferent α accordingly
zu(x y t)
zt+
zαu(x y t)
ztα
z2u(x y t)
zx2 +z2u(x y t)
zy2 + f(x y t)
u(x 0) sin(πx)sin(πy)
zu
zt | t0 sin(πx)sin(πy)
u(0 y t) u(1 y t) u(x 0 t) u(x 1 t) 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(75)
In Figure 5 h 01 and α 175 are fixed and the totalnumber of time steps N vary to plot out the CPU time (inseconds) of the FCD procedure and DCD procedure and itpresents an approximately linear computation complexityfor FCD procedure We set Δt h2 in Table 8 and h
Δt
radic
in Table 9 the discrete linfin error discrete l2 error andconvergence rates with different derivative orders α arepresented )e fourth-order space accuracy and second-order temporal accuracy can be observed clearly
7 Conclusion
In this paper we develop and analyze a fast compact finitedifference procedure for the Cattaneo equation equipped withtime-fractional derivative without singular kernel )e time-fractional derivative is of CaputondashFabrizio type with the orderof α(1lt αlt 2) Compact difference discretization is appliedto obtain a high-order approximation for spatial derivatives ofinteger order in the partial differential equation and theCaputondashFabrizio fractional derivative is discretized by means
of CrankndashNicolson approximation It has been proved thatthe proposed compact finite difference scheme has spatialaccuracy of fourth order and temporal accuracy of secondorder Since the fractional derivatives are history dependentand nonlocal huge memory for storage and computationalcost are required )is means extreme difficulty especially fora long-time simulation Enlightened by the treatment forCaputo fractional derivative [32] we develop an effective fastevaluation procedure for the new CaputondashFabrizio fractionalderivative for the compact finite difference scheme Severalnumerical experiments have been carried out to show theconvergence orders and applicability of the scheme
Inspired by the work [43] the topic about modelling andnumerical solutions of porous media flow equipped withfractional derivatives is very interesting and challenging andwill be our main research direction in the future
Data Availability
All data generated or analyzed during this study are includedin this article
Table 9 Considering h Δt
radic the discrete l2 error and convergence rates of u with different α for Example 3
Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
2minus 4 00014 mdash 00014 mdash 00015 mdash2minus 6 84738e minus 05 40463 86640e minus 05 40142 92056e minus 05 402632minus 8 52610e minus 06 40096 53795e minus 06 40095 57168e minus 06 400922minus 10 32826e minus 07 40024 33566e minus 07 40024 35673e minus 07 40023
Mathematical Problems in Engineering 15
Conflicts of Interest
)e authors declare that they have no conflicts of interest
Acknowledgments
)is work was supported in part by the National NaturalScience Foundation of China under Grant nos 91630207and 11971272
References
[1] A Chaves ldquoA fractional diffusion equation to describe Levyflightsrdquo Physics Letters A vol 239 no 1-2 pp 13ndash16 1998
[2] M Giona and H E Roman ldquoFractional diffusion equation fortransport phenomena in random mediardquo Physica A Statis-tical Mechanics and its Applications vol 185 no 1ndash4pp 87ndash97 1992
[3] R Metzler and J Klafter ldquo)e random walkrsquos guide toanomalous diffusion a fractional dynamics approachrdquoPhysics Reports vol 339 no 1 pp 1ndash77 2000
[4] R L Magin Fractional Calculus in Bioengineering BegellHouse Redding Redding CT USA 2006
[5] R L Magin O Abdullah D Baleanu and X J ZhouldquoAnomalous diffusion expressed through fractional orderdifferential operators in the Bloch-Torrey equationrdquo Journalof Magnetic Resonance vol 190 no 2 pp 255ndash270 2008
[6] I Podlubny Fractional Differential Equations An Introduc-tion to Fractional Derivatives Fractional Differential Equa-tions to Methods of Beir Solution and Some of BeirApplications Vol 198 Elsevier Amsterdam Netherlands1998
[7] M Ran and C Zhang ldquoCompact difference scheme for a classof fractional-in-space nonlinear damped wave equations intwo space dimensionsrdquo Computers amp Mathematics withApplications vol 71 no 5 pp 1151ndash1162 2016
[8] A H Bhrawy and M A Zaky ldquoHighly accurate numericalschemes for multi-dimensional space variable-order frac-tional Schrodinger equationsrdquo Computers amp Mathematicswith Applications vol 73 no 6 pp 1100ndash1117 2017
[9] S B Yuste ldquoWeighted average finite difference methods forfractional diffusion equationsrdquo Journal of ComputationalPhysics vol 216 no 1 pp 264ndash274 2006
[10] T A M Langlands and B I Henry ldquo)e accuracy andstability of an implicit solution method for the fractionaldiffusion equationrdquo Journal of Computational Physicsvol 205 no 2 pp 719ndash736 2005
[11] C-M Chen F Liu I Turner and V Anh ldquoA Fourier method forthe fractional diffusion equation describing sub-diffusionrdquo Journalof Computational Physics vol 227 no 2 pp 886ndash897 2007
[12] Y Lin and C Xu ldquoFinite differencespectral approximationsfor the time-fractional diffusion equationrdquo Journal of Com-putational Physics vol 225 no 2 pp 1533ndash1552 2007
[13] G-h Gao and Z-z Sun ldquoA compact finite difference schemefor the fractional sub-diffusion equationsrdquo Journal of Com-putational Physics vol 230 no 3 pp 586ndash595 2011
[14] F Zeng C Li F Liu and I Turner ldquoNumerical algorithms fortime-fractional subdiffusion equation with second-order ac-curacyrdquo SIAM Journal on Scientific Computing vol 37 no 1pp A55ndashA78 2015
[15] G-H Gao Z-Z Sun and H-W Zhang ldquoA new fractionalnumerical differentiation formula to approximate the Caputofractional derivative and its applicationsrdquo Journal of Com-putational Physics vol 259 no 2 pp 33ndash50 2014
[16] H Li J Cao and C Li ldquoHigh-order approximation to Caputoderivatives and Caputo-type advection-diffusion equations(III)rdquo Journal of Computational and Applied Mathematicsvol 299 pp 159ndash175 2016
[17] A A Alikhanov ldquoA new difference scheme for the timefractional diffusion equationrdquo Journal of ComputationalPhysics vol 280 pp 424ndash438 2015
[18] Z Wang and S Vong ldquoCompact difference schemes for themodified anomalous fractional sub-diffusion equation and thefractional diffusion-wave equationrdquo Journal of ComputationalPhysics vol 277 pp 1ndash15 2014
[19] H-K Pang and H-W Sun ldquoFourth order finite differenceschemes for time-space fractional sub-diffusion equationsrdquoComputers amp Mathematics with Applications vol 71 no 6pp 1287ndash1302 2016
[20] M Dehghan M Abbaszadeh and W Deng ldquoFourth-ordernumerical method for the space-time tempered fractionaldiffusion-wave equationrdquo Applied Mathematics Lettersvol 73 no 11 pp 120ndash127 2017
[21] H Sun Z-Z Sun and G-H Gao ldquoSome high order dif-ference schemes for the space and time fractional Bloch-Torrey equationsrdquo Applied Mathematics and Computationvol 281 pp 356ndash380 2016
[22] S Arshad J Huang A Q M Khaliq and Y Tang ldquoTrape-zoidal scheme for time-space fractional diffusion equationwith Riesz derivativerdquo Journal of Computational Physicsvol 350 pp 1ndash15 2017
[23] H Wang K Wang and T Sircar ldquoA direct O(Nlog2N) finitedifferencemethod for fractional diffusion equationsrdquo Journal ofComputational Physics vol 229 no 21 pp 8095ndash8104 2010
[24] H Wang and T S Basu ldquoA fast finite difference method fortwo-dimensional space-fractional diffusion equationsrdquo SIAMJournal on Scientific Computing vol 34 no 5 pp A2444ndashA2458 2012
[25] H Wang and N Du ldquoA superfast-preconditioned iterativemethod for steady-state space-fractional diffusion equationsrdquoJournal of Computational Physics vol 240 pp 49ndash57 2013
[26] H Wang and N Du ldquoA fast finite difference method forthree-dimensional time-dependent space-fractional diffusionequations and its efficient implementationrdquo Journal ofComputational Physics vol 253 pp 50ndash63 2013
[27] H Wang and N Du ldquoFast alternating-direction finite dif-ference methods for three-dimensional space-fractional dif-fusion equationsrdquo Journal of Computational Physics vol 258pp 305ndash318 2014
[28] K Wang and H Wang ldquoA fast characteristic finite differencemethod for fractional advection-diffusion equationsrdquo Ad-vances in Water Resources vol 34 no 7 pp 810ndash816 2011
[29] H Wang and H Tian ldquoA fast Galerkin method with efficientmatrix assembly and storage for a peridynamic modelrdquoJournal of Computational Physics vol 231 no 23pp 7730ndash7738 2012
[30] H Tian H Wang and W Wang ldquoAn efficient collocationmethod for a non-local diffusion modelrdquo InternationalJournal of Numerical Analysis amp Modeling vol 10 no 4pp 815ndash825 2013
[31] H Wang and H Tian ldquoA fast and faithful collocation methodwith efficient matrix assembly for a two-dimensional nonlocaldiffusion modelrdquo Computer Methods in Applied Mechanicsand Engineering vol 273 no 5 pp 19ndash36 2014
[32] S Jiang J Zhang Q Zhang and Z Zhang ldquoFast evaluation ofthe Caputo fractional derivative and its applications tofractional diffusion equationsrdquo Communications in Compu-tational Physics vol 21 no 3 pp 650ndash678 2017
16 Mathematical Problems in Engineering
[33] Y Yan Z-Z Sun and J Zhang ldquoFast evaluation of the Caputofractional derivative and its applications to fractional diffusionequations a second-order schemerdquo Communications inComputational Physics vol 22 no 4 pp 1028ndash1048 2017
[34] X Lu H-K Pang and H-W Sun ldquoFast approximate in-version of a block triangular Toeplitz matrix with applicationsto fractional sub-diffusion equationsrdquo Numerical Linear Al-gebra with Applications vol 22 no 5 pp 866ndash882 2015
[35] M Caputo and M Fabrizio ldquoA new definition of fractionalderivative without singular kernelrdquo Progress in FractionalDifferentiation and Applications vol 1 no 2 pp 73ndash85 2015
[36] A Atangana ldquoOn the new fractional derivative and appli-cation to nonlinear Fisherrsquos reaction-diffusion equationrdquoAppliedMathematics and Computation vol 273 pp 948ndash9562016
[37] M D Ortigueira and J Tenreiro Machado ldquoA critical analysisof the Caputo-Fabrizio operatorrdquo Communications in Non-linear Science and Numerical Simulation vol 59 pp 608ndash6112018
[38] V E Tarasov ldquoNo nonlocality no fractional derivativerdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 62 pp 157ndash163 2018
[39] V Tarasov ldquoCaputo-Fabrizio operator in terms of integerderivatives memory or distributed lagrdquo Computational andApplied Mathematics vol 38 p 113 2019
[40] A Giusti ldquoA comment on some new definitions of fractionalderivativerdquoNonlinear Dynamics vol 93 no 3 pp 1757ndash17632018
[41] A Atangana and J F Gomez-Aguilar ldquoFractional derivativeswith no-index law property application to chaos and sta-tisticsrdquo Chaos Solitons amp Fractals vol 114 pp 516ndash535 2018
[42] J Hristov ldquoResponse functions in linear viscoelastic consti-tutive equations and related fractional operatorsrdquo Mathe-matical Modelling of Natural Phenomena vol 14 no 3pp 1ndash34 2019
[43] L X Vivas-Cruz A Gonzalez-Calderon M A Taneco-Hernandez and D P Luis ldquo)eoretical analysis of a model offluid flow in a reservoir with the Caputo-Fabrizio operatorrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 84 Article ID 105186 2020
[44] H-L Liao and Z-Z Sun ldquoMaximum norm error bounds ofADI and compact ADI methods for solving parabolicequationsrdquo Numerical Methods for Partial DifferentialEquations vol 26 no 1 pp 37ndash60 2010
[45] Z Liu A Cheng and X Li ldquoA second order Crank-Nicolsonscheme for fractional Cattaneo equation based on newfractional derivativerdquoAppliedMathematics and Computationvol 311 pp 361ndash374 2017
[46] Z Sun and G Gao Finite Difference Method for FractionalDifferential Equations Science Press Beijing China 2015
Mathematical Problems in Engineering 17
Table 1 Considering Δt h2 the discrete linfin error and convergence rates of u with different α for Example 1
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 15320e minus 04 mdash 15941e minus 04 mdash 18076e minus 04 mdash2minus 4 95109e minus 06 40097 98981e minus 06 40094 11230e minus 05 400862minus 5 59343e minus 07 40024 61762e minus 07 40024 70079e minus 07 400222minus 6 37074e minus 08 40006 38585e minus 08 40006 43783e minus 08 40005
Table 2 Considering h Δt
radic the discrete l2 error and convergence rates of u with different α for Example 1
Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
2minus 4 00017 mdash 00018 mdash 00021 mdash2minus 6 10832e minus 04 40382 11271e minus 04 40373 12781e minus 04 403442minus 8 67252e minus 06 40097 69990e minus 06 40094 79405e minus 06 400862minus 10 41961e minus 07 40024 43672e minus 07 40023 49553e minus 07 400212minus 12 26215e minus 08 40006 27284e minus 08 40005 30959e minus 08 40005
Table 3 )e CPU time consumption of the fast compact difference scheme and direct difference scheme for Example 1
Δtα 125 α 15 α 175
DCD FCD DCD FCD DCD FCD11000 34008 18876 34632 19032 33540 1918812500 142740 46800 141960 47580 138840 4742415000 470967 93600 463010 94692 461450 9375617500 994194 139464 956754 140088 967206 138684110 000 1699318 186733 1641130 186421 1658290 184393125 000 9816362 466910 9509820 469251 9591409 460046150 000 38269853 934133 36923096 940062 37547412 919625
N times104
0
500
1000
1500
2000
2500
3000
3500
4000
CPU
tim
e
Fast evaluationDirect evaluation
0 05 1 15 2 25 3 35 4 45 5
(a)
Fast evaluationDirect evaluation
log10 (N)
ndash05
0
05
1
15
2
25
3
35
4
log1
0 (C
PU ti
me)
25 3 35 4 45 5
(b)
Figure 1 )e CPU time (a) and the log-log CPU time (b) versus the total number of time steps N for Example 1
10 Mathematical Problems in Engineering
Table 4 Considering Δt 2minus 13 the discrete linfin error and convergence rates of u with different α for Example 1
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 22486e minus 04 mdash 22733e minus 04 mdash 23705e minus 04 mdash2minus 4 13985e minus 05 40071 14140e minus 05 40069 14745e minus 05 400692minus 5 86897e minus 07 40084 87882e minus 07 40081 91726e minus 07 400682minus 6 50194e minus 08 41137 51024e minus 08 41063 54092e minus 08 40838
Table 5 Considering h 0001 the discrete linfin error and convergence rates of u with different α for Example 1
Δtα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 00046 mdash 00043 mdash 00036 mdash2minus 4 00011 20641 00011 19668 90063e minus 04 199902minus 5 28662e minus 04 19403 27171e minus 04 20174 22518e minus 04 199992minus 6 71658e minus 05 19999 67928e minus 05 20000 56295e minus 05 200002minus 7 17915e minus 05 20000 16982e minus 05 20000 14074e minus 05 20000
x
ndash02
0
02
04
06
08
1
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(a)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(b)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(c)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(d)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(e)
ndash02
0
02
04
06
08
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(f)
Figure 2 Considering c 0001 α 15 M N 100 real and numerical solutions of u at T 1 for Example 2 with the value of x0 equalto (a) 0 (b) 02 (c) 04 (d) 06 (e) 08 and (f) 1
Mathematical Problems in Engineering 11
zu(x t)
zt+
zαu(x t)
ztα
z2u(x t)
zx2 + f(x t)
u(x 0) sin(πx)zu
zt
1113868111386811138681113868111386811138681113868 t0 sin(πx)
u(0 t) u(1 t) 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(73)
In Tables 1 and 2 we take Δt h2 and h Δt
radicto ex-
amine the discrete linfin-norm (l2-norm) errors and
corresponding spatial and temporal convergence rates re-spectively We list the errors and convergence rates (order) ofthe proposed compact finite difference (CD) scheme which isalmost O(Δt2 + h4) for different α Additionally Table 3shows the CPU time (CPU) consumed by direct compact(DCD) scheme and fast compact difference (FCD) schemerespectively It is obvious that the FCD scheme has a sig-nificantly reduced CPU time over the DCD scheme Forinstance when α 15 we choose h 01 and Δt 150 000and observe that the FCD scheme consumes only 94 secondswhile the DCD scheme consumes 3692 seconds We can find
x
075
08
085
09
095
1
105
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(a)
0
01
02
03
04
05
06
07
08
09
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(b)
0
01
02
03
04
05
06
07
08
09
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(c)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(d)
Figure 3 Considering x0 05 α 15 M N 100 real and numerical solutions of u at T 1 for Example 2 with the value of c equal to(a) 1 (b) 01 (c) 001 and (d) 0001
12 Mathematical Problems in Engineering
that the performance of the FCD scheme will be moreconspicuous as the time step size Δt decreases
In Figure 1 we set h 01 and α 175 and change thetotal number of time steps N to plot out the CPU time (inseconds) of the FCD scheme and DCD scheme We canobserve that the CPU time increases almost linearly withrespect to N for the FCD scheme while the DCD schemescales like O(N2)
Tables 4 and 5 show the discrete linfin errors and con-vergence rates of the compact finite difference scheme forExample 1 )e space rates are almost O(h4) for fixedΔt 2minus 13 and the time convergence rates are always O(Δt2)for fixed h 0001 We can conclude that the numericalconvergence rates of our scheme approach almost toO(Δt2 + h4)
Example 2 )e example is described by
zu(x t)
zt+
zαu(x t)
ztα
z2u(x t)
zx2 + f(x t)
u(x 0) 0zu
zt
1113868111386811138681113868111386811138681113868t0 0
u(0 t) t2exp minusx20
c1113888 1113889 u(1 t) t2exp minus
1 minus x0( 11138572
c1113888 1113889
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(74)
Note that the exact solution of the above problem isu(x t) t2exp(minus (x minus x0)
2c) where x0 isin [0 1] and c is a
N times104
0
1000
2000
3000
4000
5000
6000
7000
8000CP
U ti
me
Fast evaluationDirect evaluation
0 05 1 15 2 25 3 35 4 45 5
(a)
Fast evaluationDirect evaluation
log10 (N)
0
05
1
15
2
25
3
35
4
log1
0 (C
PU ti
me)
25 3 35 4 45 5
(b)
Figure 4 Considering x0 05 c 001 α 15 andM 10 the CPU time (a) and the log-log CPU time (b) versus the total number oftime steps N for Example 2
Table 6 Considering Δt h2 andx0 05 the discrete linfin error and convergence rates of u with different α and c for Example 2
c hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
1
2minus 3 73224e minus 05 mdash 55643e minus 05 mdash 38478e minus 05 mdash2minus 4 45707e-06 40018 34695e minus 06 40034 23900e minus 06 400902minus 5 28558e minus 07 40004 21671e minus 07 40009 14913e minus 07 400242minus 6 17847e minus 08 40001 13543e minus 08 40001 93169e minus 09 40006
01
2minus 3 00012 mdash 00012 mdash 00012 mdash2minus 4 70462e minus 05 40900 69731e minus 05 41051 69007e minus 05 412012minus 5 43571e minus 06 40154 43110e minus 06 40157 42650e minus 06 401612minus 6 27160e minus 07 40038 26871e minus 07 40039 26583e minus 07 40040
001
2minus 4 00090 mdash 00090 mdash 00090 mdash2minus 5 49257e minus 04 41915 49235e minus 04 41922 49232e minus 04 419232minus 6 29868e minus 05 40437 29855e minus 05 40436 29854e minus 05 404362minus 7 18531e minus 06 40106 18523e minus 06 40106 18522e minus 06 40106
Mathematical Problems in Engineering 13
Table 7 Considering h Δt
radicand c 001 the discrete l2 error and convergence rates of u with different α andx0 for Example 2
x0 Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
02minus 8 00044 mdash 00044 mdash 00042 mdash2minus 10 24205e minus 04 41841 23721e minus 04 42133 23098e minus 04 418452minus 12 14684e minus 05 40430 14390e minus 05 40430 14013e minus 05 40429
0252minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14902e minus 04 41794 14894e minus 04 41802 14892e minus 04 418032minus 12 90680e minus 06 40386 90633e minus 06 40386 90616e minus 06 40386
052minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14848e minus 04 41846 14842e minus 04 41852 14842e minus 04 418522minus 12 90323e minus 06 40390 90288e minus 06 40390 90286e minus 06 40390
0752minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14902e minus 04 41794 14894e minus 04 41802 14892e minus 04 418032minus 12 90680e minus 06 40386 90633e minus 06 40386 90616e minus 06 40386
12minus 8 00044 mdash 00044 mdash 00042 mdash2minus 10 24205e minus 04 41841 23721e minus 04 42133 23098e minus 04 418452minus 12 14684e minus 05 40430 14390e minus 05 40430 14013e minus 05 40429
0 2 4 6 8 10times104
0
2000
4000
6000
8000
10000
12000
14000
N
CPU
tim
e
Fast evaluationDirect evaluation
(a)
Fast evaluationDirect evaluation
36 38 4 42 44 46 48 515
2
25
3
35
4
45
log10 (N)
log1
0 (C
PU ti
me)
(b)
Figure 5 )e CPU time (a) and the log-log CPU time (b) versus the total number of time steps N for Example 3
Table 8 Considering Δt h2 the discrete linfin error and convergence rates of u with different α for Example 3
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 2 00028 mdash 00028 mdash 00030 mdash2minus 3 16948e minus 04 40462 17328e minus 04 40142 18411e minus 04 402632minus 4 10522e minus 05 40096 10759e minus 05 40095 11434e minus 05 400922minus 5 65652e minus 07 40024 67133e minus 07 40024 71346e minus 07 40024
14 Mathematical Problems in Engineering
constant and for different α we have different f(x t)
accordinglyWe apply the fast compact difference scheme to dis-
cretize the equation In Figure 2 we setc 0001 α 15 andM N 100 and plot exact andnumerical solutions at time T 1 for Example 2 with dif-ferent x0 For x0 05 α 15 andM N 100 we alsoplot exact and numerical solutions with the different c inFigure 3 In Figure 4 for h 01 and α 15 we vary thetotal number of time steps N to plot out the CPU time (inseconds) of the FCD scheme and DCD scheme )e nu-merical experiments verified our theoretical results In Ta-ble 6 by equating Δt h2 and fixing x0 05 we compute
the discrete linfin error and convergence rates with differentfractional derivative orders α and different c It shows thatthe compact finite difference scheme has space accuracy offourth order and temporal accuracy of second order We seth
Δt
radicand fix c 001 and the discrete l2 error and
convergence rates with different α and x0 are displayed inTable 7 )ese numerical convergence rates are almostapproaching O(Δt2 + h4) as Example 1
Example 3 If the exact solution is given byu(x t) etsin(πx)sin(πy) we have different f(x y t) fordifferent α accordingly
zu(x y t)
zt+
zαu(x y t)
ztα
z2u(x y t)
zx2 +z2u(x y t)
zy2 + f(x y t)
u(x 0) sin(πx)sin(πy)
zu
zt | t0 sin(πx)sin(πy)
u(0 y t) u(1 y t) u(x 0 t) u(x 1 t) 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(75)
In Figure 5 h 01 and α 175 are fixed and the totalnumber of time steps N vary to plot out the CPU time (inseconds) of the FCD procedure and DCD procedure and itpresents an approximately linear computation complexityfor FCD procedure We set Δt h2 in Table 8 and h
Δt
radic
in Table 9 the discrete linfin error discrete l2 error andconvergence rates with different derivative orders α arepresented )e fourth-order space accuracy and second-order temporal accuracy can be observed clearly
7 Conclusion
In this paper we develop and analyze a fast compact finitedifference procedure for the Cattaneo equation equipped withtime-fractional derivative without singular kernel )e time-fractional derivative is of CaputondashFabrizio type with the orderof α(1lt αlt 2) Compact difference discretization is appliedto obtain a high-order approximation for spatial derivatives ofinteger order in the partial differential equation and theCaputondashFabrizio fractional derivative is discretized by means
of CrankndashNicolson approximation It has been proved thatthe proposed compact finite difference scheme has spatialaccuracy of fourth order and temporal accuracy of secondorder Since the fractional derivatives are history dependentand nonlocal huge memory for storage and computationalcost are required )is means extreme difficulty especially fora long-time simulation Enlightened by the treatment forCaputo fractional derivative [32] we develop an effective fastevaluation procedure for the new CaputondashFabrizio fractionalderivative for the compact finite difference scheme Severalnumerical experiments have been carried out to show theconvergence orders and applicability of the scheme
Inspired by the work [43] the topic about modelling andnumerical solutions of porous media flow equipped withfractional derivatives is very interesting and challenging andwill be our main research direction in the future
Data Availability
All data generated or analyzed during this study are includedin this article
Table 9 Considering h Δt
radic the discrete l2 error and convergence rates of u with different α for Example 3
Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
2minus 4 00014 mdash 00014 mdash 00015 mdash2minus 6 84738e minus 05 40463 86640e minus 05 40142 92056e minus 05 402632minus 8 52610e minus 06 40096 53795e minus 06 40095 57168e minus 06 400922minus 10 32826e minus 07 40024 33566e minus 07 40024 35673e minus 07 40023
Mathematical Problems in Engineering 15
Conflicts of Interest
)e authors declare that they have no conflicts of interest
Acknowledgments
)is work was supported in part by the National NaturalScience Foundation of China under Grant nos 91630207and 11971272
References
[1] A Chaves ldquoA fractional diffusion equation to describe Levyflightsrdquo Physics Letters A vol 239 no 1-2 pp 13ndash16 1998
[2] M Giona and H E Roman ldquoFractional diffusion equation fortransport phenomena in random mediardquo Physica A Statis-tical Mechanics and its Applications vol 185 no 1ndash4pp 87ndash97 1992
[3] R Metzler and J Klafter ldquo)e random walkrsquos guide toanomalous diffusion a fractional dynamics approachrdquoPhysics Reports vol 339 no 1 pp 1ndash77 2000
[4] R L Magin Fractional Calculus in Bioengineering BegellHouse Redding Redding CT USA 2006
[5] R L Magin O Abdullah D Baleanu and X J ZhouldquoAnomalous diffusion expressed through fractional orderdifferential operators in the Bloch-Torrey equationrdquo Journalof Magnetic Resonance vol 190 no 2 pp 255ndash270 2008
[6] I Podlubny Fractional Differential Equations An Introduc-tion to Fractional Derivatives Fractional Differential Equa-tions to Methods of Beir Solution and Some of BeirApplications Vol 198 Elsevier Amsterdam Netherlands1998
[7] M Ran and C Zhang ldquoCompact difference scheme for a classof fractional-in-space nonlinear damped wave equations intwo space dimensionsrdquo Computers amp Mathematics withApplications vol 71 no 5 pp 1151ndash1162 2016
[8] A H Bhrawy and M A Zaky ldquoHighly accurate numericalschemes for multi-dimensional space variable-order frac-tional Schrodinger equationsrdquo Computers amp Mathematicswith Applications vol 73 no 6 pp 1100ndash1117 2017
[9] S B Yuste ldquoWeighted average finite difference methods forfractional diffusion equationsrdquo Journal of ComputationalPhysics vol 216 no 1 pp 264ndash274 2006
[10] T A M Langlands and B I Henry ldquo)e accuracy andstability of an implicit solution method for the fractionaldiffusion equationrdquo Journal of Computational Physicsvol 205 no 2 pp 719ndash736 2005
[11] C-M Chen F Liu I Turner and V Anh ldquoA Fourier method forthe fractional diffusion equation describing sub-diffusionrdquo Journalof Computational Physics vol 227 no 2 pp 886ndash897 2007
[12] Y Lin and C Xu ldquoFinite differencespectral approximationsfor the time-fractional diffusion equationrdquo Journal of Com-putational Physics vol 225 no 2 pp 1533ndash1552 2007
[13] G-h Gao and Z-z Sun ldquoA compact finite difference schemefor the fractional sub-diffusion equationsrdquo Journal of Com-putational Physics vol 230 no 3 pp 586ndash595 2011
[14] F Zeng C Li F Liu and I Turner ldquoNumerical algorithms fortime-fractional subdiffusion equation with second-order ac-curacyrdquo SIAM Journal on Scientific Computing vol 37 no 1pp A55ndashA78 2015
[15] G-H Gao Z-Z Sun and H-W Zhang ldquoA new fractionalnumerical differentiation formula to approximate the Caputofractional derivative and its applicationsrdquo Journal of Com-putational Physics vol 259 no 2 pp 33ndash50 2014
[16] H Li J Cao and C Li ldquoHigh-order approximation to Caputoderivatives and Caputo-type advection-diffusion equations(III)rdquo Journal of Computational and Applied Mathematicsvol 299 pp 159ndash175 2016
[17] A A Alikhanov ldquoA new difference scheme for the timefractional diffusion equationrdquo Journal of ComputationalPhysics vol 280 pp 424ndash438 2015
[18] Z Wang and S Vong ldquoCompact difference schemes for themodified anomalous fractional sub-diffusion equation and thefractional diffusion-wave equationrdquo Journal of ComputationalPhysics vol 277 pp 1ndash15 2014
[19] H-K Pang and H-W Sun ldquoFourth order finite differenceschemes for time-space fractional sub-diffusion equationsrdquoComputers amp Mathematics with Applications vol 71 no 6pp 1287ndash1302 2016
[20] M Dehghan M Abbaszadeh and W Deng ldquoFourth-ordernumerical method for the space-time tempered fractionaldiffusion-wave equationrdquo Applied Mathematics Lettersvol 73 no 11 pp 120ndash127 2017
[21] H Sun Z-Z Sun and G-H Gao ldquoSome high order dif-ference schemes for the space and time fractional Bloch-Torrey equationsrdquo Applied Mathematics and Computationvol 281 pp 356ndash380 2016
[22] S Arshad J Huang A Q M Khaliq and Y Tang ldquoTrape-zoidal scheme for time-space fractional diffusion equationwith Riesz derivativerdquo Journal of Computational Physicsvol 350 pp 1ndash15 2017
[23] H Wang K Wang and T Sircar ldquoA direct O(Nlog2N) finitedifferencemethod for fractional diffusion equationsrdquo Journal ofComputational Physics vol 229 no 21 pp 8095ndash8104 2010
[24] H Wang and T S Basu ldquoA fast finite difference method fortwo-dimensional space-fractional diffusion equationsrdquo SIAMJournal on Scientific Computing vol 34 no 5 pp A2444ndashA2458 2012
[25] H Wang and N Du ldquoA superfast-preconditioned iterativemethod for steady-state space-fractional diffusion equationsrdquoJournal of Computational Physics vol 240 pp 49ndash57 2013
[26] H Wang and N Du ldquoA fast finite difference method forthree-dimensional time-dependent space-fractional diffusionequations and its efficient implementationrdquo Journal ofComputational Physics vol 253 pp 50ndash63 2013
[27] H Wang and N Du ldquoFast alternating-direction finite dif-ference methods for three-dimensional space-fractional dif-fusion equationsrdquo Journal of Computational Physics vol 258pp 305ndash318 2014
[28] K Wang and H Wang ldquoA fast characteristic finite differencemethod for fractional advection-diffusion equationsrdquo Ad-vances in Water Resources vol 34 no 7 pp 810ndash816 2011
[29] H Wang and H Tian ldquoA fast Galerkin method with efficientmatrix assembly and storage for a peridynamic modelrdquoJournal of Computational Physics vol 231 no 23pp 7730ndash7738 2012
[30] H Tian H Wang and W Wang ldquoAn efficient collocationmethod for a non-local diffusion modelrdquo InternationalJournal of Numerical Analysis amp Modeling vol 10 no 4pp 815ndash825 2013
[31] H Wang and H Tian ldquoA fast and faithful collocation methodwith efficient matrix assembly for a two-dimensional nonlocaldiffusion modelrdquo Computer Methods in Applied Mechanicsand Engineering vol 273 no 5 pp 19ndash36 2014
[32] S Jiang J Zhang Q Zhang and Z Zhang ldquoFast evaluation ofthe Caputo fractional derivative and its applications tofractional diffusion equationsrdquo Communications in Compu-tational Physics vol 21 no 3 pp 650ndash678 2017
16 Mathematical Problems in Engineering
[33] Y Yan Z-Z Sun and J Zhang ldquoFast evaluation of the Caputofractional derivative and its applications to fractional diffusionequations a second-order schemerdquo Communications inComputational Physics vol 22 no 4 pp 1028ndash1048 2017
[34] X Lu H-K Pang and H-W Sun ldquoFast approximate in-version of a block triangular Toeplitz matrix with applicationsto fractional sub-diffusion equationsrdquo Numerical Linear Al-gebra with Applications vol 22 no 5 pp 866ndash882 2015
[35] M Caputo and M Fabrizio ldquoA new definition of fractionalderivative without singular kernelrdquo Progress in FractionalDifferentiation and Applications vol 1 no 2 pp 73ndash85 2015
[36] A Atangana ldquoOn the new fractional derivative and appli-cation to nonlinear Fisherrsquos reaction-diffusion equationrdquoAppliedMathematics and Computation vol 273 pp 948ndash9562016
[37] M D Ortigueira and J Tenreiro Machado ldquoA critical analysisof the Caputo-Fabrizio operatorrdquo Communications in Non-linear Science and Numerical Simulation vol 59 pp 608ndash6112018
[38] V E Tarasov ldquoNo nonlocality no fractional derivativerdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 62 pp 157ndash163 2018
[39] V Tarasov ldquoCaputo-Fabrizio operator in terms of integerderivatives memory or distributed lagrdquo Computational andApplied Mathematics vol 38 p 113 2019
[40] A Giusti ldquoA comment on some new definitions of fractionalderivativerdquoNonlinear Dynamics vol 93 no 3 pp 1757ndash17632018
[41] A Atangana and J F Gomez-Aguilar ldquoFractional derivativeswith no-index law property application to chaos and sta-tisticsrdquo Chaos Solitons amp Fractals vol 114 pp 516ndash535 2018
[42] J Hristov ldquoResponse functions in linear viscoelastic consti-tutive equations and related fractional operatorsrdquo Mathe-matical Modelling of Natural Phenomena vol 14 no 3pp 1ndash34 2019
[43] L X Vivas-Cruz A Gonzalez-Calderon M A Taneco-Hernandez and D P Luis ldquo)eoretical analysis of a model offluid flow in a reservoir with the Caputo-Fabrizio operatorrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 84 Article ID 105186 2020
[44] H-L Liao and Z-Z Sun ldquoMaximum norm error bounds ofADI and compact ADI methods for solving parabolicequationsrdquo Numerical Methods for Partial DifferentialEquations vol 26 no 1 pp 37ndash60 2010
[45] Z Liu A Cheng and X Li ldquoA second order Crank-Nicolsonscheme for fractional Cattaneo equation based on newfractional derivativerdquoAppliedMathematics and Computationvol 311 pp 361ndash374 2017
[46] Z Sun and G Gao Finite Difference Method for FractionalDifferential Equations Science Press Beijing China 2015
Mathematical Problems in Engineering 17
Table 4 Considering Δt 2minus 13 the discrete linfin error and convergence rates of u with different α for Example 1
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 22486e minus 04 mdash 22733e minus 04 mdash 23705e minus 04 mdash2minus 4 13985e minus 05 40071 14140e minus 05 40069 14745e minus 05 400692minus 5 86897e minus 07 40084 87882e minus 07 40081 91726e minus 07 400682minus 6 50194e minus 08 41137 51024e minus 08 41063 54092e minus 08 40838
Table 5 Considering h 0001 the discrete linfin error and convergence rates of u with different α for Example 1
Δtα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 3 00046 mdash 00043 mdash 00036 mdash2minus 4 00011 20641 00011 19668 90063e minus 04 199902minus 5 28662e minus 04 19403 27171e minus 04 20174 22518e minus 04 199992minus 6 71658e minus 05 19999 67928e minus 05 20000 56295e minus 05 200002minus 7 17915e minus 05 20000 16982e minus 05 20000 14074e minus 05 20000
x
ndash02
0
02
04
06
08
1
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(a)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(b)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(c)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(d)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(e)
ndash02
0
02
04
06
08
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(f)
Figure 2 Considering c 0001 α 15 M N 100 real and numerical solutions of u at T 1 for Example 2 with the value of x0 equalto (a) 0 (b) 02 (c) 04 (d) 06 (e) 08 and (f) 1
Mathematical Problems in Engineering 11
zu(x t)
zt+
zαu(x t)
ztα
z2u(x t)
zx2 + f(x t)
u(x 0) sin(πx)zu
zt
1113868111386811138681113868111386811138681113868 t0 sin(πx)
u(0 t) u(1 t) 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(73)
In Tables 1 and 2 we take Δt h2 and h Δt
radicto ex-
amine the discrete linfin-norm (l2-norm) errors and
corresponding spatial and temporal convergence rates re-spectively We list the errors and convergence rates (order) ofthe proposed compact finite difference (CD) scheme which isalmost O(Δt2 + h4) for different α Additionally Table 3shows the CPU time (CPU) consumed by direct compact(DCD) scheme and fast compact difference (FCD) schemerespectively It is obvious that the FCD scheme has a sig-nificantly reduced CPU time over the DCD scheme Forinstance when α 15 we choose h 01 and Δt 150 000and observe that the FCD scheme consumes only 94 secondswhile the DCD scheme consumes 3692 seconds We can find
x
075
08
085
09
095
1
105
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(a)
0
01
02
03
04
05
06
07
08
09
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(b)
0
01
02
03
04
05
06
07
08
09
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(c)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(d)
Figure 3 Considering x0 05 α 15 M N 100 real and numerical solutions of u at T 1 for Example 2 with the value of c equal to(a) 1 (b) 01 (c) 001 and (d) 0001
12 Mathematical Problems in Engineering
that the performance of the FCD scheme will be moreconspicuous as the time step size Δt decreases
In Figure 1 we set h 01 and α 175 and change thetotal number of time steps N to plot out the CPU time (inseconds) of the FCD scheme and DCD scheme We canobserve that the CPU time increases almost linearly withrespect to N for the FCD scheme while the DCD schemescales like O(N2)
Tables 4 and 5 show the discrete linfin errors and con-vergence rates of the compact finite difference scheme forExample 1 )e space rates are almost O(h4) for fixedΔt 2minus 13 and the time convergence rates are always O(Δt2)for fixed h 0001 We can conclude that the numericalconvergence rates of our scheme approach almost toO(Δt2 + h4)
Example 2 )e example is described by
zu(x t)
zt+
zαu(x t)
ztα
z2u(x t)
zx2 + f(x t)
u(x 0) 0zu
zt
1113868111386811138681113868111386811138681113868t0 0
u(0 t) t2exp minusx20
c1113888 1113889 u(1 t) t2exp minus
1 minus x0( 11138572
c1113888 1113889
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(74)
Note that the exact solution of the above problem isu(x t) t2exp(minus (x minus x0)
2c) where x0 isin [0 1] and c is a
N times104
0
1000
2000
3000
4000
5000
6000
7000
8000CP
U ti
me
Fast evaluationDirect evaluation
0 05 1 15 2 25 3 35 4 45 5
(a)
Fast evaluationDirect evaluation
log10 (N)
0
05
1
15
2
25
3
35
4
log1
0 (C
PU ti
me)
25 3 35 4 45 5
(b)
Figure 4 Considering x0 05 c 001 α 15 andM 10 the CPU time (a) and the log-log CPU time (b) versus the total number oftime steps N for Example 2
Table 6 Considering Δt h2 andx0 05 the discrete linfin error and convergence rates of u with different α and c for Example 2
c hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
1
2minus 3 73224e minus 05 mdash 55643e minus 05 mdash 38478e minus 05 mdash2minus 4 45707e-06 40018 34695e minus 06 40034 23900e minus 06 400902minus 5 28558e minus 07 40004 21671e minus 07 40009 14913e minus 07 400242minus 6 17847e minus 08 40001 13543e minus 08 40001 93169e minus 09 40006
01
2minus 3 00012 mdash 00012 mdash 00012 mdash2minus 4 70462e minus 05 40900 69731e minus 05 41051 69007e minus 05 412012minus 5 43571e minus 06 40154 43110e minus 06 40157 42650e minus 06 401612minus 6 27160e minus 07 40038 26871e minus 07 40039 26583e minus 07 40040
001
2minus 4 00090 mdash 00090 mdash 00090 mdash2minus 5 49257e minus 04 41915 49235e minus 04 41922 49232e minus 04 419232minus 6 29868e minus 05 40437 29855e minus 05 40436 29854e minus 05 404362minus 7 18531e minus 06 40106 18523e minus 06 40106 18522e minus 06 40106
Mathematical Problems in Engineering 13
Table 7 Considering h Δt
radicand c 001 the discrete l2 error and convergence rates of u with different α andx0 for Example 2
x0 Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
02minus 8 00044 mdash 00044 mdash 00042 mdash2minus 10 24205e minus 04 41841 23721e minus 04 42133 23098e minus 04 418452minus 12 14684e minus 05 40430 14390e minus 05 40430 14013e minus 05 40429
0252minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14902e minus 04 41794 14894e minus 04 41802 14892e minus 04 418032minus 12 90680e minus 06 40386 90633e minus 06 40386 90616e minus 06 40386
052minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14848e minus 04 41846 14842e minus 04 41852 14842e minus 04 418522minus 12 90323e minus 06 40390 90288e minus 06 40390 90286e minus 06 40390
0752minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14902e minus 04 41794 14894e minus 04 41802 14892e minus 04 418032minus 12 90680e minus 06 40386 90633e minus 06 40386 90616e minus 06 40386
12minus 8 00044 mdash 00044 mdash 00042 mdash2minus 10 24205e minus 04 41841 23721e minus 04 42133 23098e minus 04 418452minus 12 14684e minus 05 40430 14390e minus 05 40430 14013e minus 05 40429
0 2 4 6 8 10times104
0
2000
4000
6000
8000
10000
12000
14000
N
CPU
tim
e
Fast evaluationDirect evaluation
(a)
Fast evaluationDirect evaluation
36 38 4 42 44 46 48 515
2
25
3
35
4
45
log10 (N)
log1
0 (C
PU ti
me)
(b)
Figure 5 )e CPU time (a) and the log-log CPU time (b) versus the total number of time steps N for Example 3
Table 8 Considering Δt h2 the discrete linfin error and convergence rates of u with different α for Example 3
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 2 00028 mdash 00028 mdash 00030 mdash2minus 3 16948e minus 04 40462 17328e minus 04 40142 18411e minus 04 402632minus 4 10522e minus 05 40096 10759e minus 05 40095 11434e minus 05 400922minus 5 65652e minus 07 40024 67133e minus 07 40024 71346e minus 07 40024
14 Mathematical Problems in Engineering
constant and for different α we have different f(x t)
accordinglyWe apply the fast compact difference scheme to dis-
cretize the equation In Figure 2 we setc 0001 α 15 andM N 100 and plot exact andnumerical solutions at time T 1 for Example 2 with dif-ferent x0 For x0 05 α 15 andM N 100 we alsoplot exact and numerical solutions with the different c inFigure 3 In Figure 4 for h 01 and α 15 we vary thetotal number of time steps N to plot out the CPU time (inseconds) of the FCD scheme and DCD scheme )e nu-merical experiments verified our theoretical results In Ta-ble 6 by equating Δt h2 and fixing x0 05 we compute
the discrete linfin error and convergence rates with differentfractional derivative orders α and different c It shows thatthe compact finite difference scheme has space accuracy offourth order and temporal accuracy of second order We seth
Δt
radicand fix c 001 and the discrete l2 error and
convergence rates with different α and x0 are displayed inTable 7 )ese numerical convergence rates are almostapproaching O(Δt2 + h4) as Example 1
Example 3 If the exact solution is given byu(x t) etsin(πx)sin(πy) we have different f(x y t) fordifferent α accordingly
zu(x y t)
zt+
zαu(x y t)
ztα
z2u(x y t)
zx2 +z2u(x y t)
zy2 + f(x y t)
u(x 0) sin(πx)sin(πy)
zu
zt | t0 sin(πx)sin(πy)
u(0 y t) u(1 y t) u(x 0 t) u(x 1 t) 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(75)
In Figure 5 h 01 and α 175 are fixed and the totalnumber of time steps N vary to plot out the CPU time (inseconds) of the FCD procedure and DCD procedure and itpresents an approximately linear computation complexityfor FCD procedure We set Δt h2 in Table 8 and h
Δt
radic
in Table 9 the discrete linfin error discrete l2 error andconvergence rates with different derivative orders α arepresented )e fourth-order space accuracy and second-order temporal accuracy can be observed clearly
7 Conclusion
In this paper we develop and analyze a fast compact finitedifference procedure for the Cattaneo equation equipped withtime-fractional derivative without singular kernel )e time-fractional derivative is of CaputondashFabrizio type with the orderof α(1lt αlt 2) Compact difference discretization is appliedto obtain a high-order approximation for spatial derivatives ofinteger order in the partial differential equation and theCaputondashFabrizio fractional derivative is discretized by means
of CrankndashNicolson approximation It has been proved thatthe proposed compact finite difference scheme has spatialaccuracy of fourth order and temporal accuracy of secondorder Since the fractional derivatives are history dependentand nonlocal huge memory for storage and computationalcost are required )is means extreme difficulty especially fora long-time simulation Enlightened by the treatment forCaputo fractional derivative [32] we develop an effective fastevaluation procedure for the new CaputondashFabrizio fractionalderivative for the compact finite difference scheme Severalnumerical experiments have been carried out to show theconvergence orders and applicability of the scheme
Inspired by the work [43] the topic about modelling andnumerical solutions of porous media flow equipped withfractional derivatives is very interesting and challenging andwill be our main research direction in the future
Data Availability
All data generated or analyzed during this study are includedin this article
Table 9 Considering h Δt
radic the discrete l2 error and convergence rates of u with different α for Example 3
Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
2minus 4 00014 mdash 00014 mdash 00015 mdash2minus 6 84738e minus 05 40463 86640e minus 05 40142 92056e minus 05 402632minus 8 52610e minus 06 40096 53795e minus 06 40095 57168e minus 06 400922minus 10 32826e minus 07 40024 33566e minus 07 40024 35673e minus 07 40023
Mathematical Problems in Engineering 15
Conflicts of Interest
)e authors declare that they have no conflicts of interest
Acknowledgments
)is work was supported in part by the National NaturalScience Foundation of China under Grant nos 91630207and 11971272
References
[1] A Chaves ldquoA fractional diffusion equation to describe Levyflightsrdquo Physics Letters A vol 239 no 1-2 pp 13ndash16 1998
[2] M Giona and H E Roman ldquoFractional diffusion equation fortransport phenomena in random mediardquo Physica A Statis-tical Mechanics and its Applications vol 185 no 1ndash4pp 87ndash97 1992
[3] R Metzler and J Klafter ldquo)e random walkrsquos guide toanomalous diffusion a fractional dynamics approachrdquoPhysics Reports vol 339 no 1 pp 1ndash77 2000
[4] R L Magin Fractional Calculus in Bioengineering BegellHouse Redding Redding CT USA 2006
[5] R L Magin O Abdullah D Baleanu and X J ZhouldquoAnomalous diffusion expressed through fractional orderdifferential operators in the Bloch-Torrey equationrdquo Journalof Magnetic Resonance vol 190 no 2 pp 255ndash270 2008
[6] I Podlubny Fractional Differential Equations An Introduc-tion to Fractional Derivatives Fractional Differential Equa-tions to Methods of Beir Solution and Some of BeirApplications Vol 198 Elsevier Amsterdam Netherlands1998
[7] M Ran and C Zhang ldquoCompact difference scheme for a classof fractional-in-space nonlinear damped wave equations intwo space dimensionsrdquo Computers amp Mathematics withApplications vol 71 no 5 pp 1151ndash1162 2016
[8] A H Bhrawy and M A Zaky ldquoHighly accurate numericalschemes for multi-dimensional space variable-order frac-tional Schrodinger equationsrdquo Computers amp Mathematicswith Applications vol 73 no 6 pp 1100ndash1117 2017
[9] S B Yuste ldquoWeighted average finite difference methods forfractional diffusion equationsrdquo Journal of ComputationalPhysics vol 216 no 1 pp 264ndash274 2006
[10] T A M Langlands and B I Henry ldquo)e accuracy andstability of an implicit solution method for the fractionaldiffusion equationrdquo Journal of Computational Physicsvol 205 no 2 pp 719ndash736 2005
[11] C-M Chen F Liu I Turner and V Anh ldquoA Fourier method forthe fractional diffusion equation describing sub-diffusionrdquo Journalof Computational Physics vol 227 no 2 pp 886ndash897 2007
[12] Y Lin and C Xu ldquoFinite differencespectral approximationsfor the time-fractional diffusion equationrdquo Journal of Com-putational Physics vol 225 no 2 pp 1533ndash1552 2007
[13] G-h Gao and Z-z Sun ldquoA compact finite difference schemefor the fractional sub-diffusion equationsrdquo Journal of Com-putational Physics vol 230 no 3 pp 586ndash595 2011
[14] F Zeng C Li F Liu and I Turner ldquoNumerical algorithms fortime-fractional subdiffusion equation with second-order ac-curacyrdquo SIAM Journal on Scientific Computing vol 37 no 1pp A55ndashA78 2015
[15] G-H Gao Z-Z Sun and H-W Zhang ldquoA new fractionalnumerical differentiation formula to approximate the Caputofractional derivative and its applicationsrdquo Journal of Com-putational Physics vol 259 no 2 pp 33ndash50 2014
[16] H Li J Cao and C Li ldquoHigh-order approximation to Caputoderivatives and Caputo-type advection-diffusion equations(III)rdquo Journal of Computational and Applied Mathematicsvol 299 pp 159ndash175 2016
[17] A A Alikhanov ldquoA new difference scheme for the timefractional diffusion equationrdquo Journal of ComputationalPhysics vol 280 pp 424ndash438 2015
[18] Z Wang and S Vong ldquoCompact difference schemes for themodified anomalous fractional sub-diffusion equation and thefractional diffusion-wave equationrdquo Journal of ComputationalPhysics vol 277 pp 1ndash15 2014
[19] H-K Pang and H-W Sun ldquoFourth order finite differenceschemes for time-space fractional sub-diffusion equationsrdquoComputers amp Mathematics with Applications vol 71 no 6pp 1287ndash1302 2016
[20] M Dehghan M Abbaszadeh and W Deng ldquoFourth-ordernumerical method for the space-time tempered fractionaldiffusion-wave equationrdquo Applied Mathematics Lettersvol 73 no 11 pp 120ndash127 2017
[21] H Sun Z-Z Sun and G-H Gao ldquoSome high order dif-ference schemes for the space and time fractional Bloch-Torrey equationsrdquo Applied Mathematics and Computationvol 281 pp 356ndash380 2016
[22] S Arshad J Huang A Q M Khaliq and Y Tang ldquoTrape-zoidal scheme for time-space fractional diffusion equationwith Riesz derivativerdquo Journal of Computational Physicsvol 350 pp 1ndash15 2017
[23] H Wang K Wang and T Sircar ldquoA direct O(Nlog2N) finitedifferencemethod for fractional diffusion equationsrdquo Journal ofComputational Physics vol 229 no 21 pp 8095ndash8104 2010
[24] H Wang and T S Basu ldquoA fast finite difference method fortwo-dimensional space-fractional diffusion equationsrdquo SIAMJournal on Scientific Computing vol 34 no 5 pp A2444ndashA2458 2012
[25] H Wang and N Du ldquoA superfast-preconditioned iterativemethod for steady-state space-fractional diffusion equationsrdquoJournal of Computational Physics vol 240 pp 49ndash57 2013
[26] H Wang and N Du ldquoA fast finite difference method forthree-dimensional time-dependent space-fractional diffusionequations and its efficient implementationrdquo Journal ofComputational Physics vol 253 pp 50ndash63 2013
[27] H Wang and N Du ldquoFast alternating-direction finite dif-ference methods for three-dimensional space-fractional dif-fusion equationsrdquo Journal of Computational Physics vol 258pp 305ndash318 2014
[28] K Wang and H Wang ldquoA fast characteristic finite differencemethod for fractional advection-diffusion equationsrdquo Ad-vances in Water Resources vol 34 no 7 pp 810ndash816 2011
[29] H Wang and H Tian ldquoA fast Galerkin method with efficientmatrix assembly and storage for a peridynamic modelrdquoJournal of Computational Physics vol 231 no 23pp 7730ndash7738 2012
[30] H Tian H Wang and W Wang ldquoAn efficient collocationmethod for a non-local diffusion modelrdquo InternationalJournal of Numerical Analysis amp Modeling vol 10 no 4pp 815ndash825 2013
[31] H Wang and H Tian ldquoA fast and faithful collocation methodwith efficient matrix assembly for a two-dimensional nonlocaldiffusion modelrdquo Computer Methods in Applied Mechanicsand Engineering vol 273 no 5 pp 19ndash36 2014
[32] S Jiang J Zhang Q Zhang and Z Zhang ldquoFast evaluation ofthe Caputo fractional derivative and its applications tofractional diffusion equationsrdquo Communications in Compu-tational Physics vol 21 no 3 pp 650ndash678 2017
16 Mathematical Problems in Engineering
[33] Y Yan Z-Z Sun and J Zhang ldquoFast evaluation of the Caputofractional derivative and its applications to fractional diffusionequations a second-order schemerdquo Communications inComputational Physics vol 22 no 4 pp 1028ndash1048 2017
[34] X Lu H-K Pang and H-W Sun ldquoFast approximate in-version of a block triangular Toeplitz matrix with applicationsto fractional sub-diffusion equationsrdquo Numerical Linear Al-gebra with Applications vol 22 no 5 pp 866ndash882 2015
[35] M Caputo and M Fabrizio ldquoA new definition of fractionalderivative without singular kernelrdquo Progress in FractionalDifferentiation and Applications vol 1 no 2 pp 73ndash85 2015
[36] A Atangana ldquoOn the new fractional derivative and appli-cation to nonlinear Fisherrsquos reaction-diffusion equationrdquoAppliedMathematics and Computation vol 273 pp 948ndash9562016
[37] M D Ortigueira and J Tenreiro Machado ldquoA critical analysisof the Caputo-Fabrizio operatorrdquo Communications in Non-linear Science and Numerical Simulation vol 59 pp 608ndash6112018
[38] V E Tarasov ldquoNo nonlocality no fractional derivativerdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 62 pp 157ndash163 2018
[39] V Tarasov ldquoCaputo-Fabrizio operator in terms of integerderivatives memory or distributed lagrdquo Computational andApplied Mathematics vol 38 p 113 2019
[40] A Giusti ldquoA comment on some new definitions of fractionalderivativerdquoNonlinear Dynamics vol 93 no 3 pp 1757ndash17632018
[41] A Atangana and J F Gomez-Aguilar ldquoFractional derivativeswith no-index law property application to chaos and sta-tisticsrdquo Chaos Solitons amp Fractals vol 114 pp 516ndash535 2018
[42] J Hristov ldquoResponse functions in linear viscoelastic consti-tutive equations and related fractional operatorsrdquo Mathe-matical Modelling of Natural Phenomena vol 14 no 3pp 1ndash34 2019
[43] L X Vivas-Cruz A Gonzalez-Calderon M A Taneco-Hernandez and D P Luis ldquo)eoretical analysis of a model offluid flow in a reservoir with the Caputo-Fabrizio operatorrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 84 Article ID 105186 2020
[44] H-L Liao and Z-Z Sun ldquoMaximum norm error bounds ofADI and compact ADI methods for solving parabolicequationsrdquo Numerical Methods for Partial DifferentialEquations vol 26 no 1 pp 37ndash60 2010
[45] Z Liu A Cheng and X Li ldquoA second order Crank-Nicolsonscheme for fractional Cattaneo equation based on newfractional derivativerdquoAppliedMathematics and Computationvol 311 pp 361ndash374 2017
[46] Z Sun and G Gao Finite Difference Method for FractionalDifferential Equations Science Press Beijing China 2015
Mathematical Problems in Engineering 17
zu(x t)
zt+
zαu(x t)
ztα
z2u(x t)
zx2 + f(x t)
u(x 0) sin(πx)zu
zt
1113868111386811138681113868111386811138681113868 t0 sin(πx)
u(0 t) u(1 t) 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(73)
In Tables 1 and 2 we take Δt h2 and h Δt
radicto ex-
amine the discrete linfin-norm (l2-norm) errors and
corresponding spatial and temporal convergence rates re-spectively We list the errors and convergence rates (order) ofthe proposed compact finite difference (CD) scheme which isalmost O(Δt2 + h4) for different α Additionally Table 3shows the CPU time (CPU) consumed by direct compact(DCD) scheme and fast compact difference (FCD) schemerespectively It is obvious that the FCD scheme has a sig-nificantly reduced CPU time over the DCD scheme Forinstance when α 15 we choose h 01 and Δt 150 000and observe that the FCD scheme consumes only 94 secondswhile the DCD scheme consumes 3692 seconds We can find
x
075
08
085
09
095
1
105
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(a)
0
01
02
03
04
05
06
07
08
09
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(b)
0
01
02
03
04
05
06
07
08
09
1
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(c)
0
02
04
06
08
1
12
x
Real solutionNumerical solution
0 01 02 03 04 05 06 07 08 09 1
(d)
Figure 3 Considering x0 05 α 15 M N 100 real and numerical solutions of u at T 1 for Example 2 with the value of c equal to(a) 1 (b) 01 (c) 001 and (d) 0001
12 Mathematical Problems in Engineering
that the performance of the FCD scheme will be moreconspicuous as the time step size Δt decreases
In Figure 1 we set h 01 and α 175 and change thetotal number of time steps N to plot out the CPU time (inseconds) of the FCD scheme and DCD scheme We canobserve that the CPU time increases almost linearly withrespect to N for the FCD scheme while the DCD schemescales like O(N2)
Tables 4 and 5 show the discrete linfin errors and con-vergence rates of the compact finite difference scheme forExample 1 )e space rates are almost O(h4) for fixedΔt 2minus 13 and the time convergence rates are always O(Δt2)for fixed h 0001 We can conclude that the numericalconvergence rates of our scheme approach almost toO(Δt2 + h4)
Example 2 )e example is described by
zu(x t)
zt+
zαu(x t)
ztα
z2u(x t)
zx2 + f(x t)
u(x 0) 0zu
zt
1113868111386811138681113868111386811138681113868t0 0
u(0 t) t2exp minusx20
c1113888 1113889 u(1 t) t2exp minus
1 minus x0( 11138572
c1113888 1113889
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(74)
Note that the exact solution of the above problem isu(x t) t2exp(minus (x minus x0)
2c) where x0 isin [0 1] and c is a
N times104
0
1000
2000
3000
4000
5000
6000
7000
8000CP
U ti
me
Fast evaluationDirect evaluation
0 05 1 15 2 25 3 35 4 45 5
(a)
Fast evaluationDirect evaluation
log10 (N)
0
05
1
15
2
25
3
35
4
log1
0 (C
PU ti
me)
25 3 35 4 45 5
(b)
Figure 4 Considering x0 05 c 001 α 15 andM 10 the CPU time (a) and the log-log CPU time (b) versus the total number oftime steps N for Example 2
Table 6 Considering Δt h2 andx0 05 the discrete linfin error and convergence rates of u with different α and c for Example 2
c hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
1
2minus 3 73224e minus 05 mdash 55643e minus 05 mdash 38478e minus 05 mdash2minus 4 45707e-06 40018 34695e minus 06 40034 23900e minus 06 400902minus 5 28558e minus 07 40004 21671e minus 07 40009 14913e minus 07 400242minus 6 17847e minus 08 40001 13543e minus 08 40001 93169e minus 09 40006
01
2minus 3 00012 mdash 00012 mdash 00012 mdash2minus 4 70462e minus 05 40900 69731e minus 05 41051 69007e minus 05 412012minus 5 43571e minus 06 40154 43110e minus 06 40157 42650e minus 06 401612minus 6 27160e minus 07 40038 26871e minus 07 40039 26583e minus 07 40040
001
2minus 4 00090 mdash 00090 mdash 00090 mdash2minus 5 49257e minus 04 41915 49235e minus 04 41922 49232e minus 04 419232minus 6 29868e minus 05 40437 29855e minus 05 40436 29854e minus 05 404362minus 7 18531e minus 06 40106 18523e minus 06 40106 18522e minus 06 40106
Mathematical Problems in Engineering 13
Table 7 Considering h Δt
radicand c 001 the discrete l2 error and convergence rates of u with different α andx0 for Example 2
x0 Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
02minus 8 00044 mdash 00044 mdash 00042 mdash2minus 10 24205e minus 04 41841 23721e minus 04 42133 23098e minus 04 418452minus 12 14684e minus 05 40430 14390e minus 05 40430 14013e minus 05 40429
0252minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14902e minus 04 41794 14894e minus 04 41802 14892e minus 04 418032minus 12 90680e minus 06 40386 90633e minus 06 40386 90616e minus 06 40386
052minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14848e minus 04 41846 14842e minus 04 41852 14842e minus 04 418522minus 12 90323e minus 06 40390 90288e minus 06 40390 90286e minus 06 40390
0752minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14902e minus 04 41794 14894e minus 04 41802 14892e minus 04 418032minus 12 90680e minus 06 40386 90633e minus 06 40386 90616e minus 06 40386
12minus 8 00044 mdash 00044 mdash 00042 mdash2minus 10 24205e minus 04 41841 23721e minus 04 42133 23098e minus 04 418452minus 12 14684e minus 05 40430 14390e minus 05 40430 14013e minus 05 40429
0 2 4 6 8 10times104
0
2000
4000
6000
8000
10000
12000
14000
N
CPU
tim
e
Fast evaluationDirect evaluation
(a)
Fast evaluationDirect evaluation
36 38 4 42 44 46 48 515
2
25
3
35
4
45
log10 (N)
log1
0 (C
PU ti
me)
(b)
Figure 5 )e CPU time (a) and the log-log CPU time (b) versus the total number of time steps N for Example 3
Table 8 Considering Δt h2 the discrete linfin error and convergence rates of u with different α for Example 3
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 2 00028 mdash 00028 mdash 00030 mdash2minus 3 16948e minus 04 40462 17328e minus 04 40142 18411e minus 04 402632minus 4 10522e minus 05 40096 10759e minus 05 40095 11434e minus 05 400922minus 5 65652e minus 07 40024 67133e minus 07 40024 71346e minus 07 40024
14 Mathematical Problems in Engineering
constant and for different α we have different f(x t)
accordinglyWe apply the fast compact difference scheme to dis-
cretize the equation In Figure 2 we setc 0001 α 15 andM N 100 and plot exact andnumerical solutions at time T 1 for Example 2 with dif-ferent x0 For x0 05 α 15 andM N 100 we alsoplot exact and numerical solutions with the different c inFigure 3 In Figure 4 for h 01 and α 15 we vary thetotal number of time steps N to plot out the CPU time (inseconds) of the FCD scheme and DCD scheme )e nu-merical experiments verified our theoretical results In Ta-ble 6 by equating Δt h2 and fixing x0 05 we compute
the discrete linfin error and convergence rates with differentfractional derivative orders α and different c It shows thatthe compact finite difference scheme has space accuracy offourth order and temporal accuracy of second order We seth
Δt
radicand fix c 001 and the discrete l2 error and
convergence rates with different α and x0 are displayed inTable 7 )ese numerical convergence rates are almostapproaching O(Δt2 + h4) as Example 1
Example 3 If the exact solution is given byu(x t) etsin(πx)sin(πy) we have different f(x y t) fordifferent α accordingly
zu(x y t)
zt+
zαu(x y t)
ztα
z2u(x y t)
zx2 +z2u(x y t)
zy2 + f(x y t)
u(x 0) sin(πx)sin(πy)
zu
zt | t0 sin(πx)sin(πy)
u(0 y t) u(1 y t) u(x 0 t) u(x 1 t) 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(75)
In Figure 5 h 01 and α 175 are fixed and the totalnumber of time steps N vary to plot out the CPU time (inseconds) of the FCD procedure and DCD procedure and itpresents an approximately linear computation complexityfor FCD procedure We set Δt h2 in Table 8 and h
Δt
radic
in Table 9 the discrete linfin error discrete l2 error andconvergence rates with different derivative orders α arepresented )e fourth-order space accuracy and second-order temporal accuracy can be observed clearly
7 Conclusion
In this paper we develop and analyze a fast compact finitedifference procedure for the Cattaneo equation equipped withtime-fractional derivative without singular kernel )e time-fractional derivative is of CaputondashFabrizio type with the orderof α(1lt αlt 2) Compact difference discretization is appliedto obtain a high-order approximation for spatial derivatives ofinteger order in the partial differential equation and theCaputondashFabrizio fractional derivative is discretized by means
of CrankndashNicolson approximation It has been proved thatthe proposed compact finite difference scheme has spatialaccuracy of fourth order and temporal accuracy of secondorder Since the fractional derivatives are history dependentand nonlocal huge memory for storage and computationalcost are required )is means extreme difficulty especially fora long-time simulation Enlightened by the treatment forCaputo fractional derivative [32] we develop an effective fastevaluation procedure for the new CaputondashFabrizio fractionalderivative for the compact finite difference scheme Severalnumerical experiments have been carried out to show theconvergence orders and applicability of the scheme
Inspired by the work [43] the topic about modelling andnumerical solutions of porous media flow equipped withfractional derivatives is very interesting and challenging andwill be our main research direction in the future
Data Availability
All data generated or analyzed during this study are includedin this article
Table 9 Considering h Δt
radic the discrete l2 error and convergence rates of u with different α for Example 3
Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
2minus 4 00014 mdash 00014 mdash 00015 mdash2minus 6 84738e minus 05 40463 86640e minus 05 40142 92056e minus 05 402632minus 8 52610e minus 06 40096 53795e minus 06 40095 57168e minus 06 400922minus 10 32826e minus 07 40024 33566e minus 07 40024 35673e minus 07 40023
Mathematical Problems in Engineering 15
Conflicts of Interest
)e authors declare that they have no conflicts of interest
Acknowledgments
)is work was supported in part by the National NaturalScience Foundation of China under Grant nos 91630207and 11971272
References
[1] A Chaves ldquoA fractional diffusion equation to describe Levyflightsrdquo Physics Letters A vol 239 no 1-2 pp 13ndash16 1998
[2] M Giona and H E Roman ldquoFractional diffusion equation fortransport phenomena in random mediardquo Physica A Statis-tical Mechanics and its Applications vol 185 no 1ndash4pp 87ndash97 1992
[3] R Metzler and J Klafter ldquo)e random walkrsquos guide toanomalous diffusion a fractional dynamics approachrdquoPhysics Reports vol 339 no 1 pp 1ndash77 2000
[4] R L Magin Fractional Calculus in Bioengineering BegellHouse Redding Redding CT USA 2006
[5] R L Magin O Abdullah D Baleanu and X J ZhouldquoAnomalous diffusion expressed through fractional orderdifferential operators in the Bloch-Torrey equationrdquo Journalof Magnetic Resonance vol 190 no 2 pp 255ndash270 2008
[6] I Podlubny Fractional Differential Equations An Introduc-tion to Fractional Derivatives Fractional Differential Equa-tions to Methods of Beir Solution and Some of BeirApplications Vol 198 Elsevier Amsterdam Netherlands1998
[7] M Ran and C Zhang ldquoCompact difference scheme for a classof fractional-in-space nonlinear damped wave equations intwo space dimensionsrdquo Computers amp Mathematics withApplications vol 71 no 5 pp 1151ndash1162 2016
[8] A H Bhrawy and M A Zaky ldquoHighly accurate numericalschemes for multi-dimensional space variable-order frac-tional Schrodinger equationsrdquo Computers amp Mathematicswith Applications vol 73 no 6 pp 1100ndash1117 2017
[9] S B Yuste ldquoWeighted average finite difference methods forfractional diffusion equationsrdquo Journal of ComputationalPhysics vol 216 no 1 pp 264ndash274 2006
[10] T A M Langlands and B I Henry ldquo)e accuracy andstability of an implicit solution method for the fractionaldiffusion equationrdquo Journal of Computational Physicsvol 205 no 2 pp 719ndash736 2005
[11] C-M Chen F Liu I Turner and V Anh ldquoA Fourier method forthe fractional diffusion equation describing sub-diffusionrdquo Journalof Computational Physics vol 227 no 2 pp 886ndash897 2007
[12] Y Lin and C Xu ldquoFinite differencespectral approximationsfor the time-fractional diffusion equationrdquo Journal of Com-putational Physics vol 225 no 2 pp 1533ndash1552 2007
[13] G-h Gao and Z-z Sun ldquoA compact finite difference schemefor the fractional sub-diffusion equationsrdquo Journal of Com-putational Physics vol 230 no 3 pp 586ndash595 2011
[14] F Zeng C Li F Liu and I Turner ldquoNumerical algorithms fortime-fractional subdiffusion equation with second-order ac-curacyrdquo SIAM Journal on Scientific Computing vol 37 no 1pp A55ndashA78 2015
[15] G-H Gao Z-Z Sun and H-W Zhang ldquoA new fractionalnumerical differentiation formula to approximate the Caputofractional derivative and its applicationsrdquo Journal of Com-putational Physics vol 259 no 2 pp 33ndash50 2014
[16] H Li J Cao and C Li ldquoHigh-order approximation to Caputoderivatives and Caputo-type advection-diffusion equations(III)rdquo Journal of Computational and Applied Mathematicsvol 299 pp 159ndash175 2016
[17] A A Alikhanov ldquoA new difference scheme for the timefractional diffusion equationrdquo Journal of ComputationalPhysics vol 280 pp 424ndash438 2015
[18] Z Wang and S Vong ldquoCompact difference schemes for themodified anomalous fractional sub-diffusion equation and thefractional diffusion-wave equationrdquo Journal of ComputationalPhysics vol 277 pp 1ndash15 2014
[19] H-K Pang and H-W Sun ldquoFourth order finite differenceschemes for time-space fractional sub-diffusion equationsrdquoComputers amp Mathematics with Applications vol 71 no 6pp 1287ndash1302 2016
[20] M Dehghan M Abbaszadeh and W Deng ldquoFourth-ordernumerical method for the space-time tempered fractionaldiffusion-wave equationrdquo Applied Mathematics Lettersvol 73 no 11 pp 120ndash127 2017
[21] H Sun Z-Z Sun and G-H Gao ldquoSome high order dif-ference schemes for the space and time fractional Bloch-Torrey equationsrdquo Applied Mathematics and Computationvol 281 pp 356ndash380 2016
[22] S Arshad J Huang A Q M Khaliq and Y Tang ldquoTrape-zoidal scheme for time-space fractional diffusion equationwith Riesz derivativerdquo Journal of Computational Physicsvol 350 pp 1ndash15 2017
[23] H Wang K Wang and T Sircar ldquoA direct O(Nlog2N) finitedifferencemethod for fractional diffusion equationsrdquo Journal ofComputational Physics vol 229 no 21 pp 8095ndash8104 2010
[24] H Wang and T S Basu ldquoA fast finite difference method fortwo-dimensional space-fractional diffusion equationsrdquo SIAMJournal on Scientific Computing vol 34 no 5 pp A2444ndashA2458 2012
[25] H Wang and N Du ldquoA superfast-preconditioned iterativemethod for steady-state space-fractional diffusion equationsrdquoJournal of Computational Physics vol 240 pp 49ndash57 2013
[26] H Wang and N Du ldquoA fast finite difference method forthree-dimensional time-dependent space-fractional diffusionequations and its efficient implementationrdquo Journal ofComputational Physics vol 253 pp 50ndash63 2013
[27] H Wang and N Du ldquoFast alternating-direction finite dif-ference methods for three-dimensional space-fractional dif-fusion equationsrdquo Journal of Computational Physics vol 258pp 305ndash318 2014
[28] K Wang and H Wang ldquoA fast characteristic finite differencemethod for fractional advection-diffusion equationsrdquo Ad-vances in Water Resources vol 34 no 7 pp 810ndash816 2011
[29] H Wang and H Tian ldquoA fast Galerkin method with efficientmatrix assembly and storage for a peridynamic modelrdquoJournal of Computational Physics vol 231 no 23pp 7730ndash7738 2012
[30] H Tian H Wang and W Wang ldquoAn efficient collocationmethod for a non-local diffusion modelrdquo InternationalJournal of Numerical Analysis amp Modeling vol 10 no 4pp 815ndash825 2013
[31] H Wang and H Tian ldquoA fast and faithful collocation methodwith efficient matrix assembly for a two-dimensional nonlocaldiffusion modelrdquo Computer Methods in Applied Mechanicsand Engineering vol 273 no 5 pp 19ndash36 2014
[32] S Jiang J Zhang Q Zhang and Z Zhang ldquoFast evaluation ofthe Caputo fractional derivative and its applications tofractional diffusion equationsrdquo Communications in Compu-tational Physics vol 21 no 3 pp 650ndash678 2017
16 Mathematical Problems in Engineering
[33] Y Yan Z-Z Sun and J Zhang ldquoFast evaluation of the Caputofractional derivative and its applications to fractional diffusionequations a second-order schemerdquo Communications inComputational Physics vol 22 no 4 pp 1028ndash1048 2017
[34] X Lu H-K Pang and H-W Sun ldquoFast approximate in-version of a block triangular Toeplitz matrix with applicationsto fractional sub-diffusion equationsrdquo Numerical Linear Al-gebra with Applications vol 22 no 5 pp 866ndash882 2015
[35] M Caputo and M Fabrizio ldquoA new definition of fractionalderivative without singular kernelrdquo Progress in FractionalDifferentiation and Applications vol 1 no 2 pp 73ndash85 2015
[36] A Atangana ldquoOn the new fractional derivative and appli-cation to nonlinear Fisherrsquos reaction-diffusion equationrdquoAppliedMathematics and Computation vol 273 pp 948ndash9562016
[37] M D Ortigueira and J Tenreiro Machado ldquoA critical analysisof the Caputo-Fabrizio operatorrdquo Communications in Non-linear Science and Numerical Simulation vol 59 pp 608ndash6112018
[38] V E Tarasov ldquoNo nonlocality no fractional derivativerdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 62 pp 157ndash163 2018
[39] V Tarasov ldquoCaputo-Fabrizio operator in terms of integerderivatives memory or distributed lagrdquo Computational andApplied Mathematics vol 38 p 113 2019
[40] A Giusti ldquoA comment on some new definitions of fractionalderivativerdquoNonlinear Dynamics vol 93 no 3 pp 1757ndash17632018
[41] A Atangana and J F Gomez-Aguilar ldquoFractional derivativeswith no-index law property application to chaos and sta-tisticsrdquo Chaos Solitons amp Fractals vol 114 pp 516ndash535 2018
[42] J Hristov ldquoResponse functions in linear viscoelastic consti-tutive equations and related fractional operatorsrdquo Mathe-matical Modelling of Natural Phenomena vol 14 no 3pp 1ndash34 2019
[43] L X Vivas-Cruz A Gonzalez-Calderon M A Taneco-Hernandez and D P Luis ldquo)eoretical analysis of a model offluid flow in a reservoir with the Caputo-Fabrizio operatorrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 84 Article ID 105186 2020
[44] H-L Liao and Z-Z Sun ldquoMaximum norm error bounds ofADI and compact ADI methods for solving parabolicequationsrdquo Numerical Methods for Partial DifferentialEquations vol 26 no 1 pp 37ndash60 2010
[45] Z Liu A Cheng and X Li ldquoA second order Crank-Nicolsonscheme for fractional Cattaneo equation based on newfractional derivativerdquoAppliedMathematics and Computationvol 311 pp 361ndash374 2017
[46] Z Sun and G Gao Finite Difference Method for FractionalDifferential Equations Science Press Beijing China 2015
Mathematical Problems in Engineering 17
that the performance of the FCD scheme will be moreconspicuous as the time step size Δt decreases
In Figure 1 we set h 01 and α 175 and change thetotal number of time steps N to plot out the CPU time (inseconds) of the FCD scheme and DCD scheme We canobserve that the CPU time increases almost linearly withrespect to N for the FCD scheme while the DCD schemescales like O(N2)
Tables 4 and 5 show the discrete linfin errors and con-vergence rates of the compact finite difference scheme forExample 1 )e space rates are almost O(h4) for fixedΔt 2minus 13 and the time convergence rates are always O(Δt2)for fixed h 0001 We can conclude that the numericalconvergence rates of our scheme approach almost toO(Δt2 + h4)
Example 2 )e example is described by
zu(x t)
zt+
zαu(x t)
ztα
z2u(x t)
zx2 + f(x t)
u(x 0) 0zu
zt
1113868111386811138681113868111386811138681113868t0 0
u(0 t) t2exp minusx20
c1113888 1113889 u(1 t) t2exp minus
1 minus x0( 11138572
c1113888 1113889
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(74)
Note that the exact solution of the above problem isu(x t) t2exp(minus (x minus x0)
2c) where x0 isin [0 1] and c is a
N times104
0
1000
2000
3000
4000
5000
6000
7000
8000CP
U ti
me
Fast evaluationDirect evaluation
0 05 1 15 2 25 3 35 4 45 5
(a)
Fast evaluationDirect evaluation
log10 (N)
0
05
1
15
2
25
3
35
4
log1
0 (C
PU ti
me)
25 3 35 4 45 5
(b)
Figure 4 Considering x0 05 c 001 α 15 andM 10 the CPU time (a) and the log-log CPU time (b) versus the total number oftime steps N for Example 2
Table 6 Considering Δt h2 andx0 05 the discrete linfin error and convergence rates of u with different α and c for Example 2
c hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
1
2minus 3 73224e minus 05 mdash 55643e minus 05 mdash 38478e minus 05 mdash2minus 4 45707e-06 40018 34695e minus 06 40034 23900e minus 06 400902minus 5 28558e minus 07 40004 21671e minus 07 40009 14913e minus 07 400242minus 6 17847e minus 08 40001 13543e minus 08 40001 93169e minus 09 40006
01
2minus 3 00012 mdash 00012 mdash 00012 mdash2minus 4 70462e minus 05 40900 69731e minus 05 41051 69007e minus 05 412012minus 5 43571e minus 06 40154 43110e minus 06 40157 42650e minus 06 401612minus 6 27160e minus 07 40038 26871e minus 07 40039 26583e minus 07 40040
001
2minus 4 00090 mdash 00090 mdash 00090 mdash2minus 5 49257e minus 04 41915 49235e minus 04 41922 49232e minus 04 419232minus 6 29868e minus 05 40437 29855e minus 05 40436 29854e minus 05 404362minus 7 18531e minus 06 40106 18523e minus 06 40106 18522e minus 06 40106
Mathematical Problems in Engineering 13
Table 7 Considering h Δt
radicand c 001 the discrete l2 error and convergence rates of u with different α andx0 for Example 2
x0 Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
02minus 8 00044 mdash 00044 mdash 00042 mdash2minus 10 24205e minus 04 41841 23721e minus 04 42133 23098e minus 04 418452minus 12 14684e minus 05 40430 14390e minus 05 40430 14013e minus 05 40429
0252minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14902e minus 04 41794 14894e minus 04 41802 14892e minus 04 418032minus 12 90680e minus 06 40386 90633e minus 06 40386 90616e minus 06 40386
052minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14848e minus 04 41846 14842e minus 04 41852 14842e minus 04 418522minus 12 90323e minus 06 40390 90288e minus 06 40390 90286e minus 06 40390
0752minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14902e minus 04 41794 14894e minus 04 41802 14892e minus 04 418032minus 12 90680e minus 06 40386 90633e minus 06 40386 90616e minus 06 40386
12minus 8 00044 mdash 00044 mdash 00042 mdash2minus 10 24205e minus 04 41841 23721e minus 04 42133 23098e minus 04 418452minus 12 14684e minus 05 40430 14390e minus 05 40430 14013e minus 05 40429
0 2 4 6 8 10times104
0
2000
4000
6000
8000
10000
12000
14000
N
CPU
tim
e
Fast evaluationDirect evaluation
(a)
Fast evaluationDirect evaluation
36 38 4 42 44 46 48 515
2
25
3
35
4
45
log10 (N)
log1
0 (C
PU ti
me)
(b)
Figure 5 )e CPU time (a) and the log-log CPU time (b) versus the total number of time steps N for Example 3
Table 8 Considering Δt h2 the discrete linfin error and convergence rates of u with different α for Example 3
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 2 00028 mdash 00028 mdash 00030 mdash2minus 3 16948e minus 04 40462 17328e minus 04 40142 18411e minus 04 402632minus 4 10522e minus 05 40096 10759e minus 05 40095 11434e minus 05 400922minus 5 65652e minus 07 40024 67133e minus 07 40024 71346e minus 07 40024
14 Mathematical Problems in Engineering
constant and for different α we have different f(x t)
accordinglyWe apply the fast compact difference scheme to dis-
cretize the equation In Figure 2 we setc 0001 α 15 andM N 100 and plot exact andnumerical solutions at time T 1 for Example 2 with dif-ferent x0 For x0 05 α 15 andM N 100 we alsoplot exact and numerical solutions with the different c inFigure 3 In Figure 4 for h 01 and α 15 we vary thetotal number of time steps N to plot out the CPU time (inseconds) of the FCD scheme and DCD scheme )e nu-merical experiments verified our theoretical results In Ta-ble 6 by equating Δt h2 and fixing x0 05 we compute
the discrete linfin error and convergence rates with differentfractional derivative orders α and different c It shows thatthe compact finite difference scheme has space accuracy offourth order and temporal accuracy of second order We seth
Δt
radicand fix c 001 and the discrete l2 error and
convergence rates with different α and x0 are displayed inTable 7 )ese numerical convergence rates are almostapproaching O(Δt2 + h4) as Example 1
Example 3 If the exact solution is given byu(x t) etsin(πx)sin(πy) we have different f(x y t) fordifferent α accordingly
zu(x y t)
zt+
zαu(x y t)
ztα
z2u(x y t)
zx2 +z2u(x y t)
zy2 + f(x y t)
u(x 0) sin(πx)sin(πy)
zu
zt | t0 sin(πx)sin(πy)
u(0 y t) u(1 y t) u(x 0 t) u(x 1 t) 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(75)
In Figure 5 h 01 and α 175 are fixed and the totalnumber of time steps N vary to plot out the CPU time (inseconds) of the FCD procedure and DCD procedure and itpresents an approximately linear computation complexityfor FCD procedure We set Δt h2 in Table 8 and h
Δt
radic
in Table 9 the discrete linfin error discrete l2 error andconvergence rates with different derivative orders α arepresented )e fourth-order space accuracy and second-order temporal accuracy can be observed clearly
7 Conclusion
In this paper we develop and analyze a fast compact finitedifference procedure for the Cattaneo equation equipped withtime-fractional derivative without singular kernel )e time-fractional derivative is of CaputondashFabrizio type with the orderof α(1lt αlt 2) Compact difference discretization is appliedto obtain a high-order approximation for spatial derivatives ofinteger order in the partial differential equation and theCaputondashFabrizio fractional derivative is discretized by means
of CrankndashNicolson approximation It has been proved thatthe proposed compact finite difference scheme has spatialaccuracy of fourth order and temporal accuracy of secondorder Since the fractional derivatives are history dependentand nonlocal huge memory for storage and computationalcost are required )is means extreme difficulty especially fora long-time simulation Enlightened by the treatment forCaputo fractional derivative [32] we develop an effective fastevaluation procedure for the new CaputondashFabrizio fractionalderivative for the compact finite difference scheme Severalnumerical experiments have been carried out to show theconvergence orders and applicability of the scheme
Inspired by the work [43] the topic about modelling andnumerical solutions of porous media flow equipped withfractional derivatives is very interesting and challenging andwill be our main research direction in the future
Data Availability
All data generated or analyzed during this study are includedin this article
Table 9 Considering h Δt
radic the discrete l2 error and convergence rates of u with different α for Example 3
Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
2minus 4 00014 mdash 00014 mdash 00015 mdash2minus 6 84738e minus 05 40463 86640e minus 05 40142 92056e minus 05 402632minus 8 52610e minus 06 40096 53795e minus 06 40095 57168e minus 06 400922minus 10 32826e minus 07 40024 33566e minus 07 40024 35673e minus 07 40023
Mathematical Problems in Engineering 15
Conflicts of Interest
)e authors declare that they have no conflicts of interest
Acknowledgments
)is work was supported in part by the National NaturalScience Foundation of China under Grant nos 91630207and 11971272
References
[1] A Chaves ldquoA fractional diffusion equation to describe Levyflightsrdquo Physics Letters A vol 239 no 1-2 pp 13ndash16 1998
[2] M Giona and H E Roman ldquoFractional diffusion equation fortransport phenomena in random mediardquo Physica A Statis-tical Mechanics and its Applications vol 185 no 1ndash4pp 87ndash97 1992
[3] R Metzler and J Klafter ldquo)e random walkrsquos guide toanomalous diffusion a fractional dynamics approachrdquoPhysics Reports vol 339 no 1 pp 1ndash77 2000
[4] R L Magin Fractional Calculus in Bioengineering BegellHouse Redding Redding CT USA 2006
[5] R L Magin O Abdullah D Baleanu and X J ZhouldquoAnomalous diffusion expressed through fractional orderdifferential operators in the Bloch-Torrey equationrdquo Journalof Magnetic Resonance vol 190 no 2 pp 255ndash270 2008
[6] I Podlubny Fractional Differential Equations An Introduc-tion to Fractional Derivatives Fractional Differential Equa-tions to Methods of Beir Solution and Some of BeirApplications Vol 198 Elsevier Amsterdam Netherlands1998
[7] M Ran and C Zhang ldquoCompact difference scheme for a classof fractional-in-space nonlinear damped wave equations intwo space dimensionsrdquo Computers amp Mathematics withApplications vol 71 no 5 pp 1151ndash1162 2016
[8] A H Bhrawy and M A Zaky ldquoHighly accurate numericalschemes for multi-dimensional space variable-order frac-tional Schrodinger equationsrdquo Computers amp Mathematicswith Applications vol 73 no 6 pp 1100ndash1117 2017
[9] S B Yuste ldquoWeighted average finite difference methods forfractional diffusion equationsrdquo Journal of ComputationalPhysics vol 216 no 1 pp 264ndash274 2006
[10] T A M Langlands and B I Henry ldquo)e accuracy andstability of an implicit solution method for the fractionaldiffusion equationrdquo Journal of Computational Physicsvol 205 no 2 pp 719ndash736 2005
[11] C-M Chen F Liu I Turner and V Anh ldquoA Fourier method forthe fractional diffusion equation describing sub-diffusionrdquo Journalof Computational Physics vol 227 no 2 pp 886ndash897 2007
[12] Y Lin and C Xu ldquoFinite differencespectral approximationsfor the time-fractional diffusion equationrdquo Journal of Com-putational Physics vol 225 no 2 pp 1533ndash1552 2007
[13] G-h Gao and Z-z Sun ldquoA compact finite difference schemefor the fractional sub-diffusion equationsrdquo Journal of Com-putational Physics vol 230 no 3 pp 586ndash595 2011
[14] F Zeng C Li F Liu and I Turner ldquoNumerical algorithms fortime-fractional subdiffusion equation with second-order ac-curacyrdquo SIAM Journal on Scientific Computing vol 37 no 1pp A55ndashA78 2015
[15] G-H Gao Z-Z Sun and H-W Zhang ldquoA new fractionalnumerical differentiation formula to approximate the Caputofractional derivative and its applicationsrdquo Journal of Com-putational Physics vol 259 no 2 pp 33ndash50 2014
[16] H Li J Cao and C Li ldquoHigh-order approximation to Caputoderivatives and Caputo-type advection-diffusion equations(III)rdquo Journal of Computational and Applied Mathematicsvol 299 pp 159ndash175 2016
[17] A A Alikhanov ldquoA new difference scheme for the timefractional diffusion equationrdquo Journal of ComputationalPhysics vol 280 pp 424ndash438 2015
[18] Z Wang and S Vong ldquoCompact difference schemes for themodified anomalous fractional sub-diffusion equation and thefractional diffusion-wave equationrdquo Journal of ComputationalPhysics vol 277 pp 1ndash15 2014
[19] H-K Pang and H-W Sun ldquoFourth order finite differenceschemes for time-space fractional sub-diffusion equationsrdquoComputers amp Mathematics with Applications vol 71 no 6pp 1287ndash1302 2016
[20] M Dehghan M Abbaszadeh and W Deng ldquoFourth-ordernumerical method for the space-time tempered fractionaldiffusion-wave equationrdquo Applied Mathematics Lettersvol 73 no 11 pp 120ndash127 2017
[21] H Sun Z-Z Sun and G-H Gao ldquoSome high order dif-ference schemes for the space and time fractional Bloch-Torrey equationsrdquo Applied Mathematics and Computationvol 281 pp 356ndash380 2016
[22] S Arshad J Huang A Q M Khaliq and Y Tang ldquoTrape-zoidal scheme for time-space fractional diffusion equationwith Riesz derivativerdquo Journal of Computational Physicsvol 350 pp 1ndash15 2017
[23] H Wang K Wang and T Sircar ldquoA direct O(Nlog2N) finitedifferencemethod for fractional diffusion equationsrdquo Journal ofComputational Physics vol 229 no 21 pp 8095ndash8104 2010
[24] H Wang and T S Basu ldquoA fast finite difference method fortwo-dimensional space-fractional diffusion equationsrdquo SIAMJournal on Scientific Computing vol 34 no 5 pp A2444ndashA2458 2012
[25] H Wang and N Du ldquoA superfast-preconditioned iterativemethod for steady-state space-fractional diffusion equationsrdquoJournal of Computational Physics vol 240 pp 49ndash57 2013
[26] H Wang and N Du ldquoA fast finite difference method forthree-dimensional time-dependent space-fractional diffusionequations and its efficient implementationrdquo Journal ofComputational Physics vol 253 pp 50ndash63 2013
[27] H Wang and N Du ldquoFast alternating-direction finite dif-ference methods for three-dimensional space-fractional dif-fusion equationsrdquo Journal of Computational Physics vol 258pp 305ndash318 2014
[28] K Wang and H Wang ldquoA fast characteristic finite differencemethod for fractional advection-diffusion equationsrdquo Ad-vances in Water Resources vol 34 no 7 pp 810ndash816 2011
[29] H Wang and H Tian ldquoA fast Galerkin method with efficientmatrix assembly and storage for a peridynamic modelrdquoJournal of Computational Physics vol 231 no 23pp 7730ndash7738 2012
[30] H Tian H Wang and W Wang ldquoAn efficient collocationmethod for a non-local diffusion modelrdquo InternationalJournal of Numerical Analysis amp Modeling vol 10 no 4pp 815ndash825 2013
[31] H Wang and H Tian ldquoA fast and faithful collocation methodwith efficient matrix assembly for a two-dimensional nonlocaldiffusion modelrdquo Computer Methods in Applied Mechanicsand Engineering vol 273 no 5 pp 19ndash36 2014
[32] S Jiang J Zhang Q Zhang and Z Zhang ldquoFast evaluation ofthe Caputo fractional derivative and its applications tofractional diffusion equationsrdquo Communications in Compu-tational Physics vol 21 no 3 pp 650ndash678 2017
16 Mathematical Problems in Engineering
[33] Y Yan Z-Z Sun and J Zhang ldquoFast evaluation of the Caputofractional derivative and its applications to fractional diffusionequations a second-order schemerdquo Communications inComputational Physics vol 22 no 4 pp 1028ndash1048 2017
[34] X Lu H-K Pang and H-W Sun ldquoFast approximate in-version of a block triangular Toeplitz matrix with applicationsto fractional sub-diffusion equationsrdquo Numerical Linear Al-gebra with Applications vol 22 no 5 pp 866ndash882 2015
[35] M Caputo and M Fabrizio ldquoA new definition of fractionalderivative without singular kernelrdquo Progress in FractionalDifferentiation and Applications vol 1 no 2 pp 73ndash85 2015
[36] A Atangana ldquoOn the new fractional derivative and appli-cation to nonlinear Fisherrsquos reaction-diffusion equationrdquoAppliedMathematics and Computation vol 273 pp 948ndash9562016
[37] M D Ortigueira and J Tenreiro Machado ldquoA critical analysisof the Caputo-Fabrizio operatorrdquo Communications in Non-linear Science and Numerical Simulation vol 59 pp 608ndash6112018
[38] V E Tarasov ldquoNo nonlocality no fractional derivativerdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 62 pp 157ndash163 2018
[39] V Tarasov ldquoCaputo-Fabrizio operator in terms of integerderivatives memory or distributed lagrdquo Computational andApplied Mathematics vol 38 p 113 2019
[40] A Giusti ldquoA comment on some new definitions of fractionalderivativerdquoNonlinear Dynamics vol 93 no 3 pp 1757ndash17632018
[41] A Atangana and J F Gomez-Aguilar ldquoFractional derivativeswith no-index law property application to chaos and sta-tisticsrdquo Chaos Solitons amp Fractals vol 114 pp 516ndash535 2018
[42] J Hristov ldquoResponse functions in linear viscoelastic consti-tutive equations and related fractional operatorsrdquo Mathe-matical Modelling of Natural Phenomena vol 14 no 3pp 1ndash34 2019
[43] L X Vivas-Cruz A Gonzalez-Calderon M A Taneco-Hernandez and D P Luis ldquo)eoretical analysis of a model offluid flow in a reservoir with the Caputo-Fabrizio operatorrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 84 Article ID 105186 2020
[44] H-L Liao and Z-Z Sun ldquoMaximum norm error bounds ofADI and compact ADI methods for solving parabolicequationsrdquo Numerical Methods for Partial DifferentialEquations vol 26 no 1 pp 37ndash60 2010
[45] Z Liu A Cheng and X Li ldquoA second order Crank-Nicolsonscheme for fractional Cattaneo equation based on newfractional derivativerdquoAppliedMathematics and Computationvol 311 pp 361ndash374 2017
[46] Z Sun and G Gao Finite Difference Method for FractionalDifferential Equations Science Press Beijing China 2015
Mathematical Problems in Engineering 17
Table 7 Considering h Δt
radicand c 001 the discrete l2 error and convergence rates of u with different α andx0 for Example 2
x0 Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
02minus 8 00044 mdash 00044 mdash 00042 mdash2minus 10 24205e minus 04 41841 23721e minus 04 42133 23098e minus 04 418452minus 12 14684e minus 05 40430 14390e minus 05 40430 14013e minus 05 40429
0252minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14902e minus 04 41794 14894e minus 04 41802 14892e minus 04 418032minus 12 90680e minus 06 40386 90633e minus 06 40386 90616e minus 06 40386
052minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14848e minus 04 41846 14842e minus 04 41852 14842e minus 04 418522minus 12 90323e minus 06 40390 90288e minus 06 40390 90286e minus 06 40390
0752minus 8 00027 mdash 00027 mdash 00027 mdash2minus 10 14902e minus 04 41794 14894e minus 04 41802 14892e minus 04 418032minus 12 90680e minus 06 40386 90633e minus 06 40386 90616e minus 06 40386
12minus 8 00044 mdash 00044 mdash 00042 mdash2minus 10 24205e minus 04 41841 23721e minus 04 42133 23098e minus 04 418452minus 12 14684e minus 05 40430 14390e minus 05 40430 14013e minus 05 40429
0 2 4 6 8 10times104
0
2000
4000
6000
8000
10000
12000
14000
N
CPU
tim
e
Fast evaluationDirect evaluation
(a)
Fast evaluationDirect evaluation
36 38 4 42 44 46 48 515
2
25
3
35
4
45
log10 (N)
log1
0 (C
PU ti
me)
(b)
Figure 5 )e CPU time (a) and the log-log CPU time (b) versus the total number of time steps N for Example 3
Table 8 Considering Δt h2 the discrete linfin error and convergence rates of u with different α for Example 3
hα 125 α 15 α 175
eNinfin Order eNinfin Order eNinfin Order
2minus 2 00028 mdash 00028 mdash 00030 mdash2minus 3 16948e minus 04 40462 17328e minus 04 40142 18411e minus 04 402632minus 4 10522e minus 05 40096 10759e minus 05 40095 11434e minus 05 400922minus 5 65652e minus 07 40024 67133e minus 07 40024 71346e minus 07 40024
14 Mathematical Problems in Engineering
constant and for different α we have different f(x t)
accordinglyWe apply the fast compact difference scheme to dis-
cretize the equation In Figure 2 we setc 0001 α 15 andM N 100 and plot exact andnumerical solutions at time T 1 for Example 2 with dif-ferent x0 For x0 05 α 15 andM N 100 we alsoplot exact and numerical solutions with the different c inFigure 3 In Figure 4 for h 01 and α 15 we vary thetotal number of time steps N to plot out the CPU time (inseconds) of the FCD scheme and DCD scheme )e nu-merical experiments verified our theoretical results In Ta-ble 6 by equating Δt h2 and fixing x0 05 we compute
the discrete linfin error and convergence rates with differentfractional derivative orders α and different c It shows thatthe compact finite difference scheme has space accuracy offourth order and temporal accuracy of second order We seth
Δt
radicand fix c 001 and the discrete l2 error and
convergence rates with different α and x0 are displayed inTable 7 )ese numerical convergence rates are almostapproaching O(Δt2 + h4) as Example 1
Example 3 If the exact solution is given byu(x t) etsin(πx)sin(πy) we have different f(x y t) fordifferent α accordingly
zu(x y t)
zt+
zαu(x y t)
ztα
z2u(x y t)
zx2 +z2u(x y t)
zy2 + f(x y t)
u(x 0) sin(πx)sin(πy)
zu
zt | t0 sin(πx)sin(πy)
u(0 y t) u(1 y t) u(x 0 t) u(x 1 t) 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(75)
In Figure 5 h 01 and α 175 are fixed and the totalnumber of time steps N vary to plot out the CPU time (inseconds) of the FCD procedure and DCD procedure and itpresents an approximately linear computation complexityfor FCD procedure We set Δt h2 in Table 8 and h
Δt
radic
in Table 9 the discrete linfin error discrete l2 error andconvergence rates with different derivative orders α arepresented )e fourth-order space accuracy and second-order temporal accuracy can be observed clearly
7 Conclusion
In this paper we develop and analyze a fast compact finitedifference procedure for the Cattaneo equation equipped withtime-fractional derivative without singular kernel )e time-fractional derivative is of CaputondashFabrizio type with the orderof α(1lt αlt 2) Compact difference discretization is appliedto obtain a high-order approximation for spatial derivatives ofinteger order in the partial differential equation and theCaputondashFabrizio fractional derivative is discretized by means
of CrankndashNicolson approximation It has been proved thatthe proposed compact finite difference scheme has spatialaccuracy of fourth order and temporal accuracy of secondorder Since the fractional derivatives are history dependentand nonlocal huge memory for storage and computationalcost are required )is means extreme difficulty especially fora long-time simulation Enlightened by the treatment forCaputo fractional derivative [32] we develop an effective fastevaluation procedure for the new CaputondashFabrizio fractionalderivative for the compact finite difference scheme Severalnumerical experiments have been carried out to show theconvergence orders and applicability of the scheme
Inspired by the work [43] the topic about modelling andnumerical solutions of porous media flow equipped withfractional derivatives is very interesting and challenging andwill be our main research direction in the future
Data Availability
All data generated or analyzed during this study are includedin this article
Table 9 Considering h Δt
radic the discrete l2 error and convergence rates of u with different α for Example 3
Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
2minus 4 00014 mdash 00014 mdash 00015 mdash2minus 6 84738e minus 05 40463 86640e minus 05 40142 92056e minus 05 402632minus 8 52610e minus 06 40096 53795e minus 06 40095 57168e minus 06 400922minus 10 32826e minus 07 40024 33566e minus 07 40024 35673e minus 07 40023
Mathematical Problems in Engineering 15
Conflicts of Interest
)e authors declare that they have no conflicts of interest
Acknowledgments
)is work was supported in part by the National NaturalScience Foundation of China under Grant nos 91630207and 11971272
References
[1] A Chaves ldquoA fractional diffusion equation to describe Levyflightsrdquo Physics Letters A vol 239 no 1-2 pp 13ndash16 1998
[2] M Giona and H E Roman ldquoFractional diffusion equation fortransport phenomena in random mediardquo Physica A Statis-tical Mechanics and its Applications vol 185 no 1ndash4pp 87ndash97 1992
[3] R Metzler and J Klafter ldquo)e random walkrsquos guide toanomalous diffusion a fractional dynamics approachrdquoPhysics Reports vol 339 no 1 pp 1ndash77 2000
[4] R L Magin Fractional Calculus in Bioengineering BegellHouse Redding Redding CT USA 2006
[5] R L Magin O Abdullah D Baleanu and X J ZhouldquoAnomalous diffusion expressed through fractional orderdifferential operators in the Bloch-Torrey equationrdquo Journalof Magnetic Resonance vol 190 no 2 pp 255ndash270 2008
[6] I Podlubny Fractional Differential Equations An Introduc-tion to Fractional Derivatives Fractional Differential Equa-tions to Methods of Beir Solution and Some of BeirApplications Vol 198 Elsevier Amsterdam Netherlands1998
[7] M Ran and C Zhang ldquoCompact difference scheme for a classof fractional-in-space nonlinear damped wave equations intwo space dimensionsrdquo Computers amp Mathematics withApplications vol 71 no 5 pp 1151ndash1162 2016
[8] A H Bhrawy and M A Zaky ldquoHighly accurate numericalschemes for multi-dimensional space variable-order frac-tional Schrodinger equationsrdquo Computers amp Mathematicswith Applications vol 73 no 6 pp 1100ndash1117 2017
[9] S B Yuste ldquoWeighted average finite difference methods forfractional diffusion equationsrdquo Journal of ComputationalPhysics vol 216 no 1 pp 264ndash274 2006
[10] T A M Langlands and B I Henry ldquo)e accuracy andstability of an implicit solution method for the fractionaldiffusion equationrdquo Journal of Computational Physicsvol 205 no 2 pp 719ndash736 2005
[11] C-M Chen F Liu I Turner and V Anh ldquoA Fourier method forthe fractional diffusion equation describing sub-diffusionrdquo Journalof Computational Physics vol 227 no 2 pp 886ndash897 2007
[12] Y Lin and C Xu ldquoFinite differencespectral approximationsfor the time-fractional diffusion equationrdquo Journal of Com-putational Physics vol 225 no 2 pp 1533ndash1552 2007
[13] G-h Gao and Z-z Sun ldquoA compact finite difference schemefor the fractional sub-diffusion equationsrdquo Journal of Com-putational Physics vol 230 no 3 pp 586ndash595 2011
[14] F Zeng C Li F Liu and I Turner ldquoNumerical algorithms fortime-fractional subdiffusion equation with second-order ac-curacyrdquo SIAM Journal on Scientific Computing vol 37 no 1pp A55ndashA78 2015
[15] G-H Gao Z-Z Sun and H-W Zhang ldquoA new fractionalnumerical differentiation formula to approximate the Caputofractional derivative and its applicationsrdquo Journal of Com-putational Physics vol 259 no 2 pp 33ndash50 2014
[16] H Li J Cao and C Li ldquoHigh-order approximation to Caputoderivatives and Caputo-type advection-diffusion equations(III)rdquo Journal of Computational and Applied Mathematicsvol 299 pp 159ndash175 2016
[17] A A Alikhanov ldquoA new difference scheme for the timefractional diffusion equationrdquo Journal of ComputationalPhysics vol 280 pp 424ndash438 2015
[18] Z Wang and S Vong ldquoCompact difference schemes for themodified anomalous fractional sub-diffusion equation and thefractional diffusion-wave equationrdquo Journal of ComputationalPhysics vol 277 pp 1ndash15 2014
[19] H-K Pang and H-W Sun ldquoFourth order finite differenceschemes for time-space fractional sub-diffusion equationsrdquoComputers amp Mathematics with Applications vol 71 no 6pp 1287ndash1302 2016
[20] M Dehghan M Abbaszadeh and W Deng ldquoFourth-ordernumerical method for the space-time tempered fractionaldiffusion-wave equationrdquo Applied Mathematics Lettersvol 73 no 11 pp 120ndash127 2017
[21] H Sun Z-Z Sun and G-H Gao ldquoSome high order dif-ference schemes for the space and time fractional Bloch-Torrey equationsrdquo Applied Mathematics and Computationvol 281 pp 356ndash380 2016
[22] S Arshad J Huang A Q M Khaliq and Y Tang ldquoTrape-zoidal scheme for time-space fractional diffusion equationwith Riesz derivativerdquo Journal of Computational Physicsvol 350 pp 1ndash15 2017
[23] H Wang K Wang and T Sircar ldquoA direct O(Nlog2N) finitedifferencemethod for fractional diffusion equationsrdquo Journal ofComputational Physics vol 229 no 21 pp 8095ndash8104 2010
[24] H Wang and T S Basu ldquoA fast finite difference method fortwo-dimensional space-fractional diffusion equationsrdquo SIAMJournal on Scientific Computing vol 34 no 5 pp A2444ndashA2458 2012
[25] H Wang and N Du ldquoA superfast-preconditioned iterativemethod for steady-state space-fractional diffusion equationsrdquoJournal of Computational Physics vol 240 pp 49ndash57 2013
[26] H Wang and N Du ldquoA fast finite difference method forthree-dimensional time-dependent space-fractional diffusionequations and its efficient implementationrdquo Journal ofComputational Physics vol 253 pp 50ndash63 2013
[27] H Wang and N Du ldquoFast alternating-direction finite dif-ference methods for three-dimensional space-fractional dif-fusion equationsrdquo Journal of Computational Physics vol 258pp 305ndash318 2014
[28] K Wang and H Wang ldquoA fast characteristic finite differencemethod for fractional advection-diffusion equationsrdquo Ad-vances in Water Resources vol 34 no 7 pp 810ndash816 2011
[29] H Wang and H Tian ldquoA fast Galerkin method with efficientmatrix assembly and storage for a peridynamic modelrdquoJournal of Computational Physics vol 231 no 23pp 7730ndash7738 2012
[30] H Tian H Wang and W Wang ldquoAn efficient collocationmethod for a non-local diffusion modelrdquo InternationalJournal of Numerical Analysis amp Modeling vol 10 no 4pp 815ndash825 2013
[31] H Wang and H Tian ldquoA fast and faithful collocation methodwith efficient matrix assembly for a two-dimensional nonlocaldiffusion modelrdquo Computer Methods in Applied Mechanicsand Engineering vol 273 no 5 pp 19ndash36 2014
[32] S Jiang J Zhang Q Zhang and Z Zhang ldquoFast evaluation ofthe Caputo fractional derivative and its applications tofractional diffusion equationsrdquo Communications in Compu-tational Physics vol 21 no 3 pp 650ndash678 2017
16 Mathematical Problems in Engineering
[33] Y Yan Z-Z Sun and J Zhang ldquoFast evaluation of the Caputofractional derivative and its applications to fractional diffusionequations a second-order schemerdquo Communications inComputational Physics vol 22 no 4 pp 1028ndash1048 2017
[34] X Lu H-K Pang and H-W Sun ldquoFast approximate in-version of a block triangular Toeplitz matrix with applicationsto fractional sub-diffusion equationsrdquo Numerical Linear Al-gebra with Applications vol 22 no 5 pp 866ndash882 2015
[35] M Caputo and M Fabrizio ldquoA new definition of fractionalderivative without singular kernelrdquo Progress in FractionalDifferentiation and Applications vol 1 no 2 pp 73ndash85 2015
[36] A Atangana ldquoOn the new fractional derivative and appli-cation to nonlinear Fisherrsquos reaction-diffusion equationrdquoAppliedMathematics and Computation vol 273 pp 948ndash9562016
[37] M D Ortigueira and J Tenreiro Machado ldquoA critical analysisof the Caputo-Fabrizio operatorrdquo Communications in Non-linear Science and Numerical Simulation vol 59 pp 608ndash6112018
[38] V E Tarasov ldquoNo nonlocality no fractional derivativerdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 62 pp 157ndash163 2018
[39] V Tarasov ldquoCaputo-Fabrizio operator in terms of integerderivatives memory or distributed lagrdquo Computational andApplied Mathematics vol 38 p 113 2019
[40] A Giusti ldquoA comment on some new definitions of fractionalderivativerdquoNonlinear Dynamics vol 93 no 3 pp 1757ndash17632018
[41] A Atangana and J F Gomez-Aguilar ldquoFractional derivativeswith no-index law property application to chaos and sta-tisticsrdquo Chaos Solitons amp Fractals vol 114 pp 516ndash535 2018
[42] J Hristov ldquoResponse functions in linear viscoelastic consti-tutive equations and related fractional operatorsrdquo Mathe-matical Modelling of Natural Phenomena vol 14 no 3pp 1ndash34 2019
[43] L X Vivas-Cruz A Gonzalez-Calderon M A Taneco-Hernandez and D P Luis ldquo)eoretical analysis of a model offluid flow in a reservoir with the Caputo-Fabrizio operatorrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 84 Article ID 105186 2020
[44] H-L Liao and Z-Z Sun ldquoMaximum norm error bounds ofADI and compact ADI methods for solving parabolicequationsrdquo Numerical Methods for Partial DifferentialEquations vol 26 no 1 pp 37ndash60 2010
[45] Z Liu A Cheng and X Li ldquoA second order Crank-Nicolsonscheme for fractional Cattaneo equation based on newfractional derivativerdquoAppliedMathematics and Computationvol 311 pp 361ndash374 2017
[46] Z Sun and G Gao Finite Difference Method for FractionalDifferential Equations Science Press Beijing China 2015
Mathematical Problems in Engineering 17
constant and for different α we have different f(x t)
accordinglyWe apply the fast compact difference scheme to dis-
cretize the equation In Figure 2 we setc 0001 α 15 andM N 100 and plot exact andnumerical solutions at time T 1 for Example 2 with dif-ferent x0 For x0 05 α 15 andM N 100 we alsoplot exact and numerical solutions with the different c inFigure 3 In Figure 4 for h 01 and α 15 we vary thetotal number of time steps N to plot out the CPU time (inseconds) of the FCD scheme and DCD scheme )e nu-merical experiments verified our theoretical results In Ta-ble 6 by equating Δt h2 and fixing x0 05 we compute
the discrete linfin error and convergence rates with differentfractional derivative orders α and different c It shows thatthe compact finite difference scheme has space accuracy offourth order and temporal accuracy of second order We seth
Δt
radicand fix c 001 and the discrete l2 error and
convergence rates with different α and x0 are displayed inTable 7 )ese numerical convergence rates are almostapproaching O(Δt2 + h4) as Example 1
Example 3 If the exact solution is given byu(x t) etsin(πx)sin(πy) we have different f(x y t) fordifferent α accordingly
zu(x y t)
zt+
zαu(x y t)
ztα
z2u(x y t)
zx2 +z2u(x y t)
zy2 + f(x y t)
u(x 0) sin(πx)sin(πy)
zu
zt | t0 sin(πx)sin(πy)
u(0 y t) u(1 y t) u(x 0 t) u(x 1 t) 0
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(75)
In Figure 5 h 01 and α 175 are fixed and the totalnumber of time steps N vary to plot out the CPU time (inseconds) of the FCD procedure and DCD procedure and itpresents an approximately linear computation complexityfor FCD procedure We set Δt h2 in Table 8 and h
Δt
radic
in Table 9 the discrete linfin error discrete l2 error andconvergence rates with different derivative orders α arepresented )e fourth-order space accuracy and second-order temporal accuracy can be observed clearly
7 Conclusion
In this paper we develop and analyze a fast compact finitedifference procedure for the Cattaneo equation equipped withtime-fractional derivative without singular kernel )e time-fractional derivative is of CaputondashFabrizio type with the orderof α(1lt αlt 2) Compact difference discretization is appliedto obtain a high-order approximation for spatial derivatives ofinteger order in the partial differential equation and theCaputondashFabrizio fractional derivative is discretized by means
of CrankndashNicolson approximation It has been proved thatthe proposed compact finite difference scheme has spatialaccuracy of fourth order and temporal accuracy of secondorder Since the fractional derivatives are history dependentand nonlocal huge memory for storage and computationalcost are required )is means extreme difficulty especially fora long-time simulation Enlightened by the treatment forCaputo fractional derivative [32] we develop an effective fastevaluation procedure for the new CaputondashFabrizio fractionalderivative for the compact finite difference scheme Severalnumerical experiments have been carried out to show theconvergence orders and applicability of the scheme
Inspired by the work [43] the topic about modelling andnumerical solutions of porous media flow equipped withfractional derivatives is very interesting and challenging andwill be our main research direction in the future
Data Availability
All data generated or analyzed during this study are includedin this article
Table 9 Considering h Δt
radic the discrete l2 error and convergence rates of u with different α for Example 3
Δtα 125 α 15 α 175
eN2 Order eN2 Order eN2 Order
2minus 4 00014 mdash 00014 mdash 00015 mdash2minus 6 84738e minus 05 40463 86640e minus 05 40142 92056e minus 05 402632minus 8 52610e minus 06 40096 53795e minus 06 40095 57168e minus 06 400922minus 10 32826e minus 07 40024 33566e minus 07 40024 35673e minus 07 40023
Mathematical Problems in Engineering 15
Conflicts of Interest
)e authors declare that they have no conflicts of interest
Acknowledgments
)is work was supported in part by the National NaturalScience Foundation of China under Grant nos 91630207and 11971272
References
[1] A Chaves ldquoA fractional diffusion equation to describe Levyflightsrdquo Physics Letters A vol 239 no 1-2 pp 13ndash16 1998
[2] M Giona and H E Roman ldquoFractional diffusion equation fortransport phenomena in random mediardquo Physica A Statis-tical Mechanics and its Applications vol 185 no 1ndash4pp 87ndash97 1992
[3] R Metzler and J Klafter ldquo)e random walkrsquos guide toanomalous diffusion a fractional dynamics approachrdquoPhysics Reports vol 339 no 1 pp 1ndash77 2000
[4] R L Magin Fractional Calculus in Bioengineering BegellHouse Redding Redding CT USA 2006
[5] R L Magin O Abdullah D Baleanu and X J ZhouldquoAnomalous diffusion expressed through fractional orderdifferential operators in the Bloch-Torrey equationrdquo Journalof Magnetic Resonance vol 190 no 2 pp 255ndash270 2008
[6] I Podlubny Fractional Differential Equations An Introduc-tion to Fractional Derivatives Fractional Differential Equa-tions to Methods of Beir Solution and Some of BeirApplications Vol 198 Elsevier Amsterdam Netherlands1998
[7] M Ran and C Zhang ldquoCompact difference scheme for a classof fractional-in-space nonlinear damped wave equations intwo space dimensionsrdquo Computers amp Mathematics withApplications vol 71 no 5 pp 1151ndash1162 2016
[8] A H Bhrawy and M A Zaky ldquoHighly accurate numericalschemes for multi-dimensional space variable-order frac-tional Schrodinger equationsrdquo Computers amp Mathematicswith Applications vol 73 no 6 pp 1100ndash1117 2017
[9] S B Yuste ldquoWeighted average finite difference methods forfractional diffusion equationsrdquo Journal of ComputationalPhysics vol 216 no 1 pp 264ndash274 2006
[10] T A M Langlands and B I Henry ldquo)e accuracy andstability of an implicit solution method for the fractionaldiffusion equationrdquo Journal of Computational Physicsvol 205 no 2 pp 719ndash736 2005
[11] C-M Chen F Liu I Turner and V Anh ldquoA Fourier method forthe fractional diffusion equation describing sub-diffusionrdquo Journalof Computational Physics vol 227 no 2 pp 886ndash897 2007
[12] Y Lin and C Xu ldquoFinite differencespectral approximationsfor the time-fractional diffusion equationrdquo Journal of Com-putational Physics vol 225 no 2 pp 1533ndash1552 2007
[13] G-h Gao and Z-z Sun ldquoA compact finite difference schemefor the fractional sub-diffusion equationsrdquo Journal of Com-putational Physics vol 230 no 3 pp 586ndash595 2011
[14] F Zeng C Li F Liu and I Turner ldquoNumerical algorithms fortime-fractional subdiffusion equation with second-order ac-curacyrdquo SIAM Journal on Scientific Computing vol 37 no 1pp A55ndashA78 2015
[15] G-H Gao Z-Z Sun and H-W Zhang ldquoA new fractionalnumerical differentiation formula to approximate the Caputofractional derivative and its applicationsrdquo Journal of Com-putational Physics vol 259 no 2 pp 33ndash50 2014
[16] H Li J Cao and C Li ldquoHigh-order approximation to Caputoderivatives and Caputo-type advection-diffusion equations(III)rdquo Journal of Computational and Applied Mathematicsvol 299 pp 159ndash175 2016
[17] A A Alikhanov ldquoA new difference scheme for the timefractional diffusion equationrdquo Journal of ComputationalPhysics vol 280 pp 424ndash438 2015
[18] Z Wang and S Vong ldquoCompact difference schemes for themodified anomalous fractional sub-diffusion equation and thefractional diffusion-wave equationrdquo Journal of ComputationalPhysics vol 277 pp 1ndash15 2014
[19] H-K Pang and H-W Sun ldquoFourth order finite differenceschemes for time-space fractional sub-diffusion equationsrdquoComputers amp Mathematics with Applications vol 71 no 6pp 1287ndash1302 2016
[20] M Dehghan M Abbaszadeh and W Deng ldquoFourth-ordernumerical method for the space-time tempered fractionaldiffusion-wave equationrdquo Applied Mathematics Lettersvol 73 no 11 pp 120ndash127 2017
[21] H Sun Z-Z Sun and G-H Gao ldquoSome high order dif-ference schemes for the space and time fractional Bloch-Torrey equationsrdquo Applied Mathematics and Computationvol 281 pp 356ndash380 2016
[22] S Arshad J Huang A Q M Khaliq and Y Tang ldquoTrape-zoidal scheme for time-space fractional diffusion equationwith Riesz derivativerdquo Journal of Computational Physicsvol 350 pp 1ndash15 2017
[23] H Wang K Wang and T Sircar ldquoA direct O(Nlog2N) finitedifferencemethod for fractional diffusion equationsrdquo Journal ofComputational Physics vol 229 no 21 pp 8095ndash8104 2010
[24] H Wang and T S Basu ldquoA fast finite difference method fortwo-dimensional space-fractional diffusion equationsrdquo SIAMJournal on Scientific Computing vol 34 no 5 pp A2444ndashA2458 2012
[25] H Wang and N Du ldquoA superfast-preconditioned iterativemethod for steady-state space-fractional diffusion equationsrdquoJournal of Computational Physics vol 240 pp 49ndash57 2013
[26] H Wang and N Du ldquoA fast finite difference method forthree-dimensional time-dependent space-fractional diffusionequations and its efficient implementationrdquo Journal ofComputational Physics vol 253 pp 50ndash63 2013
[27] H Wang and N Du ldquoFast alternating-direction finite dif-ference methods for three-dimensional space-fractional dif-fusion equationsrdquo Journal of Computational Physics vol 258pp 305ndash318 2014
[28] K Wang and H Wang ldquoA fast characteristic finite differencemethod for fractional advection-diffusion equationsrdquo Ad-vances in Water Resources vol 34 no 7 pp 810ndash816 2011
[29] H Wang and H Tian ldquoA fast Galerkin method with efficientmatrix assembly and storage for a peridynamic modelrdquoJournal of Computational Physics vol 231 no 23pp 7730ndash7738 2012
[30] H Tian H Wang and W Wang ldquoAn efficient collocationmethod for a non-local diffusion modelrdquo InternationalJournal of Numerical Analysis amp Modeling vol 10 no 4pp 815ndash825 2013
[31] H Wang and H Tian ldquoA fast and faithful collocation methodwith efficient matrix assembly for a two-dimensional nonlocaldiffusion modelrdquo Computer Methods in Applied Mechanicsand Engineering vol 273 no 5 pp 19ndash36 2014
[32] S Jiang J Zhang Q Zhang and Z Zhang ldquoFast evaluation ofthe Caputo fractional derivative and its applications tofractional diffusion equationsrdquo Communications in Compu-tational Physics vol 21 no 3 pp 650ndash678 2017
16 Mathematical Problems in Engineering
[33] Y Yan Z-Z Sun and J Zhang ldquoFast evaluation of the Caputofractional derivative and its applications to fractional diffusionequations a second-order schemerdquo Communications inComputational Physics vol 22 no 4 pp 1028ndash1048 2017
[34] X Lu H-K Pang and H-W Sun ldquoFast approximate in-version of a block triangular Toeplitz matrix with applicationsto fractional sub-diffusion equationsrdquo Numerical Linear Al-gebra with Applications vol 22 no 5 pp 866ndash882 2015
[35] M Caputo and M Fabrizio ldquoA new definition of fractionalderivative without singular kernelrdquo Progress in FractionalDifferentiation and Applications vol 1 no 2 pp 73ndash85 2015
[36] A Atangana ldquoOn the new fractional derivative and appli-cation to nonlinear Fisherrsquos reaction-diffusion equationrdquoAppliedMathematics and Computation vol 273 pp 948ndash9562016
[37] M D Ortigueira and J Tenreiro Machado ldquoA critical analysisof the Caputo-Fabrizio operatorrdquo Communications in Non-linear Science and Numerical Simulation vol 59 pp 608ndash6112018
[38] V E Tarasov ldquoNo nonlocality no fractional derivativerdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 62 pp 157ndash163 2018
[39] V Tarasov ldquoCaputo-Fabrizio operator in terms of integerderivatives memory or distributed lagrdquo Computational andApplied Mathematics vol 38 p 113 2019
[40] A Giusti ldquoA comment on some new definitions of fractionalderivativerdquoNonlinear Dynamics vol 93 no 3 pp 1757ndash17632018
[41] A Atangana and J F Gomez-Aguilar ldquoFractional derivativeswith no-index law property application to chaos and sta-tisticsrdquo Chaos Solitons amp Fractals vol 114 pp 516ndash535 2018
[42] J Hristov ldquoResponse functions in linear viscoelastic consti-tutive equations and related fractional operatorsrdquo Mathe-matical Modelling of Natural Phenomena vol 14 no 3pp 1ndash34 2019
[43] L X Vivas-Cruz A Gonzalez-Calderon M A Taneco-Hernandez and D P Luis ldquo)eoretical analysis of a model offluid flow in a reservoir with the Caputo-Fabrizio operatorrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 84 Article ID 105186 2020
[44] H-L Liao and Z-Z Sun ldquoMaximum norm error bounds ofADI and compact ADI methods for solving parabolicequationsrdquo Numerical Methods for Partial DifferentialEquations vol 26 no 1 pp 37ndash60 2010
[45] Z Liu A Cheng and X Li ldquoA second order Crank-Nicolsonscheme for fractional Cattaneo equation based on newfractional derivativerdquoAppliedMathematics and Computationvol 311 pp 361ndash374 2017
[46] Z Sun and G Gao Finite Difference Method for FractionalDifferential Equations Science Press Beijing China 2015
Mathematical Problems in Engineering 17
Conflicts of Interest
)e authors declare that they have no conflicts of interest
Acknowledgments
)is work was supported in part by the National NaturalScience Foundation of China under Grant nos 91630207and 11971272
References
[1] A Chaves ldquoA fractional diffusion equation to describe Levyflightsrdquo Physics Letters A vol 239 no 1-2 pp 13ndash16 1998
[2] M Giona and H E Roman ldquoFractional diffusion equation fortransport phenomena in random mediardquo Physica A Statis-tical Mechanics and its Applications vol 185 no 1ndash4pp 87ndash97 1992
[3] R Metzler and J Klafter ldquo)e random walkrsquos guide toanomalous diffusion a fractional dynamics approachrdquoPhysics Reports vol 339 no 1 pp 1ndash77 2000
[4] R L Magin Fractional Calculus in Bioengineering BegellHouse Redding Redding CT USA 2006
[5] R L Magin O Abdullah D Baleanu and X J ZhouldquoAnomalous diffusion expressed through fractional orderdifferential operators in the Bloch-Torrey equationrdquo Journalof Magnetic Resonance vol 190 no 2 pp 255ndash270 2008
[6] I Podlubny Fractional Differential Equations An Introduc-tion to Fractional Derivatives Fractional Differential Equa-tions to Methods of Beir Solution and Some of BeirApplications Vol 198 Elsevier Amsterdam Netherlands1998
[7] M Ran and C Zhang ldquoCompact difference scheme for a classof fractional-in-space nonlinear damped wave equations intwo space dimensionsrdquo Computers amp Mathematics withApplications vol 71 no 5 pp 1151ndash1162 2016
[8] A H Bhrawy and M A Zaky ldquoHighly accurate numericalschemes for multi-dimensional space variable-order frac-tional Schrodinger equationsrdquo Computers amp Mathematicswith Applications vol 73 no 6 pp 1100ndash1117 2017
[9] S B Yuste ldquoWeighted average finite difference methods forfractional diffusion equationsrdquo Journal of ComputationalPhysics vol 216 no 1 pp 264ndash274 2006
[10] T A M Langlands and B I Henry ldquo)e accuracy andstability of an implicit solution method for the fractionaldiffusion equationrdquo Journal of Computational Physicsvol 205 no 2 pp 719ndash736 2005
[11] C-M Chen F Liu I Turner and V Anh ldquoA Fourier method forthe fractional diffusion equation describing sub-diffusionrdquo Journalof Computational Physics vol 227 no 2 pp 886ndash897 2007
[12] Y Lin and C Xu ldquoFinite differencespectral approximationsfor the time-fractional diffusion equationrdquo Journal of Com-putational Physics vol 225 no 2 pp 1533ndash1552 2007
[13] G-h Gao and Z-z Sun ldquoA compact finite difference schemefor the fractional sub-diffusion equationsrdquo Journal of Com-putational Physics vol 230 no 3 pp 586ndash595 2011
[14] F Zeng C Li F Liu and I Turner ldquoNumerical algorithms fortime-fractional subdiffusion equation with second-order ac-curacyrdquo SIAM Journal on Scientific Computing vol 37 no 1pp A55ndashA78 2015
[15] G-H Gao Z-Z Sun and H-W Zhang ldquoA new fractionalnumerical differentiation formula to approximate the Caputofractional derivative and its applicationsrdquo Journal of Com-putational Physics vol 259 no 2 pp 33ndash50 2014
[16] H Li J Cao and C Li ldquoHigh-order approximation to Caputoderivatives and Caputo-type advection-diffusion equations(III)rdquo Journal of Computational and Applied Mathematicsvol 299 pp 159ndash175 2016
[17] A A Alikhanov ldquoA new difference scheme for the timefractional diffusion equationrdquo Journal of ComputationalPhysics vol 280 pp 424ndash438 2015
[18] Z Wang and S Vong ldquoCompact difference schemes for themodified anomalous fractional sub-diffusion equation and thefractional diffusion-wave equationrdquo Journal of ComputationalPhysics vol 277 pp 1ndash15 2014
[19] H-K Pang and H-W Sun ldquoFourth order finite differenceschemes for time-space fractional sub-diffusion equationsrdquoComputers amp Mathematics with Applications vol 71 no 6pp 1287ndash1302 2016
[20] M Dehghan M Abbaszadeh and W Deng ldquoFourth-ordernumerical method for the space-time tempered fractionaldiffusion-wave equationrdquo Applied Mathematics Lettersvol 73 no 11 pp 120ndash127 2017
[21] H Sun Z-Z Sun and G-H Gao ldquoSome high order dif-ference schemes for the space and time fractional Bloch-Torrey equationsrdquo Applied Mathematics and Computationvol 281 pp 356ndash380 2016
[22] S Arshad J Huang A Q M Khaliq and Y Tang ldquoTrape-zoidal scheme for time-space fractional diffusion equationwith Riesz derivativerdquo Journal of Computational Physicsvol 350 pp 1ndash15 2017
[23] H Wang K Wang and T Sircar ldquoA direct O(Nlog2N) finitedifferencemethod for fractional diffusion equationsrdquo Journal ofComputational Physics vol 229 no 21 pp 8095ndash8104 2010
[24] H Wang and T S Basu ldquoA fast finite difference method fortwo-dimensional space-fractional diffusion equationsrdquo SIAMJournal on Scientific Computing vol 34 no 5 pp A2444ndashA2458 2012
[25] H Wang and N Du ldquoA superfast-preconditioned iterativemethod for steady-state space-fractional diffusion equationsrdquoJournal of Computational Physics vol 240 pp 49ndash57 2013
[26] H Wang and N Du ldquoA fast finite difference method forthree-dimensional time-dependent space-fractional diffusionequations and its efficient implementationrdquo Journal ofComputational Physics vol 253 pp 50ndash63 2013
[27] H Wang and N Du ldquoFast alternating-direction finite dif-ference methods for three-dimensional space-fractional dif-fusion equationsrdquo Journal of Computational Physics vol 258pp 305ndash318 2014
[28] K Wang and H Wang ldquoA fast characteristic finite differencemethod for fractional advection-diffusion equationsrdquo Ad-vances in Water Resources vol 34 no 7 pp 810ndash816 2011
[29] H Wang and H Tian ldquoA fast Galerkin method with efficientmatrix assembly and storage for a peridynamic modelrdquoJournal of Computational Physics vol 231 no 23pp 7730ndash7738 2012
[30] H Tian H Wang and W Wang ldquoAn efficient collocationmethod for a non-local diffusion modelrdquo InternationalJournal of Numerical Analysis amp Modeling vol 10 no 4pp 815ndash825 2013
[31] H Wang and H Tian ldquoA fast and faithful collocation methodwith efficient matrix assembly for a two-dimensional nonlocaldiffusion modelrdquo Computer Methods in Applied Mechanicsand Engineering vol 273 no 5 pp 19ndash36 2014
[32] S Jiang J Zhang Q Zhang and Z Zhang ldquoFast evaluation ofthe Caputo fractional derivative and its applications tofractional diffusion equationsrdquo Communications in Compu-tational Physics vol 21 no 3 pp 650ndash678 2017
16 Mathematical Problems in Engineering
[33] Y Yan Z-Z Sun and J Zhang ldquoFast evaluation of the Caputofractional derivative and its applications to fractional diffusionequations a second-order schemerdquo Communications inComputational Physics vol 22 no 4 pp 1028ndash1048 2017
[34] X Lu H-K Pang and H-W Sun ldquoFast approximate in-version of a block triangular Toeplitz matrix with applicationsto fractional sub-diffusion equationsrdquo Numerical Linear Al-gebra with Applications vol 22 no 5 pp 866ndash882 2015
[35] M Caputo and M Fabrizio ldquoA new definition of fractionalderivative without singular kernelrdquo Progress in FractionalDifferentiation and Applications vol 1 no 2 pp 73ndash85 2015
[36] A Atangana ldquoOn the new fractional derivative and appli-cation to nonlinear Fisherrsquos reaction-diffusion equationrdquoAppliedMathematics and Computation vol 273 pp 948ndash9562016
[37] M D Ortigueira and J Tenreiro Machado ldquoA critical analysisof the Caputo-Fabrizio operatorrdquo Communications in Non-linear Science and Numerical Simulation vol 59 pp 608ndash6112018
[38] V E Tarasov ldquoNo nonlocality no fractional derivativerdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 62 pp 157ndash163 2018
[39] V Tarasov ldquoCaputo-Fabrizio operator in terms of integerderivatives memory or distributed lagrdquo Computational andApplied Mathematics vol 38 p 113 2019
[40] A Giusti ldquoA comment on some new definitions of fractionalderivativerdquoNonlinear Dynamics vol 93 no 3 pp 1757ndash17632018
[41] A Atangana and J F Gomez-Aguilar ldquoFractional derivativeswith no-index law property application to chaos and sta-tisticsrdquo Chaos Solitons amp Fractals vol 114 pp 516ndash535 2018
[42] J Hristov ldquoResponse functions in linear viscoelastic consti-tutive equations and related fractional operatorsrdquo Mathe-matical Modelling of Natural Phenomena vol 14 no 3pp 1ndash34 2019
[43] L X Vivas-Cruz A Gonzalez-Calderon M A Taneco-Hernandez and D P Luis ldquo)eoretical analysis of a model offluid flow in a reservoir with the Caputo-Fabrizio operatorrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 84 Article ID 105186 2020
[44] H-L Liao and Z-Z Sun ldquoMaximum norm error bounds ofADI and compact ADI methods for solving parabolicequationsrdquo Numerical Methods for Partial DifferentialEquations vol 26 no 1 pp 37ndash60 2010
[45] Z Liu A Cheng and X Li ldquoA second order Crank-Nicolsonscheme for fractional Cattaneo equation based on newfractional derivativerdquoAppliedMathematics and Computationvol 311 pp 361ndash374 2017
[46] Z Sun and G Gao Finite Difference Method for FractionalDifferential Equations Science Press Beijing China 2015
Mathematical Problems in Engineering 17
[33] Y Yan Z-Z Sun and J Zhang ldquoFast evaluation of the Caputofractional derivative and its applications to fractional diffusionequations a second-order schemerdquo Communications inComputational Physics vol 22 no 4 pp 1028ndash1048 2017
[34] X Lu H-K Pang and H-W Sun ldquoFast approximate in-version of a block triangular Toeplitz matrix with applicationsto fractional sub-diffusion equationsrdquo Numerical Linear Al-gebra with Applications vol 22 no 5 pp 866ndash882 2015
[35] M Caputo and M Fabrizio ldquoA new definition of fractionalderivative without singular kernelrdquo Progress in FractionalDifferentiation and Applications vol 1 no 2 pp 73ndash85 2015
[36] A Atangana ldquoOn the new fractional derivative and appli-cation to nonlinear Fisherrsquos reaction-diffusion equationrdquoAppliedMathematics and Computation vol 273 pp 948ndash9562016
[37] M D Ortigueira and J Tenreiro Machado ldquoA critical analysisof the Caputo-Fabrizio operatorrdquo Communications in Non-linear Science and Numerical Simulation vol 59 pp 608ndash6112018
[38] V E Tarasov ldquoNo nonlocality no fractional derivativerdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 62 pp 157ndash163 2018
[39] V Tarasov ldquoCaputo-Fabrizio operator in terms of integerderivatives memory or distributed lagrdquo Computational andApplied Mathematics vol 38 p 113 2019
[40] A Giusti ldquoA comment on some new definitions of fractionalderivativerdquoNonlinear Dynamics vol 93 no 3 pp 1757ndash17632018
[41] A Atangana and J F Gomez-Aguilar ldquoFractional derivativeswith no-index law property application to chaos and sta-tisticsrdquo Chaos Solitons amp Fractals vol 114 pp 516ndash535 2018
[42] J Hristov ldquoResponse functions in linear viscoelastic consti-tutive equations and related fractional operatorsrdquo Mathe-matical Modelling of Natural Phenomena vol 14 no 3pp 1ndash34 2019
[43] L X Vivas-Cruz A Gonzalez-Calderon M A Taneco-Hernandez and D P Luis ldquo)eoretical analysis of a model offluid flow in a reservoir with the Caputo-Fabrizio operatorrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 84 Article ID 105186 2020
[44] H-L Liao and Z-Z Sun ldquoMaximum norm error bounds ofADI and compact ADI methods for solving parabolicequationsrdquo Numerical Methods for Partial DifferentialEquations vol 26 no 1 pp 37ndash60 2010
[45] Z Liu A Cheng and X Li ldquoA second order Crank-Nicolsonscheme for fractional Cattaneo equation based on newfractional derivativerdquoAppliedMathematics and Computationvol 311 pp 361ndash374 2017
[46] Z Sun and G Gao Finite Difference Method for FractionalDifferential Equations Science Press Beijing China 2015
Mathematical Problems in Engineering 17
