Fast Evaluation of the Caputo Fractional Derivative ...jiang/Papers/2017CiCP.pdf... approximation...

Commun Comput Physdoi 104208cicpOA-2016-0136

Vol 21 No 3 pp 650-678March 2017

Fast Evaluation of the Caputo Fractional Derivative

and its Applications to Fractional Diffusion Equations

Shidong Jiang1 Jiwei Zhang2lowast Qian Zhang2 and Zhimin Zhang23

1 Department of Mathematical Sciences New Jersey Institute of Technology NewarkNJ 07102 USA2 Beijing Computational Science Research Center Beijing 100093 China3 Department of Mathematics Wayne State University Detroit MI 48202 USA

Communicated by Tao Zhou

Received 19 August 2016 Accepted (in revised version) 23 November 2016

Abstract The computational work and storage of numerically solving the time frac-tional PDEs are generally huge for the traditional direct methods since they requiretotal O(NSNT) memory and O(NSN2

T) work where NT and NS represent the totalnumber of time steps and grid points in space respectively To overcome this difficultywe present an efficient algorithm for the evaluation of the Caputo fractional derivativeC0 Dα

t f (t) of order αisin (01) The algorithm is based on an efficient sum-of-exponentials

(SOE) approximation for the kernel tminus1minusα on the interval [∆tT] with a uniform abso-lute error ε We give the theoretical analysis to show that the number of exponentials

Nexp needed is of order O(logNT) for T≫1 or O(log2 NT) for Tasymp1 for fixed accuracyε The resulting algorithm requires only O(NSNexp) storage and O(NSNT Nexp) workwhen numerically solving the time fractional PDEs Furthermore we also give the sta-bility and error analysis of the new scheme and present several numerical examplesto demonstrate the performance of our scheme

AMS subject classifications 33C10 33F05 35Q40 35Q55 34A08 35R11 26A33

Key words Fractional derivative Caputo derivative sum-of-exponentials approximation frac-tional diffusion equation fast convolution algorithm stability analysis

1 Introduction

Over the last few decades the fractional calculus has received much attention of bothphysical scientists and mathematicians since they can faithfully capture the dynamics of

lowastCorresponding author Email addresses shidongjiangnjitedu (S Jiang) jwzhangcsrcaccn (JZhang) qianzhangcsrcaccn (Q Zhang) zmzhangcsrcaccn (Z Zhang)

httpwwwglobal-scicom 650 ccopy2017 Global-Science Press

httpswwwcambridgeorgcoreterms httpsdoiorg104208cicpOA-2016-0136Downloaded from httpswwwcambridgeorgcore New Jersey Institute of Technology on 16 Mar 2017 at 130234 subject to the Cambridge Core terms of use available at

S Jiang et al Commun Comput Phys 21 (2017) pp 650-678 651

physical process in many applied sciences including biology ecology and control systemThe anomalous diffusion also referred to as the non-Gaussian process has been observedand validated in many phenomena with accurate physical measurement [192829464851] The mathematical and numerical analysis of the factional calculus became a subjectof intensive investigations

In this paper we consider a fast evaluation of the following fractional partial differ-ential equation

C0 Dα

t u(xt)=∆u(xt)+F(uxt) 0ltαlt1 (11)

where the Caputo fractional derivative C0 Dα

t u(xt) is defined by the formula

C0 Dα

t u(xt)=1

Γ(mminusα)

int t

0

u(m)(xτ)

(tminusτ)α+1minusmdτ mminus1ltαltm misinZ (12)

The existing schemes for solving (11) require the storage of the solution at all previoustime steps and the computational complexity of these schemes is O(N2

T NS) with NT thetotal number of time steps and NS the number of grid points in space This is in darkcontrast with the usual diffusion equations where one only needs to store the solution ata fixed number of time steps and the computational complexity is linear with respect toNT

It is easy to see that the difficulty is caused by the Caputo factional derivative ap-peared in (11) Indeed one of the popular schemes of discretizing the Caputo fractionalderivative is the so-called L1 approximation [15 16 23 30 32 36 39ndash41 52] which is sim-ply based on the piecewise linear interpolation of u on each subinterval For 0ltαlt1 theorder of accuracy of the L1 approximation is 2minusα There are also high-order discretiza-tion schemes by using piecewise high-order polynomial interpolation of u [10 17 33 47]and [9 34 37] For each spatial point x these methods require the storage of all previousfunction values u(0)u(t1)middotmiddotmiddot u(tn) and O(n) flops at the nth step Thus it requires onaverage O(NT) storage and the total computational cost is O(N2

T) which forms a bot-tleneck for long time simulations especially when one tries to solve the time fractionalpartial differential equations (PDEs)

Here we present an efficient scheme for solving the fractional PDEs (11) Our key ob-servation is that the Caputo derivative can be evaluated almost as efficient as the usuallyderivatives (besides some logarithmic factors) We first split the convolution integral in(12) into two parts - a local part containing the integral from tminus∆t to t and a history partcontaining the integral from 0 to tminus∆t The local part is approximated using the stan-dard L1 approximation For the history part integration by parts leads to a convolutionintegral of u with the kernel tminus1minusα We show that tminus1minusα (0 lt α lt 1) admits an efficientsum-of-exponentials (SOE) approximation on the interval [δT] with δ = ∆t a uniformabsolute error ε and the number of exponentials needed is of the order

Nexp =O(

log1

ε

(

loglog1

ε+log

T

δ

)

+log1

δ

(

loglog1

ε+log

1

δ

))

(13)


652 S Jiang et al Commun Comput Phys 21 (2017) pp 650-678

That is for fixed precision ε we have Nexp =O(logNT) for T ≫ 1 or Nexp =O(log2 NT)

for T asymp 1 assuming that NT = T∆t The approximation can be used to accelerate the

evaluation of the convolution via the standard recurrence relation The resulting al-gorithm has nearly optimal complexity ndash O(NT Nexp) work and O(Nexp) storage Wewould like to remark here that the SOE approximations have been applied to speedup the evaluation of the convolution integrals in many applications (see for example[1 6 20 21 24 26 31 38 45 50 53 54 58 59] and references therein)

When we incorporate the fast evaluation scheme of the Caputo fractional derivativeto solve the fractional diffusion equations (both linear and nonlinear) the resulting algo-rithm for solving the fractional PDEs is both efficient and stable The computational costof the new algorithm is O(NS NT Nexp) as compared with O(NSN2

T) for direct methodsand the storage requirement is only O(NSNexp) as compared with O(NS NT) for directmethods since one needs to store the solution in the whole computational spatial domainat all times Furthermore we have carried out a rigorous and detailed analysis to provethat our scheme is unconditionally stable with respect to arbitrary step sizes With thesetwo properties our scheme provides an efficient and reliable tool for long time large scalesimulation of fractional PDEs In literatures there are many other efforts to speed up theevaluation of weakly singular kernel Lubich and Schadle [44] present a fast convolutionfor non-reflecting boundary conditions Baffet and Hesthaven [3] compressed the kernelin the Laplace domain and obtained a sum-of-poles approximation for the Laplace trans-form of the kernel And other fast approximation to time-fractional derivative is based onthe block triangular Toeplitz matrix or block triangular Toeplitz-like matrix see [27 43]

The paper is organized as follows In Section 2 we describe the fast algorithm for theevaluation of the Caputo fractional derivative and provide rigorous error analysis of ourdiscretization scheme In Section 3 we apply our fast algorithm to solve the linear frac-tional diffusion PDEs and present the stability and error analysis for the overall schemeIn Section 4 we study the nonlinear fractional diffusion PDEs and demonstrate that ourfast algorithm has the same order of convergence as the direct method in this case Fi-nally we conclude our paper with a brief discussion on the extension and generalizationof our scheme

2 Fast evaluation of the Caputo fractional derivative

In this section we consider the fast evaluation of the Caputo fractional derivative for0ltαlt1 Suppose that we would like to evaluate the Caputo fractional derivative on theinterval [0T] over a set of grid points Ωt =tn n=01middotmiddotmiddot NT with t0 =0 tNT

=T and∆tn = tnminustnminus1

We first split the convolution integral in (12) into a sum of local part and history partthat is

C0 Dα

t u(t)|t=tn =1

Γ(1minusα)

int tn

0

uprime(s)ds

(tnminuss)α



=1

Γ(1minusα)

int tn

tnminus1

uprime(s)ds

(tnminuss)α+

1

Γ(1minusα)

int tnminus1

0

uprime(s)ds

(tnminuss)α

=Cl(tn)+Ch(tn) (21)

where the last equality defines the local part and the history part respectively For thelocal part we apply the standard L1 approximation which approximates u(s) on [tnminus1tn]by a linear polynomial (with u(tnminus1) and u(tn) as the interpolation nodes) or uprime(s) by a

constant u(tn)minusu(tnminus1)∆tn

We have

Cl(tn)asympu(tn)minusu(tnminus1)

∆tnΓ(1minusα)

int tn

tnminus1

1

(tnminuss)αds=

u(tn)minusu(tnminus1)

∆tαnΓ(2minusα)

(22)

For the history part we apply the integration by parts to eliminate uprime(s) and have

Ch(tn)=1

Γ(1minusα)

int tnminus1

0

uprime(s)ds

(tnminuss)α

=1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαint tnminus1

0

u(s)ds

(tnminuss)1+α

]

(23)

21 Efficient sum-of-exponentials approximation for the power function

We now show that the kernel tminusβ (0ltβlt2) can be approximated via a sum-of-exponentialsapproximation efficiently on the interval [δT] with δ=min1lenleN ∆tn and the absolute er-ror ε That is there exist positive real numbers si and wi (i = 1middotmiddotmiddot Nexp) such that for0ltβlt2

∣

∣

∣

∣

∣

1

tβminus

Nexp

sumi=1

ωieminussit

∣

∣

∣

∣

∣

le ε tisin [δT] (24)

where Nexp is given by (13) The evaluation of the history part via (23) only needsthe case 1 lt β lt 2 However one may also evaluate the history part directly withoutintegration by parts and the case 0 lt β lt 1 is then needed Meanwhile one can ob-tain the code for si and ωi in (24) from the homepage httpswebnjitedu~jiang

or httpwwwcsrcaccn~jwzhangWe start our proof from the following integral representation of the power function

Lemma 21 For any βgt01

tβ=

1

Γ(β)

int infin

0eminustmiddotssβminus1ds (25)

Proof This follows from a change of variable x= tmiddots and the integral definition of the Γ

function [49]

Eq (25) can be viewed as a representation of tminusβ using an infinitely many (continu-ous) exponentials In order to obtain an efficient SOE approximation we first truncate the



integral to a finite interval then subdivide the finite interval into a set of dyadic intervalsand discretize the integral on each dyadic interval with proper quadratures

We now assume 0ltβlt2

Lemma 22 For tgeδgt0

∣

∣

∣

∣

1

Γ(β)

int infin

peminustssβminus1ds

∣

∣

∣

∣

le

eminusδp pβminus1

Γ(β)1δ 0ltβle1

eminusδp2βminus1(

pβ

Γ(β)+ 1

δβ

)

1ltβlt2(26)

Proof For the case of 0ltβle1∣

∣

∣

∣

1

Γ(β)

int infin


∣

∣

∣

∣

=

∣

∣

∣

∣

1

Γ(β)eminustp

int infin

0eminustx(x+p)βminus1dx

∣

∣

∣

∣

le eminusδp pβminus1

Γ(β)

int infin

0eminusδxdx


Γ(β)

1

δ (27)

For the case of 1ltβlt2∣

∣

∣

∣

1

Γ(β)

int infin


∣

∣

∣

∣

=

∣

∣

∣

∣

1

Γ(β)eminustp

int infin


∣

∣

∣

∣

le eminusδp 1

Γ(β)

(

int p

0(2p)βminus1dx+

int infin

peminusδx(2x)βminus1dx

)

le eminusδp 2βminus1

Γ(β)

(

pβ+int infin

0eminusδxxβminus1dx

)

le eminusδp2βminus1

(

pβ

Γ(β)+

1

δβ

)

(28)

This completes the proof

Remark 21 The truncation error can be made arbitrarily small for fixed δ by choosingsufficiently large p Usually we have δlt1 and if one would like to bound the truncationerror by εlt1e then δpgt1 or pgt1δ When 0ltβle1

eminusδp pβminus1

Γ(β)

1

δlt eminusδp pβ

Γ(β)pδlt eminusδp p (29)

when 1ltβlt2

eminusδp2βminus1

(

pβ

Γ(β)+

1

δβ

)

lt eminusδp2βminus1

(

pβ

Γ(β)+pβ

)

lt5eminusδp p2 (210)

Thus p=O(

log(

1εδ

)

δ)

is sufficient to bound the truncation error by ε



We now proceed to discuss the discretization error for the integral on the interval[0p] On each dyadic interval we will discretize the integral via standard Gauss-Legendrequadrature and analyze the discretization error The error can be controlled more or lessuniformly since each dyadic interval is separated from the singular point ie the originby its own length

Lemma 23 Consider a dyadic interval [ab]= [2j 2j+1] and let s1middotmiddotmiddot sn and w1middotmiddotmiddot wn be thenodes and weights for n-point Gauss-Legendre quadrature on the interval Then for βisin(02) andngt1

∣

∣

∣

∣

∣

int b

aeminustssβminus1dsminus

n

sumk=1

wksβminus1k eminusskt

∣

∣

∣

∣

∣

lt252 πaβ

(

e1e

4

)2n

(211)

Proof For any interval [ab] the standard estimate for n-point Gauss-Legendre quadra-ture [49] yields

∣

∣

∣

∣

∣

int b


n

sumk=1


∣

∣

∣

∣

∣

le (bminusa)2n+1

2n+1

(n)4

[(2n)]3max

sisin(ab)

∣

∣

∣D2n

s (eminusstsβminus1)∣

∣

∣ (212)

where Ds denotes the derivative with respect to s Applying Stirlingrsquos approximation [49]radic

2πnn+12eminusnltnlt2

radicπnn+12eminusn (213)

we obtain(n)4

[(2n)]3lt2

radicπ( e

8

)2nradic

n

n2n (214)

Observe now |(βminus1)middotmiddotmiddot(βminusk)|le k for kgt1 and thus∣

∣

∣Dk

s sβminus1∣

∣

∣= sβminus1le2

radicπradic

2n(2n)keminusksβminuskminus1 for k=0∣

∣

∣Dk

s sβminus1∣

∣

∣=∣

∣

∣(βminus1)sβminuskminus1

∣

∣

∣le2

radicπradic


∣

∣Dk

s sβminus1∣

∣

∣=∣

∣

∣(βminus1)middotmiddotmiddot(βminusk)sβminuskminus1

∣

∣

∣le ksβminuskminus1

le2radic

πkk+12eminusksβminuskminus1le2radic

πradic

2n(2n)keminusksβminuskminus1 for kgt1

(215)

We also haveD2nminusk

s eminusst=(minust)2nminuskeminusst (216)

Combining (215) (216) with the Leibniz rule we obtain

∣

∣

∣D2n


∣

∣=

∣

∣

∣

∣

∣

2n

sumk=0

(

2nk

)

(

D2nminusks eminusst

)(

Dks sβminus1

)

∣

∣

∣

∣

∣

le2radic

πradic

2nsβminus1eminusst2n

sumk=0

(

2nk

)

t2nminusk(2n)keminusksminusk

=2radic

πradic

2nsβminus1eminusst

(

t+2n

es

)2n

(217)



Combining (212) (214) (217) and bminusa= a we have

∣

∣

∣

∣

∣

int b


n

sumk=1


∣

∣

∣

∣

∣

le maxalesleb

2radic

2π(bminusa)sβminus1eminusst

(

e(bminusa)t

8n+

bminusa

4s

)2n

le252 πaβeminusat

(

eat

8n+

1

4

)2n

(218)

And (211) follows from the fact

maxxgt0

eminusx

(

ex

8n+

1

4

)2n

=

(

e1e

4

)2n

nge2 (219)


We now consider the end interval [0a] We will use Gauss-Jacobi quadrature with theweight function sβminus1 to discretize the integral Under this weight the function to be in-tegrated is a simple exponential function Thus the discretization error can be controlledto be arbitrarily small with properly chosen parameters

Lemma 24 Let s1middotmiddotmiddot sn and w1middotmiddotmiddot wn (nge 2) be the nodes and weights for n-point Gauss-Jacobi quadrature with the weight function sβminus1 on the interval Then for 0lt tltT βisin(02) andngt1

∣

∣

∣

∣

∣

int a

0eminustssβminus1dsminus

n

sumk=1

wkeminusskt

∣

∣

∣

∣

∣

lt4radic

πaβn32( e

8

)2n(

aT

nminus1

)2n

(220)

Proof The standard estimate for n-point Gauss-Jacobi quadrature [49] yields

∣

∣

∣

∣

∣

int a


n

sumk=1

wkeminusskt

∣

∣

∣

∣

∣

le a2n+β

2n+β

(n)2[Γ(n+β)]2

(2n)[Γ(2n+β)]2max

sisin(0a)

∣

∣D2ns eminusst

∣

∣ (221)

For ngt 1 we have Γ(n+β)lt Γ(n+lceilβrceil)= (n+lceilβrceilminus1) Γ(2n+β)gt Γ(2n+lfloorβrfloor)= (2n+lfloorβrfloorminus1) 2n+βgt2n+lfloorβrfloor Thus

∣

∣

∣

∣

∣

int a


n

sumk=1

wkeminusskt

∣

∣

∣

∣

∣

le max0leslea

a2n+β

2n+lfloorβrfloor(n)2[(n+lceilβrceilminus1)]2

(2n)[(2n+lfloorβrfloorminus1)]2t2neminusst

le aβ

2n

(n)2[(n+lceilβrceilminus1)]2

(2n)[(2n+lfloorβrfloorminus1)]2(aT)2n

le4radic

πaβn32( e

8

)2n(

aT

nminus1

)2n

(222)

where the last inequality follows from Stirlingrsquos approximation (213) and the fact thatlceilβrceilminuslfloorβrfloorle1



We are now in a position to combine the last three lemmas to give an efficient SOEapproximation for tminusβ on [δT] for βisin (02) as follows

Theorem 21 Let 0lt δ le t le T (δ le 1 and T ge 1) let εgt 0 be the desired precision let no =O(log 1

ε ) let M=O(logT) and let N=O(loglog 1ε +log 1

δ ) Furthermore let so1middotmiddotmiddot sono andwo1middotmiddotmiddot wono be the nodes and weights for the no-point Gauss-Jacobi quadrature on the interval[02minusM] let sj1middotmiddotmiddot sjns

and wj1middotmiddotmiddot wjnsbe the nodes and weights for ns-point Gauss-Legendre

quadrature on small intervals [2j2j+1] j=minusMmiddotmiddotmiddot minus1 where ns=O(

log 1ε

)

and let sj1middotmiddotmiddot sjnl

and wj1middotmiddotmiddot wjnsl be the nodes and weights for nl-point Gauss-Legendre quadrature on large

intervals [2j2j+1] j=0middotmiddotmiddot N where nl =O(

log 1ε +log 1

δ

)

Then for tisin [δT] and βisin (02)

∣

∣

∣

∣

∣

1

tβminus(

no

sumk=1

eminussoktwok+minus1

sumj=minusM

ns

sumk=1

eminuss jktsβminus1jk wjk+

N

sumj=0

nl

sumk=1

eminuss jktsβminus1jk wjk

)∣

∣

∣

∣

∣

le ε (223)

Remark 22 The important fact which emerges from this theorem is that the total numberof exponentials needed to approximate tminusβ for 0 lt δ le t le T with an absolute error ε isgiven by the formula (13) which gives the right complexity estimate for Nexp Howevereven though the proof of Theorem 21 is explicitly constructive and one may even obtainthe concrete number of exponentials needed by more detailed calculation the resultingnumber of exponentials following the construction in Theorem 21 is unnecessarily largeThis is because the construction is based on locally optimal quadrature on each dyadicinterval leading to a large prefactor in the O complexity

As a global after-processing optimization one may apply modified Pronyrsquos methodin [5] to reduce the number of exponentials for nodes on the interval (01) while standardmodel reduction method in [55] can be applied to reduce the number of exponentials fornodes on the interval [1p] Our numerical experiments indicate that there is about a fac-tor of ten in the reduction of the number of exponentials after these further optimizationsteps Thus we state Theorem 21 using the O notation

Table 1 and Table 2 list the actual number of exponentials needed to approximatetminus1minusα with various ǫ and NT =T∆t after applying the reduction algorithms in Remark22 We observe that the number of exponentials needed is very modest even for highaccuracy approximations Indeed one needs less than 90 terms in order to march onemillion steps with 9-digit or 8-digit accuracy

22 Fast evaluation of the history part

We replace the convolution kernel tminus1minusα by its SOE approximation in (24) to approximatethe history part defined in (23) as follows



Table 1 Number of exponentials needed to approximate tminus1minusα with fixed ∆t=10minus3

∆t=10minus3 α=02 α=05

ǫ

NexpT∆t 103 104 105 106 103 104 105 106

10minus3 25 29 33 37 30 35 40 44

10minus6 34 40 45 51 39 46 52 59

10minus9 43 51 58 66 49 57 65 74

Table 2 Number of exponentials needed to approximate tminus1minusα with fixed T=1

T=1 α=02 α=05

ǫ

Nexp ∆t10minus3 10minus4 10minus5 10minus6 10minus3 10minus4 10minus5 10minus6

10minus3 25 34 43 55 30 41 52 66

10minus6 34 45 58 69 39 51 65 81

10minus8 40 51 64 79 43 57 72 84

Ch(tn)asymp1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωi

int tnminus1

0eminus(tnminusτ)siu(τ)dτ

]

=1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

(224)

Define

Uhisti(tn)=int tnminus1


To evaluate Uhisti(tn) for n=12middotmiddotmiddot NT we observe the following simple recurrence rela-tion

Uhisti(tn)= eminussi∆tnUhisti(tnminus1)+int tnminus1

tnminus2

eminussi(tnminusτ)u(τ)dτ (225)

At each time step we only need O(1) work to compute Uhisti(tn) since Uhisti(tnminus1) isknown at that point Thus the total work is reduced from O(N2

T) to O(NT Nexp) and thetotal memory requirement is reduced from O(NT) to O(Nexp)

One may compute the integral on the right hand side of (225) by interpolating u viaa linear function and then evaluating the resulting approximation analytically We have

int tnminus1

tnminus2

eminussi(tnminusτ)u(τ)dτasymp eminussi∆tn

s2i ∆tnminus1

[

(eminussi∆tnminus1minus1+si∆tnminus1)u(tnminus1)

+ (1minuseminussi∆tnminus1minuseminussi∆tnminus1si∆tnminus1)u(tnminus2)]

(226)



We note that the weights in front of u(tnminus1) and u(tnminus2) in (226) are subject to significantcancellation error when si∆tnminus1 is small In that case we can compute the weights by aTaylor expansion of exponentials with a small number of terms

Remark 23 From above analysis the fast evaluation of the Caputo fractional derivativeis valid for non-uniform mesh For simplicity we will use uniform mesh in time with∆tn =∆t in the remainder of the section

23 Error analysis

Define mesh functionsun=u(tn) 1lenleNT

It is straightforward to verify that our scheme of evaluating the Caputo fractional deriva-tive is equivalent to the following formula

C0 Dα

t u(t)|t=tn asympunminusunminus1

∆tαΓ(2minusα)+

1

Γ(1minusα)

[

unminus1

∆tαminus u0

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

FC0 D

αt un for ngt2 (227)

Noting that Uhisti(tn)=0 when n=1 we have

FC0 D

αt u1=

∆tminusα

Γ(2minusα)(u1minusu0) (228)

Recall that the L1-approximation (based on the linear interpolation of the function u) ofthe Caputo derivative C

0 Dαt u (see for example [42 48]) is defined by the formula

C0 D

αt un=

∆tminusα

Γ(2minusα)

[

a(α)0 unminus

nminus1

sumk=1

(a(α)nminuskminus1minusa

(α)nminusk)u

kminusa(α)nminus1u0

]

(229)

where a(α)k =(k+1)1minusαminusk1minusα The following lemma which can be found in [52] estab-

lishes an error bound for the L1-approximation (229)

Lemma 25 (see [40 52]) Suppose that f (t)isinC2[0tn] and

Rnu =C0 Dα

t u(t)∣

∣

t=tnminusC

0 Dαt un (230)

where 0ltαlt1 Then

|Rnu|le ∆t2minusα

Γ(2minusα)

(

1minusα

12+

22minusα

2minusαminus(1+2minusα)

)

max0letletn

|uprimeprime(t)| (231)

The following lemma provides an error bound for our fast evaluation scheme de-noted by FC

0 Dαt un in (227) and (228)



Lemma 26 Suppose that u(t)isinC2[0tn] and let

FRnu =C0 Dα

t u(t)∣

∣

t=tnminusFC

0 Dαt un (232)

where 0ltαlt1 Then

|FRnu|le ∆t2minusα

Γ(2minusα)

(

1minusα

12+

22minusα


)

max0letletn

|uprimeprime(t)|+ αεtnminus1

Γ(1minusα)max

0letletnminus1

|u(t)| (233)

Proof Obviously the only difference between our approximation FC0 D

αt un and the L1-

approximation C0 D

αt un is that the convolution kernel admits an absolute error bounded

by ε in its sum-of-exponentials approximation (24) namely

∣

∣

∣

FC0 D

αt unminusC

0 Dαt un∣

∣

∣le αε

Γ(1minusα)

nminus1

suml=1

int tl

tlminus1

|Π1lu(s)|ds (234)

where Π1lu(t)=u(tlminus1)tlminust∆t +u(tl)

tminustlminus1

∆t And the triangle inequality leads to

|FRnu|le |CRnu|+ αε

Γ(1minusα)

nminus1

suml=1

int tl

tlminus1

|Π1lu(s)|ds (235)

where

nminus1

suml=1

int tl

tlminus1

|Π1lu(s)|dsle max0letletnminus1

|u(t)|tnminus1 (236)

Combining Lemma 25 and (236) we obtain Lemma 26

We also have the following useful inequality The proof is given in Appendix A

Lemma 27 For any mesh functions g=gk |0lekleN defined on Ωt the following inequalityholds

∆tn

sumk=1

(FC0 D

αt gk)gk ge tminusα

n minus2αεtnminus1

2Γ(1minusα)∆t

n

sumk=1

(gk)2minus t1minusαn +α(1minusα)εtnminus1∆t

2Γ(2minusα)(g0)2 (237)

The results in Lemmas 26 and 27 are useful in proving the stability and convergenceproperties of the overall numerical scheme for solving time-fractional PDEs when ourfast evaluation scheme is applied to compute the Caputo fractional derivative



3 Application I Linear fractional diffusion equation

We now consider the computation of the initial value (IV) problem of the linear fractionaldiffusion equation given by

C0 Dα

t u(xt)=uxx(xt)+ f (xt) xisinR tgt0 (31)

u(x0)=u0(x) xisinR (32)

u(xt)rarr0 when |x|rarrinfin (33)

where the initial data u0 and the source term f (xt) are assumed to be compactly sup-ported in the interval Ωi =x|xl ltxltxr To solve this problem using a finite differencescheme one needs to truncate the computational domain to a finite interval and imposesome boundary conditions at the end points see [24714ndash161822] The exact nonreflect-ing boundary conditions for the above problem have been derived in [15] via standardLaplace transform method and it is shown in [15] that the above problem is equivalent tothe following initial-boundary value problem (IBVP)

C0 Dα

t u(xt)=uxx(xt)+ f (xt) xisinΩi tgt0 (34)

u(x0)=u0(x) xisinΩi (35)

partu(xt)

partx=

1

Γ(1minus α2 )

int t

0

us(xs)

(tminuss)α2

ds =C0 D

α2

t u(xt) x= xl (36)

partu(xt)

partx=minus 1

Γ(1minus α2 )

int t

0

us(xs)

(tminuss)α2

ds =minusC0 D

α2

t u(xt) x= xr (37)

We now incorporate the fast evaluation of the Caputo fractional derivative into the exist-ing finite difference (FD) scheme introduced in [15] to numerically solve the IBVP (34)-(37) establish the corresponding stability and convergence properties and present anexample to demonstrate the performance of our scheme by comparing it with the directscheme given in [15]

31 Construction of the finite difference scheme with fast evaluation

We first introduce some standard notations For two given positive integers NT and NS

let tnNTn=0 be a equidistant partition of [0T] with tn=n∆t and ∆t=TNT and let xjNS

j=0

be a partition of (xlxr) with xi = xl+ih and h=(xrminusxl)NS Denote

δtuni =

uni minusunminus1

i

∆t δxun

i+ 12=

uni+1minusun

i

h

δ2xun

i =δxui+ 1

2minusδxuiminus 1

2

h δxun

i =un

i+1minusuniminus1

2h



Lemma 31 (see [52]) Suppose that uisinC3[xl xr] Then

uxx(x0)minus2

h

[

δxu 12minusux(x0)

]

=minush

3uxxx(x0+θ1h) θ1 isin (01) (38)

uxx(xNS)minus 2

h

[

ux(xNS)minusδxuNSminus 1

2

]

=h

3uxxx(xNS

minusθ2h) θ2isin (01) (39)

The FD scheme in [15] for the problem (34)-(37) can be written by

C0 D

αt un

i =δ2xun

i + f ni 1le ileNSminus1 1lenleNT (310)

C0 D

αt un

0 =2

h

[

δxun12minusC

0 Dα2t un

0

]

+ f n0 1lenleNT (311)

C0 D

αt un

NS=

2

h

[

minusδxunNSminus 1

2minusC

0 Dα2t un

NS

]

+ f nNS

1lenleNT (312)

u0i =u0(xi) 0le ileNS (313)

Replacing the standard L1-approximation C0 D for the Caputo derivative by our fast eval-

uation scheme FC0 D we obtain a fast FD scheme of the following form

FC0 D

αt un

i =δ2xun


FC0 D

αt un

0 =2

h

[

δxun12minusFC

0 Dα2t un

0

]


FC0 D

αt un

NS=

2

h

[

minusδxunNSminus 1

2minusFC

0 Dα2t un

NS

]

+ f nNS

1lenleNT (316)


32 Stability and error analysis of the new scheme

The stability and error analysis of the scheme (310)-(313) have been given in [15] In thissection we study the corresponding theoretical analysis for our new scheme (314)-(317)We first denote Sh =u|u=(u0u1middotmiddotmiddot uNs)

Lemma 32 ( [15]) For any mesh function uisinSh the following inequality holds

u2infin le θδxu2+

(1

θ+

1

L

)

u2 forallθgt0

where L is the length of the computational domain and here L= xrminusxl

We now show the prior estimate for the solution of the scheme (314)-(317)

Theorem 31 (Prior Estimate) Suppose uki |0le ileNS 0lekleNT is the solution of the finite

difference scheme (314)-(317) Then for any 1lenleNT

∆tn

sumk=1

∥

∥

∥uk∥

∥

∥

2

infinle2(

1+radic

1+L2micro)

Lmicro

(

ρ∥

∥u0∥

∥

2+[

(u00)

2+(u0NS)2]

+∆t

8ν

n

sumk=1

[

(h f k0 )

2+(h f kNS)2]

+∆t

micro

n

sumk=1

hNSminus1

sumi=1

( f ki )

2

)

(318)



where

ρ=T1minusα

2Γ(2minusα) micro=

Tminusαminus2αεT

Γ(1minusα)

=T1minus α

2

2Γ(2minus α2 )

ν=Tminus α

2 minusαεT

Γ(1minus α2 )

(319)

Proof Multiplying huki on both sides of (314) and summing up for i from 1 to NSminus1 we

have

hNSminus1

sumi=1

(FC0 D

αt uk

i )uki minush

NSminus1

sumi=1

(δ2xuk

i )uki =h

NSminus1

sumi=1

f ki uk

i

Multiplying h2 uk

0 and h2 uk

NSon both sides of (315) and (316) respectively then adding the

results with the above identity we obtain

(FC0 D

αt ukuk)+

[

minus(δxvk12)uk

0minushNSminus1

sumi=1

(δ2xuk

i )uki +(δxuk

NSminus 12)uk

NS

]

+(FC0 D

α2t uk

0)uk0+(FC

0 D

α2t uk

NS)uk

NS=

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS (320)

Applying the summation by parts we have

minus(δxvk12)uk

0minushNSminus1

sumi=1

(δ2xuk

i )uki +(δxuk

NSminus 12)uk

NS=∥

∥

∥δxuk

∥

∥

∥

2 (321)

Substituting (321) into (320) and multiplying ∆t on both sides of the resulting identityand summing up for k from 1 to n it follows from Lemma 27 that

∆ttminusαn minus2αεtnminus1

2Γ(1minusα)

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

tminus α

2n minusαεtnminus1

2Γ(1minus α2 )

n

sumk=1

[

(uk0)

2+(ukNS)2]

+∆tn

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2

le t1minusαn +α(1minusα)εtnminus1∆t

2Γ(2minusα)

∥

∥u0∥

∥

2+

t1minus α

2n + α

2 (1minus α2 )εtnminus1∆t

2Γ(2minus α2 )

[

(u00)

2+(u0NS)2]

+∆tn

sumk=1

[

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS

]

(322)

Applying the Cauchy-Schwarz inequality we obtain

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS



le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α

2n minusαεtnminus1)

[

(h f k0 )

2+(h f kNS)2]

+hNSminus1

sumi=1

[

tminusαn minus2αεtnminus1

4Γ(1minusα)(uk

i )2+

Γ(1minusα)


( f ki )

2

]

le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]

+tminusαn minus2αεtnminus1

4Γ(1minusα)

∥

∥

∥uk∥

∥

∥

2+h

NSminus1

sumi=1

Γ(1minusα)


( f ki )

2 (323)

The substitution of (323) into (322) produces

micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2leρn

∥

∥u0∥

∥

2+n

[

(u00)

2+(u0NS)2]

+∆t

8νn

n

sumk=1

[

(h f k0 )

2+(h f kNS)2]

+∆t

micron

n

sumk=1

hNSminus1

sumi=1

( f ki )

2 (324)

where

ρn =t1minusαn +α(1minusα)εtnminus1∆t

2Γ(2minusα) micron =


Γ(1minusα)

n =t1minus α

2n + α


2Γ(2minus α2 )

νn =tminus α


Γ(1minus α2 )

(325)

Taking θngt 0 such that 1θn+1L

θn = micron

4

(

ie θn = 2(

1+radic

1+L2micron)

(Lmicron))

and followingfrom Lemma 32 we have

∆tn

sumk=1

∥

∥

∥uk∥

∥

∥

2

infinle 2(

1+radic

1+L2micron)

Lmicron

(

micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2)

(326)

Combining (324) with (326) and notice

ρn leρ n le micron gemicro νn geν

we obtain the inequality (318)

The priori estimate leads to the stability of the scheme (314)-(317)

Theorem 32 (Stability) The scheme (314)-(317) is unconditionally stable for any given com-pactly supported initial data and source term

We now present the error analysis of the scheme (314)-(317) whose proof can befound in Appendix B



Theorem 33 (Error Analysis) Suppose u(xt)isinC42xt ([xl xr]times[0T]) and uk

i |0leileNS 0lekleNT are solutions of the problem (34)-(37) and the difference scheme (314)-(317) respectivelyLet ek

i =uki minusu(xitk) Then there exists a positive constant c2 such that

radic

∆tn

sumk=1

ek2infin le c2(h

2+∆t2minusα+ε) 1lenleNT (327)

where c22 =

4c21T(

1+radic

1+L2micro)

Lmicro

(

1ν +

Lmicro

)

with c1 is a positive constant (see (B5)-(B7)) and micro ν are

defined in (319)

Remark 31 The convergence rate established in [15] for the scheme (310)-(313) is C(h2+∆t2minusα) which differs from the error estimate in (327) by a tolerance error ε see Fig 1 forexample This implies that the fast scheme (314)-(317) has almost the same accuracy andconvergence rate as the direct scheme (310)-(313) when ε is chosen to be smaller thanh2+∆t2minusα but is much faster (O(NS NT Nexp) vs O(NSN2

T) work) and needs less storage(O(NS Nexp) vs O(NSNT) memory) when the total time step number NT is moderate andlarge

33 Numerical results

In this section we illustrate the performance of our scheme (314)-(317) by comparing itwith the scheme (310)-(313) from various aspects ndash convergence rate and computationalcost We observe that the scheme (314)-(317) has the same convergence rate O(h2+∆t2minusα) but is much faster than the scheme (310)-(313) even when NT is of moderatesize

Example 31 Now we use the example in [15] with the exact solution of the problem(31)-(33)

u(xt)= x4(πminusx)4[

exp(minusx)t3+α+1]

(xt)isinΩitimes[0T] (328)

and the source term f (xt) in the form of

f (xt)=

Γ(4+α)x4(πminusx)4exp(minusx)t36minusx2(πminusx)2t3+α exp(minusx)[

x2(56minus16x+x2)minus2πx(28minus12x+x2)+π2(12minus8x+x2)]

+4(3π2minus14πx+14x2) (xt)isinΩitimes[0T]

0 (xt) isinΩitimes[0T]

We define the maximum norm of the error and the convergence rates with respect totemporal and spatial sizes respectively by the formulas

E(h∆t)=

radic

radic

radic

radic∆tNT

sumk=1

ek2infin rt = log2

E(h∆t)

E(h∆t2) rs = log2

E(h∆t)

E(h2∆t)



Table 3 Errors and convergence orders in time with fixed spatial size h=π20000 for the fast scheme (314)-(317) and the direct scheme (310)-(313)

∆t

α=02 α=05

Fast scheme Direct scheme Fast scheme Direct scheme

E(hτ) rt E(hτ) rt E(hτ) rt E(hτ) rt

110 1570e-02 170 1570e-02 170 8151e-02 148 8151e-02 148

120 4846e-03 170 4844e-03 170 2922e-02 147 2923e-02 147

140 1489e-03 172 1488e-03 171 1052e-02 148 1052e-02 148

180 4524e-04 170 4541e-04 172 3781e-03 148 3784e-03 148

1160 1395e-04 1375e-04 1357e-03 1357e-03

Table 4 The errors convergence orders in space and CPU time in seconds with fixed temporal step size∆t=130000 for the fast scheme (314)-(317) and the direct scheme (310)-(313)

h

α=02 α=05


E(hτ) rs E(hτ) rs E(hτ) rs E(hτ) rs

π10 8862e-01 206 8632e-01 203 8652e-01 205 8035e-01 199

π20 2122e-01 201 2107e-01 201 2087e-01 201 2016e-01 199

π40 5258e-02 200 5244e-02 200 5177e-02 200 5064e-02 199

π80 1312e-02 200 1311e-02 200 1292e-02 200 1272e-02 199

π160 3280e-03 3277e-03 3229e-03 3195e-03

CPU(s) 4337 330466 4365 222606

where the error ek is measured against the exact solution (328)

In our calculation we set the computational domain Ωi=[0π] T=1 and the toleranceprecision ε=10minus7 for the SOE approximation Table 3 shows the errors in maximum normand temporal convergence rate O(∆t2minusα) for α=0205 computed by the schemes (314)-(317) and (310)-(313) respectively where the spatial mesh size is fixed at h= π

20000 AndTable 4 shows the relative Linfin errors spatial convergence rate O(h2) and CPU time forα=02 and α=05 where the temporal mesh size is fixed at ∆t= 1

30000

From Tables 3 and 4 it is not surprising to see that both schemes have the same errorsand convergence rate O(h2+∆t2minusα) The reason is that as shown in Theorem 33 theinfluence of the SOE approximation error ε is negligible when it is chosen to be verysmall To further investigate the conclusion of Theorem 33 we plot in Fig 1 the evolutionof the difference of the solutions to the schemes (314)-(317) and (310)-(313) using thesame mesh sizes One can see that the difference of the solutions is smaller than the SOEapproximation precision

In Fig 2 we plot out the CPU time (in seconds) of the two schemes by fixing all otherparameters and only varying the total number of time steps NT We observe that whilethe direct scheme scales like O(N2

T) the CPU time increases almost linearly with the totalnumber of time steps NT for the fast scheme



Figure 1 Example 31 The evolution of the difference (ie Error=FUnh minusDUn

h ) of the solutions to the schemes

(314)-(317) (denote by DUnh ) and (310)-(313) (denote by FUn

h ) for ε= 1e-71e-81e-91e-10 respectively Here

we set T=2 Ωi =[0π] M=80 N=2000 and α=05

42 44 46 48 5 5205

1

15

2

25

3

35

4

Log10

(NT)

Log 10

(cpu

time)

Direct EvaluationFast Evaluation

Figure 2 Example 31 The log-log (in base 10) plot of the CPU time (in seconds) versus the total number oftime steps NT for Application I Here NS =30 and α=05

4 Application II Nonlinear fractional diffusion equation

In this section we apply the fast evaluation of the Caputo derivative to solve the follow-ing nonlinear fractional diffusion equation

C0 Dα

t u(xt)=uxx+ f (u)

u(x0)=u0(x)(41)



If f (u) = minusu(1minusu) or f (u) = minus01u(1minusu)(uminus0001) (41) is called the time fractionalFisher equation or Huxley equation respectively

Suppose that the initial value u0(x) is compactly supported on Ωi = [xl xr] Efficientnonlinear absorbing boundary conditions (ABCs) have been derived in [8355657] Withthese ABCs the IVP (41) on unbounded domain can be reformulated into an IBVP on abounded domain given by

C0 Dα

t u(xt)=uxx+ f (u) 0lt tleT xisinΩi (42)


(partx+3sα20 )

C0 Dα

t u+(3sα0partx+s

3α2

0 )u=(partx+3sα20 ) f (u) x= xr (44)

(partxminus3sα20 )

C0 Dα

t u+(3sα0partxminuss

3α2

0 )u=(partxminus3sα20 ) f (u) x= xl (45)

where s0 is the Pade expansion point that is chosen in such a way that it is close to thefrequency of the wave touching on the artificial boundaries

The discretization of the problem (42)-(45) is given as

C0 D

αt Un

i =δ2xUn

i + f (Unminus1i ) 1le ileMminus1 1lenleNT (46)

(δx+3sα20 )

C0 D

αt Un

Mminus1+(3sα0 δx+s

3α2

0 )UnMminus1=(δx+3s

α20 ) f (Unminus1

Mminus1) 1lenleNT (47)

(δxminus3sα20 )

C0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2

0 )Unminus11 =(δxminus3s

α20 ) f (Unminus1

1 ) 1lenleNT (48)

Under the assumption that f (u) isin C2([0T]) it has been shown in [35] that the finitedifferent scheme (46)-(48) has the convergence rate of O(h2+∆t) in Linfin norm wherethe Linfin norm is defined by einfin =max0leileM |ei| Here we focus on the speed-up of theevaluation of the time fractional PDEs Thus we replace the approximation C

0 Dαt by our

fast evaluation scheme FC0 D

αt and have a new discretized scheme given by

FC0 D

αt Un

i =δ2xUn


(δx+3sα20 )

FC0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

FC0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (411)

Using the similar process of numerical analysis as those in [35] we can obtain the conver-gence rate of O(h2+∆t+ε) in Linfin norm for the fast scheme (49)-(411) In fact one onlyneeds to replace the error estimate of C

0 Dαt in Lemma 25 by the one in Lemma 26

41 Numerical examples

We will give two examples ndash the Fisher equation and the Huxley equation to illustratethe performance of our scheme For both examples in order to investigate the con-vergence orders of our scheme the reference solution is computed over a large intervalΩr=[minus1212] with very small mesh sizes h=2minus10 and ∆t=2minus14 We then set Ωi=[minus66]



and the precision for the sum-of-exponentials approximation of the convolution kernelto ε=10minus9 and T=1 The temporal step size is fixed at ∆t=2minus14 when testing the order ofconvergence in space and the spatial step size is fixed at h=2minus10 when testing the orderof convergence in time

Example 41 We consider the time fractional Fisher equation (ie f (u) =minusu(1minusu) in(41)) with the double Gaussian initial value

u(x0)=exp(minus10(xminus05)2)+exp(minus10(x+05)2) (412)

Tables 5 and 6 show again the same accuracy and convergence order O(h2+∆t) in Linfin

norm for both schemes (46)-(48) and (49)-(411) with α=02 05 We plot out in Fig 4 theevolution of the difference of two solutions by taking ε= 1e-71e-81e-91e-10 respectivelyWe observe that the difference decreases as ε decreases Fig 3 shows the CPU time inseconds for both schemes versus NT Again one can see that the fast scheme has almostlinear complexity in NT and is much faster than the direct scheme

Table 5 Errors and convergence orders in time for the fractional Fisher equation by fixing h=2minus10

∆t

α=02 α=05


eninfin rt eninfin rt eninfin rt eninfin rt

110 8601e-04 102 8601e-04 102 2194e-03 109 2194e-03 109

120 4243e-04 101 4243e-04 101 1030e-03 106 1030e-03 106

140 2104e-04 101 2104e-04 101 4934e-04 104 4934e-04 104

180 1046e-04 1046e-04 2392e-04 2392e-04

Table 6 Errors convergence orders in space and CPU time for fractional Fisher equation by fixing ∆t=2minus14

h

α=02 α=05


eninfin rs eninfin rs eninfin rs eninfin rs

180 8196e-04 196 8196e-04 196 6045e-04 197 6045e-04 197

1160 2112e-04 201 2112e-04 201 1547e-04 201 1547e-04 201

1320 5246e-05 200 5246e-05 200 3842e-05 200 3842e-05 200

1640 1316e-05 1316e-05 9649e-06 9649e-06

CPU(s) 3880 107191 4022 85058

Example 42 We consider the fractional Huxley equation (ie f (u) =minus01u(1minusu)(uminus0001) in (41)) with the double Gaussian initial value see Eq (412) Again Tables 7 and8 present the numerical results for α=02 05 which show that our fast scheme (49)-(411)has the same convergence order O(h2+∆t) in Linfin norm as the direct scheme (46)-(48)



4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)


4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)


Figure 3 The log-log (in base 10) plot of the CPU time (in seconds) versus the total number of time steps NTfor two schemes Here NS = 80 and α= 05 The left panel shows the results for the Fisher equation and theright panel shows the results for the Huxley equation

Figure 4 Example 41 (Fisher equation) The evolution of the difference (ie Error= FUnh minusDUn

h ) between the

solution to the scheme (49)-(411) (denote by DUnh ) and the solution to the scheme (46)-(48) (denote by FUn

h )

for ε= 1e-71e-81e-91e-10 respectively In the calculation we take T=5[xl xr]= [minus55] M=160N=2000and α=05

but takes much less computational time in Fig 3 The dependence of ε is plotted in Fig 5to show the difference of the numerical solutions to both schemes Again the abovenumerical tests demonstrate our theoretical analysis



Table 7 Errors and convergence orders in time for fractional Huxley equation with fixed spatial step size h=2minus10

∆t

α=02 α=05



110 1089e-03 104 1089e-03 104 2961e-03 106 2961e-03 106

120 5304e-04 102 5304-04 102 1418e-03 104 1418e-03 104

140 2614e-04 101 2614e-04 101 6896e-04 103 6896e-04 103

180 1294e-04 1294e-04 3379e-04 3379e-04

Table 8 Errors convergence orders in space and CPU time for fractional Huxley equation by fixing ∆t=2minus14

h

α=02 α=05



180 9051e-04 196 9051e-04 196 6882e-04 198 6882e-04 198

1160 2319e-04 201 2319e-04 201 1746e-04 201 1746e-04 201

1320 5762e-05 200 5762e-05 200 4344e-05 200 4344e-05 200

1640 1447e-05 1447e-05 1091e-05 1091e-05

CPU(s) 3869 110658 4014 85587

Figure 5 Example 42 (Huxley equation) The evolution of the difference (ie Error=FUnh minusDUn

h ) between the


h )




5 Conclusions

We have presented an efficient algorithm for solving fractional partial differential equa-tions The algorithm combines the fast evaluation scheme for Caputo fractional deriva-tive C

0 Dαt f (t) and standard finite difference scheme We have proved a prior estimate and

error bound about the solution of our new FD scheme which leads to the stability of thenew scheme The numerical results on linear and nonlinear fraction PDEs show that ournew scheme has the same order of convergence as the existing standard FD schemes butwith nearly optimal complexity in CPU time and storage

Our work can be extended along several directions First it is straightforward to de-sign high order schemes for the evaluation of fractional derivatives Second one maydevelop fast high-order algorithms for solving fractional PDEs which contains fractionalderivatives in both time and space when the current scheme is combined with other ex-isting schemes [11ndash13 25] Third the scheme can be easily incorporated with other dis-cretization schemes for the spatial variable such as finite element methods Fourth effi-cient and stable artificial boundary conditions can be designed using similar techniquesin [24] for solving fractional PDEs in high dimensions These issues are currently underinvestigation and the results will be reported on a later date

Acknowledgments

Shidong Jiang is supported by NSF under grant DMS-1418918 Jiwei Zhang is partiallysupported by NSFC under grants 91430216 and U1530401 Zhimin Zhang is partiallysupported by NSFC grants 11471031 91430216 U1530401 and by NSF under grant DMS-1419040 The authors would like to thank the referees for helpful comments

A The proof of Lemma 27

Proof We rewrite the definition of (227) by

FC0 D

αt un=


(1minusα)∆tαnΓ(1minusα)

+1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

=∆tminusα

n

Γ(1minusα)

(

un

1minusαminus(

α

1minusα+a0)u

nminus1minusnminus2

suml=1

(anminuslminus1+bnminuslminus2)ulminus(bnminus2+

1

nα)u0

)

(A1)

where

an =α∆tαNexp

sumj=1

ωjeminusns j∆tλ1

j bn =α∆tαNexp

sumj=1


j



λ1j =

eminuss j∆t

s2j ∆t

(eminuss j∆tminus1+sj∆t) λ2j =

eminuss j∆t

s2j ∆t

(1minuseminuss j∆tminuseminuss j∆tsj∆t)

Applying the definition (A1) and the Cauchy-Schwarz inequality we have

(FC0 D

αt gk)gk =

1

∆tαΓ(1minusα)

(

1

1minusα(gk)2minus

( α

1minusα+a0

)

gkminus1gk

minuskminus2

suml=1

(akminuslminus1+bkminuslminus2)gl gkminus(

bkminus2+1

kα

)

g0gk

)

ge 1

∆tαΓ(1minusα)

[

(

2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minus1

2

( α

1minusα+a0

)

(gkminus1)2minus 1

2

kminus2

suml=1

(akminuslminus1+bkminuslminus2)(gl)2

minus1

2

(

bkminus2+1

kα

)

(g0)2

)]

(A2)

Summing the above inequality from k=1 to n we obtain

∆tn

sumk=1

(FC0 D

αt gk)gk ge ∆t1minusα

Γ(1minusα)

[

n

sumk=2

(

1

1minusαminus α

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minusn

sumk=2

(

α

2(1minusα)+

a0

2

)

(gkminus1)2minus 1

2

n

sumk=2

kminus2

suml=1

(

akminuslminus1+bkminuslminus2

)

(gl)2

+1

2(1minusα)(g1)2minus 1


2

n

sumk=2

(

bkminus2+1

kα

)

(g0)2

]

=∆t1minusα

Γ(1minusα)

n

sumk=1

(

Ck(gk)2minusC0(g0)2

)

(A3)

where the coefficients Ck (k=01middotmiddotmiddot n) are given by the formula

Cnk =

1

2(1minusα)+

1

2

n

sumk=2

(

bkminus2+1

kα

)

k=0

1

2minus 1

2

nminus2

suml=0

(al+bl)+1

2bnminus2 k=1

1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1

2bnminuskminus1 2le kltn

2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα k=n

(A4)



From (24) we have the estimate

1

t1+αminusεle

Nexp

sumj=1

ωjeminuss jtle 1

t1+α+ε (A5)

It is also straightforward to verify that

kminus2

suml=0

(al+bl)=α∆tαint k∆t

∆t

Nexp

sumj=1

ωjeminuss jtdt (A6)

Combining (A5) and (A6) we obtain

(

1minus 1

kα

)

minusα∆tαtkminus1εlekminus2

suml=0

(al+bl)le(

1minus 1

kα

)

+α∆tαtkminus1ε (A7)

Substituting (A7) into (A4) yields the following estimates

Cn0 le

n1minusα

2(1minusα)+

α∆tαtnminus1ε

2

Cn1 ge

1

2minus 1

2

nminus2

suml=0

(al+bl)ge1

2nαminusα∆tαtnminus1ε

Cnk =1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1

2bnminuskminus1

ge 12nα minusα∆tαtnminus1ε 2le klenminus1

Cnn ge

2minusα

2(1minusα)minus

nminus2

suml=0

(al+bl)minus1

2nαge 1


(A8)

Combining (A3) and (A8) we obtain Lemma 27

B The proof of Theorem 33

Proof We observe that the error eki satisfies the following FD scheme

FC0 D

αt ek

i =δ2xek

i +Tki 1le ileNSminus1 1le kleNT (B1)

FC0 D

αt ek

0=2

h

[

δxek12minusFC

0 Dα2t ek

0

]

+Tk0 (B2)

FC0 D

αt ek

NS=

2

h

[

minusδxenNSminus 1

2minusFC

0 Dα2t ek

NS

]

+TkNS

(B3)

e0i =0 0le ileNS (B4)



where the truncation terms Tk at the interior and boundary points are given by the for-mulas

Tki =minus

[

C0 Dα

t u(xitk)minusFC0 D

αt Uk

i

]

+[

uxx(xitk)minusδ2xUk

i

]

1le ileNSminus1 1le kleNT

Tk0 =

uxx(x0tk)minus2

h

[

δxUk12minusux(x0tk)

]

minus 2

h

[

C0 D

α2t u(x0tk)minusFC

0 D

α2t Uk

0

]

minus[

C0 Dα

t u(x0tk)minusFC0 D

αt Uk

0

]

1le kleNT

TkNS

=

uxx(xNStk)minus

2

h

[

ux(xNStk)minusδxUk

NSminus 12

]

minus 2

h

[

C0 D

α2t u(xNS

tk)minusFC0 D

α2t Uk

NS

]

minus[

C0 Dα

t u(xNStk)minusFC

0 Dαt Uk

NS

]

1le kleNT

Using Lemma 31 and Taylor expansion it is easy to show that the truncation terms Tk

satisfy the following error bounds

∣

∣

∣Tk

i

∣

∣

∣le c1(∆t2minusα+h2+ε) 1le ileNSminus1 1le kleNT (B5)

∣

∣

∣Tk

0

∣

∣

∣le c1(∆t2minusα+h+

∆t2minusα2

h+

ε

h) 1le kleNT (B6)

∣

∣

∣Tk

NS

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B7)

with c1 some positive constant Thus for hle1 and ∆tle1 we have

1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

le c21

2ν

(

h∆t2minusα+∆t2minus α2 +h2+ε

)2+

2c21L

micro

(

∆t2minusα+h2+ε)2

le2c21

ν

(


+2c2

1L

micro

(


le4c21

(1

ν+

L

micro

)

(∆t2minusα+h2)+4c21

(1

ν+

L

micro

)

ε2 (B8)

A direct application of Theorem 31 to the system (B1)-(B4) produces

∆tn

sumk=1

ek2infin le ∆t(1+

radic

1+L2micro)

Lmicro

n

sumk=1

(

1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

)

(B9)

Substituting (B8) into (B9) simplifying the resulting expressions and taking the squareroot for both sides we obtain (327)



References

[1] B Alpert L Greengard and T Hagstrom Rapid evaluation of nonreflecting boundary ker-nels for time-domain wave propagation SIAM J Numer Anal 37 (2000) 1138ndash1164

[2] A A Awotunde R A Ghanam and N Tatar Artificial boundary condition for a modifiedfractional diffusion problem Bound Value Probl 1 (2015) 1ndash17

[3] D Baffet and Jan S Hesthaven A Laplace transform based kernel reduction scheme forfractional differential equations httpsinfoscienceepflchrecord212955 2016

[4] H Brunner H Han and D Yin Artificial boundary conditions and finite difference approx-imations for a time-fractional diffusion-wave equation on a two-dimensional unboundedspatial domain J Comput Phys 276 (2014) 541ndash562

[5] G Beylkin and L Monzon On approximation of functions by exponential sums Appl Com-put Harmon Anal 19 (2005) 17ndash48

[6] G Beylkin and L Monzon Approximation by exponential sums revisited Appl ComputHarmon Anal 28 (2010) 131ndash149

[7] H Brunner H Han and D Yin The maximum principle for time-fractional diffusion equa-tions and its application Numer Funct Anal Optim 36 (2015) 1307ndash1321

[8] H Brunner X Wu and J Zhang Computational Solution of Blow-Up Problems for Semilin-ear Parabolic PDEs on Unbounded Domains SIAM J Sci Comput 31(2010) 4478ndash4496

[9] JX Cao CP Li and YQ Chen High-order approximation to Caputo derivatives andCaputo-type advection-diffusion equations (II) Frac Calc and Appl Anal 183 (2015) 735ndash761

[10] J Cao and C Xu A high order scheme for the numerical solution of the fractional ordinarydifferential equations J Comput Phys 238 (2013) 154ndash168

[11] C Chen F Liu V Anh and I Turner Numerical methods for solving a two-dimensionalvariable-order anomalous subdiffusion equation Math Comp 81 (2012) 345ndash366

[12] M Cui Compact finite difference method for the fractional diffusion equation J ComputPhys 228 (2009) 7792ndash7804

[13] M Cui Compact alternating direction implicit method for two-dimensional time fractionaldiffusion equation J Comput Phys 231 (2012) 2621ndash2633

[14] J R Dea Absorbing boundary conditions for the fractional wave equation Appl MathComput 219 (2013) 9810ndash9820

[15] G H Gao Z Z Sun and Y N Zhang A finite difference scheme for fractional sub-diffusionequations on an unbounded domain using artificial boundary conditions J Comput Phys231 (2012) 2865ndash2879

[16] G H Gao and Z Z Sun The finite difference approximation for a class of fractional sub-diffusion equations on a space unbounded domain J Comput Phys 236 (2013) 443ndash460

[17] G H Gao and Z Z Sun A new fractional numerical differentiation formula to approximatethe Caputo fractional derivative and its applications J Comput Phys 259 (2014) 33ndash50

[18] R Ghaffari and S M Hosseini Obtaining artificial boundary conditions for fractional sub-diffusion equation on space two-dimensional unbounded domains Comput Math Appl68 (2014) 13ndash26

[19] B L Guo Q Xu and A L Zhu A second-order finite difference method for two-dimensionalfractional percolation equations Commun Comput Phys 19 (2016) 733-757

[20] L Greengard and P Lin Spectral approximation of the free-space heat kernel Appl Com-put Harmon Anal 9 (2000) 83ndash97

[21] L Greengard and J Strain A fast algorithm for the evaluation of heat potentials Comm



Pure Appl Math 43 (1990) 949ndash963[22] H Han and X Wu Artificial Boundary Method Tsinghua Univ Press 2013[23] X Huang and D Yin Artificial boundary conditions and finite difference approximations for

a time-fractional diffusion-wave equation on high dimensional unbounded spatial domainJ Comput Phys to appear

[24] S Jiang and L Greengard Efficient representation of nonreflecting boundary conditions forthe time-dependent Schrodinger equation in two dimensions Comm Pure Appl Math 61(2008) 261ndash288

[25] S Jiang L Greengard and W Bao Fast and accurate evaluation of nonlocal Coulomb anddipole-dipole interactions via the nonuniform FFT SIAM J Sci Comput 365 (2014) B777-B794

[26] S Jiang L Greengard and S Wang Efficient sum-of-exponentials approximations for theheat kernel and their applications Adv Comput Math 413 (2015) 529ndash551

[27] R Ke M Ng and H Sun A fast direct method for block triangular Toeplitz-like with tridi-agonal block systems from time-fractional partial differential equations J Comput Phys303 (2015) 203-211

[28] A Kilbas H Srivastava and J Trujillo Theory and Applications of Fractional DifferentialEquations Elesvier Science and Technology Boston 2006

[29] T Langlands and B Henry Fractional chemotaxis diffusion equations Phys Rev E 81(2010) 051102

[30] T Langlands and B Henry The accuracy and stability of an implicit solution method for thefractional diffusion equation J Comput Phys 205 (2005) 719ndash736

[31] S Lei and H Sun A circulant preconditioner for fractional diffusion equations J ComputPhys 242 (2013) 715ndash725

[32] C Li W Deng and Y Wu Numerical analysis and physical simulations for the time frac-tional radial diffusion equation Comput Math Appl 62 (2011) 1024ndash1037

[33] C Li A Chen and J Ye Numerical approaches to fractional calculus and fractional ordinarydifferential equation J Comput Phys 230 (2011) 3352ndash3368

[34] C Li R Wu and H Ding High-order approximation to Caputo derivatives and Caputo-typeadvection-diffusion equations Commun Appl Indu Math 62 e-536 (2014) 1-32

[35] D Li and J Zhang Efficient implementation to numerically solve the nonlinear time frac-tional parabolic problems on unbounded spatial domain J Comput Phys 322 (2016) 415-428

[36] D Li C Zhang M Ran A linear finite difference scheme for generalized time fractionalBurgers equation Appl Math Model 40 (2016) 6069-6081

[37] H Li J Cao C Li High-order approximation to Caputo derivatives and Caputo-typeadvection-diffusion equations (III) J Comput Appl Math 299 (2016) 159-175

[38] J Li A fast time stepping method for evaluating fractional integrals SIAM J Sci Comput31 (2010) 4696ndash4714

[39] LM Li D Xu Alternating direction implicit Galerkin finite element method for the two-dimensional time fractional evolution equation Numer Math Theor Meth Appl 7 (2014)41-57

[40] Y Lin and C Xu Finite differencespectral approximations for the time-fractional diffusionequation J Comput Phys 225 (2007) 1533ndash1552

[41] Y Lin X Li and C Xu Finite differencespectral approximations for the fractional cableequation Math Comp 80 (2011) 1369ndash1396

[42] Y Lin and C Xu Finite differencespectral approximations for time-fractional diffusion



equation J Comput Phys 225 (2007) 1533ndash1552[43] X Lu H Sun and H Pang Fast approximate inversion of a block triangular Toeplitz matrix

with applications to fractional subdiffusion equations J Numer Lin Alg Appl 22 (2015)866-882

[44] C Lubich and A Schadle Fast convolution for nonreflecting boundary conditions SIAM JSci Comput 24 (2002) 161ndash182

[45] W McLean Fast summation by interval clustering for an evolution equation with memorySIAM J Numer Anal 346 (2012) 3039ndash3056

[46] R Metzler and J Klafter The random walks guide to anomalous diffusion a fractionaldynamics approach Phys Rep 339 (2000)

[47] Z Odibat Approximations of fractional integrals and Caputo fractional derivatives ApplMath Comput 178 (2006) 527ndash533

[48] K B Oldham and J Spanier The Fractional Calculus Academic Press New York 1974[49] F W J Olver D W Lozier R F Boisvert and C W Clark editors NIST Handbook of

Mathematical Functions Cambridge University Press New York NY 2010[50] H Pang and H Sun Multigrid method for fractional diffusion equations J Comput Phys

231 (2012) 693ndash703[51] I Podlubny Fractional Differential Equations Academic Press San Diego 1999[52] Z Sun and X Wu A fully discrete difference scheme for a diffusion-wave system Appl

Numer Math 56 (2006) 193ndash209[53] H Wang K Wang and T Sircar A direct O(Nlog2N) finite difference method for fractional

diffusion equations J Comput Phys 229 (2010) 8095ndash8104[54] H Wang and S B Treena A fast finite difference method for two-dimensional space-

fractional diffusion equations SIAM J Sci Comput 34 (2012) 2444ndash2458[55] K Xu and S Jiang A bootstrap method for sum-of-poles approximations J Sci Comput

55 (2013) 16ndash39[56] J Zhang Z Xu X Wu Unified approach to split absorbing boundary conditions for nonlin-

ear Schrodinger equations Phys Rev E 78 (2008) 026709[57] J Zhang Z Xu X Wu Unified approach to split absorbing boundary conditions for nonlin-

ear Schrodinger equations Two dimensional case Phys Rev E 79 (2009) 046711[58] Q Zhang J Zhang S Jiang Z Zhang Numerical solution to a linearized time fractional

KdV equation on unbounded domains accepted by Math Comput 2016[59] C Zheng Approximation stability and fast evaluation of exact artificial boundary condition

for one-dimensional heat equation J Comput Math 25 (2007) 730ndash745



physical process in many applied sciences including biology ecology and control systemThe anomalous diffusion also referred to as the non-Gaussian process has been observedand validated in many phenomena with accurate physical measurement [192829464851] The mathematical and numerical analysis of the factional calculus became a subjectof intensive investigations

In this paper we consider a fast evaluation of the following fractional partial differ-ential equation

C0 Dα

t u(xt)=∆u(xt)+F(uxt) 0ltαlt1 (11)

where the Caputo fractional derivative C0 Dα

t u(xt) is defined by the formula

C0 Dα

t u(xt)=1

Γ(mminusα)

int t

0

u(m)(xτ)

(tminusτ)α+1minusmdτ mminus1ltαltm misinZ (12)

The existing schemes for solving (11) require the storage of the solution at all previoustime steps and the computational complexity of these schemes is O(N2

T NS) with NT thetotal number of time steps and NS the number of grid points in space This is in darkcontrast with the usual diffusion equations where one only needs to store the solution ata fixed number of time steps and the computational complexity is linear with respect toNT

It is easy to see that the difficulty is caused by the Caputo factional derivative ap-peared in (11) Indeed one of the popular schemes of discretizing the Caputo fractionalderivative is the so-called L1 approximation [15 16 23 30 32 36 39ndash41 52] which is sim-ply based on the piecewise linear interpolation of u on each subinterval For 0ltαlt1 theorder of accuracy of the L1 approximation is 2minusα There are also high-order discretiza-tion schemes by using piecewise high-order polynomial interpolation of u [10 17 33 47]and [9 34 37] For each spatial point x these methods require the storage of all previousfunction values u(0)u(t1)middotmiddotmiddot u(tn) and O(n) flops at the nth step Thus it requires onaverage O(NT) storage and the total computational cost is O(N2

T) which forms a bot-tleneck for long time simulations especially when one tries to solve the time fractionalpartial differential equations (PDEs)

Here we present an efficient scheme for solving the fractional PDEs (11) Our key ob-servation is that the Caputo derivative can be evaluated almost as efficient as the usuallyderivatives (besides some logarithmic factors) We first split the convolution integral in(12) into two parts - a local part containing the integral from tminus∆t to t and a history partcontaining the integral from 0 to tminus∆t The local part is approximated using the stan-dard L1 approximation For the history part integration by parts leads to a convolutionintegral of u with the kernel tminus1minusα We show that tminus1minusα (0 lt α lt 1) admits an efficientsum-of-exponentials (SOE) approximation on the interval [δT] with δ = ∆t a uniformabsolute error ε and the number of exponentials needed is of the order

Nexp =O(

log1

ε

(

loglog1

ε+log

T

δ

)

+log1

δ

(

loglog1

ε+log

1

δ

))

(13)













C0 Dα

t u(t)|t=tn =1

Γ(1minusα)

int tn

0

uprime(s)ds

(tnminuss)α



=1

Γ(1minusα)

int tn

tnminus1

uprime(s)ds

(tnminuss)α+

1

Γ(1minusα)

int tnminus1

0

uprime(s)ds

(tnminuss)α

=Cl(tn)+Ch(tn) (21)



We have


∆tnΓ(1minusα)

int tn

tnminus1

1

(tnminuss)αds=


∆tαnΓ(2minusα)

(22)


Ch(tn)=1

Γ(1minusα)

int tnminus1

0

uprime(s)ds

(tnminuss)α

=1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαint tnminus1

0

u(s)ds

(tnminuss)1+α

]

(23)



∣

∣

∣

∣

∣

1

tβminus

Nexp

sumi=1

ωieminussit

∣

∣

∣

∣

∣





tβ=

1

Γ(β)

int infin



function [49]







∣

∣

∣

∣

1

Γ(β)

int infin


∣

∣

∣

∣

le

eminusδp pβminus1

Γ(β)1δ 0ltβle1

eminusδp2βminus1(

pβ

Γ(β)+ 1

δβ

)

1ltβlt2(26)


∣

∣

∣

1

Γ(β)

int infin


∣

∣

∣

∣

=

∣

∣

∣

∣

1

Γ(β)eminustp

int infin


∣

∣

∣

∣


Γ(β)

int infin

0eminusδxdx


Γ(β)

1

δ (27)


∣

∣

∣

1

Γ(β)

int infin


∣

∣

∣

∣

=

∣

∣

∣

∣

1

Γ(β)eminustp

int infin


∣

∣

∣

∣

le eminusδp 1

Γ(β)

(

int p

0(2p)βminus1dx+

int infin


)


Γ(β)

(

pβ+int infin


)


(

pβ

Γ(β)+

1

δβ

)

(28)



eminusδp pβminus1

Γ(β)

1

δlt eminusδp pβ


when 1ltβlt2

eminusδp2βminus1

(

pβ

Γ(β)+

1

δβ

)


(

pβ

Γ(β)+pβ

)


Thus p=O(

log(

1εδ

)

δ)






∣

∣

∣

∣

∣

int b


n

sumk=1


∣

∣

∣

∣

∣

lt252 πaβ

(

e1e

4

)2n

(211)


∣

∣

∣

∣

∣

int b


n

sumk=1


∣

∣

∣

∣

∣

le (bminusa)2n+1

2n+1

(n)4

[(2n)]3max

sisin(ab)

∣

∣

∣D2n


∣

∣ (212)




we obtain(n)4

[(2n)]3lt2

radicπ( e

8

)2nradic

n

n2n (214)


∣

∣Dk

s sβminus1∣

∣

∣= sβminus1le2

radicπradic


∣

∣Dk

s sβminus1∣

∣

∣=∣

∣


∣

∣

∣le2

radicπradic


∣

∣Dk

s sβminus1∣

∣

∣=∣

∣


∣

∣


le2radic


πradic


(215)




∣

∣

∣D2n


∣

∣=

∣

∣

∣

∣

∣

2n

sumk=0

(

2nk

)

(

D2nminusks eminusst

)(

Dks sβminus1

)

∣

∣

∣

∣

∣

le2radic

πradic


sumk=0

(

2nk

)


=2radic

πradic

2nsβminus1eminusst

(

t+2n

es

)2n

(217)




∣

∣

∣

∣

∣

int b


n

sumk=1


∣

∣

∣

∣

∣

le maxalesleb

2radic


(

e(bminusa)t

8n+

bminusa

4s

)2n

le252 πaβeminusat

(

eat

8n+

1

4

)2n

(218)


maxxgt0

eminusx

(

ex

8n+

1

4

)2n

=

(

e1e

4

)2n

nge2 (219)




∣

∣

∣

∣

∣

int a


n

sumk=1

wkeminusskt

∣

∣

∣

∣

∣

lt4radic

πaβn32( e

8

)2n(

aT

nminus1

)2n

(220)


∣

∣

∣

∣

∣

int a


n

sumk=1

wkeminusskt

∣

∣

∣

∣

∣

le a2n+β

2n+β

(n)2[Γ(n+β)]2

(2n)[Γ(2n+β)]2max

sisin(0a)

∣

∣D2ns eminusst

∣

∣ (221)


∣

∣

∣

∣

∣

int a


n

sumk=1

wkeminusskt

∣

∣

∣

∣

∣

le max0leslea

a2n+β



le aβ

2n



le4radic

πaβn32( e

8

)2n(

aT

nminus1

)2n

(222)










log 1ε

)




log 1ε +log 1

δ

)


∣

∣

∣

∣

∣

1

tβminus(

no

sumk=1


sumj=minusM

ns

sumk=1


N

sumj=0

nl

sumk=1


)∣

∣

∣

∣

∣

le ε (223)









∆t=10minus3 α=02 α=05

ǫ

NexpT∆t 103 104 105 106 103 104 105 106

10minus3 25 29 33 37 30 35 40 44

10minus6 34 40 45 51 39 46 52 59

10minus9 43 51 58 66 49 57 65 74


T=1 α=02 α=05

ǫ


10minus3 25 34 43 55 30 41 52 66

10minus6 34 45 58 69 39 51 65 81

10minus8 40 51 64 79 43 57 72 84

Ch(tn)asymp1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωi

int tnminus1


]

=1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

(224)

Define





tnminus2





int tnminus1

tnminus2


s2i ∆tnminus1

[



(226)





23 Error analysis



C0 Dα


∆tαΓ(2minusα)+

1

Γ(1minusα)

[

unminus1

∆tαminus u0

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

FC0 D



FC0 D

αt u1=

∆tminusα




C0 D

αt un=

∆tminusα

Γ(2minusα)

[

a(α)0 unminus

nminus1

sumk=1


(α)nminusk)u


]

(229)




Rnu =C0 Dα

t u(t)∣

∣

t=tnminusC

0 Dαt un (230)

where 0ltαlt1 Then


Γ(2minusα)

(

1minusα

12+

22minusα


)

max0letletn



0 Dαt un in (227) and (228)




FRnu =C0 Dα

t u(t)∣

∣

t=tnminusFC

0 Dαt un (232)

where 0ltαlt1 Then


Γ(2minusα)

(

1minusα

12+

22minusα


)

max0letletn


Γ(1minusα)max

0letletnminus1

|u(t)| (233)


αt un and the L1-

approximation C0 D



∣

∣

∣

FC0 D

αt unminusC

0 Dαt un∣

∣

∣le αε

Γ(1minusα)

nminus1

suml=1

int tl

tlminus1

|Π1lu(s)|ds (234)


tminustlminus1



Γ(1minusα)

nminus1

suml=1

int tl

tlminus1

|Π1lu(s)|ds (235)

where

nminus1

suml=1

int tl

tlminus1






∆tn

sumk=1

(FC0 D



2Γ(1minusα)∆t

n

sumk=1


2Γ(2minusα)(g0)2 (237)






C0 Dα





C0 Dα



partu(xt)

partx=

1

Γ(1minus α2 )

int t

0

us(xs)

(tminuss)α2

ds =C0 D

α2

t u(xt) x= xl (36)

partu(xt)

partx=minus 1

Γ(1minus α2 )

int t

0

us(xs)

(tminuss)α2

ds =minusC0 D

α2

t u(xt) x= xr (37)





j=0


δtuni =

uni minusunminus1

i

∆t δxun

i+ 12=

uni+1minusun

i

h

δ2xun

i =δxui+ 1

2minusδxuiminus 1

2

h δxun

i =un

i+1minusuniminus1

2h




uxx(x0)minus2

h

[

δxu 12minusux(x0)

]

=minush

3uxxx(x0+θ1h) θ1 isin (01) (38)

uxx(xNS)minus 2

h

[


2

]

=h

3uxxx(xNS



C0 D

αt un

i =δ2xun


C0 D

αt un

0 =2

h

[

δxun12minusC

0 Dα2t un

0

]


C0 D

αt un

NS=

2

h

[

minusδxunNSminus 1

2minusC

0 Dα2t un

NS

]

+ f nNS

1lenleNT (312)




FC0 D

αt un

i =δ2xun


FC0 D

αt un

0 =2

h

[

δxun12minusFC

0 Dα2t un

0

]


FC0 D

αt un

NS=

2

h

[

minusδxunNSminus 1

2minusFC

0 Dα2t un

NS

]

+ f nNS

1lenleNT (316)





u2infin le θδxu2+

(1

θ+

1

L

)

u2 forallθgt0





∆tn

sumk=1

∥

∥

∥uk∥

∥

∥

2

infinle2(

1+radic

1+L2micro)

Lmicro

(

ρ∥

∥u0∥

∥

2+[

(u00)

2+(u0NS)2]

+∆t

8ν

n

sumk=1

[

(h f k0 )

2+(h f kNS)2]

+∆t

micro

n

sumk=1

hNSminus1

sumi=1

( f ki )

2

)

(318)



where

ρ=T1minusα


Tminusαminus2αεT

Γ(1minusα)

=T1minus α

2

2Γ(2minus α2 )

ν=Tminus α

2 minusαεT

Γ(1minus α2 )

(319)


have

hNSminus1

sumi=1

(FC0 D

αt uk

i )uki minush

NSminus1

sumi=1

(δ2xuk

i )uki =h

NSminus1

sumi=1

f ki uk

i

Multiplying h2 uk

0 and h2 uk



(FC0 D

αt ukuk)+

[

minus(δxvk12)uk

0minushNSminus1

sumi=1

(δ2xuk

i )uki +(δxuk

NSminus 12)uk

NS

]

+(FC0 D

α2t uk

0)uk0+(FC

0 D

α2t uk

NS)uk

NS=

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS (320)


minus(δxvk12)uk

0minushNSminus1

sumi=1

(δ2xuk

i )uki +(δxuk

NSminus 12)uk

NS=∥

∥

∥δxuk

∥

∥

∥

2 (321)



2Γ(1minusα)

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

tminus α


2Γ(1minus α2 )

n

sumk=1

[

(uk0)

2+(ukNS)2]

+∆tn

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2


2Γ(2minusα)

∥

∥u0∥

∥

2+

t1minus α

2n + α


2Γ(2minus α2 )

[

(u00)

2+(u0NS)2]

+∆tn

sumk=1

[

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS

]

(322)


1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS



le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]

+hNSminus1

sumi=1

[


4Γ(1minusα)(uk

i )2+

Γ(1minusα)


( f ki )

2

]

le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]


4Γ(1minusα)

∥

∥

∥uk∥

∥

∥

2+h

NSminus1

sumi=1

Γ(1minusα)


( f ki )

2 (323)


micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2leρn

∥

∥u0∥

∥

2+n

[

(u00)

2+(u0NS)2]

+∆t

8νn

n

sumk=1

[

(h f k0 )

2+(h f kNS)2]

+∆t

micron

n

sumk=1

hNSminus1

sumi=1

( f ki )

2 (324)

where




Γ(1minusα)

n =t1minus α

2n + α


2Γ(2minus α2 )

νn =tminus α


Γ(1minus α2 )

(325)


θn = micron

4

(

ie θn = 2(

1+radic

1+L2micron)

(Lmicron))


∆tn

sumk=1

∥

∥

∥uk∥

∥

∥

2

infinle 2(

1+radic

1+L2micron)

Lmicron

(

micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2)

(326)












radic

∆tn

sumk=1

ek2infin le c2(h


where c22 =

4c21T(

1+radic

1+L2micro)

Lmicro

(

1ν +

Lmicro

)


defined in (319)







exp(minusx)t3+α+1]



f (xt)=






E(h∆t)=

radic

radic

radic

radic∆tNT

sumk=1

ek2infin rt = log2

E(h∆t)

E(h∆t2) rs = log2

E(h∆t)

E(h2∆t)




∆t

α=02 α=05



110 1570e-02 170 1570e-02 170 8151e-02 148 8151e-02 148

120 4846e-03 170 4844e-03 170 2922e-02 147 2923e-02 147

140 1489e-03 172 1488e-03 171 1052e-02 148 1052e-02 148

180 4524e-04 170 4541e-04 172 3781e-03 148 3784e-03 148

1160 1395e-04 1375e-04 1357e-03 1357e-03


h

α=02 α=05



π10 8862e-01 206 8632e-01 203 8652e-01 205 8035e-01 199

π20 2122e-01 201 2107e-01 201 2087e-01 201 2016e-01 199

π40 5258e-02 200 5244e-02 200 5177e-02 200 5064e-02 199

π80 1312e-02 200 1311e-02 200 1292e-02 200 1272e-02 199

π160 3280e-03 3277e-03 3229e-03 3195e-03

CPU(s) 4337 330466 4365 222606




30000











42 44 46 48 5 5205

1

15

2

25

3

35

4

Log10

(NT)

Log 10

(cpu

time)





C0 Dα

t u(xt)=uxx+ f (u)

u(x0)=u0(x)(41)





C0 Dα



(partx+3sα20 )

C0 Dα

t u+(3sα0partx+s

3α2

0 )u=(partx+3sα20 ) f (u) x= xr (44)

(partxminus3sα20 )

C0 Dα


3α2




C0 D

αt Un

i =δ2xUn


(δx+3sα20 )

C0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

C0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (48)


0 Dαt by our



FC0 D

αt Un

i =δ2xUn


(δx+3sα20 )

FC0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

FC0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (411)













∆t

α=02 α=05



110 8601e-04 102 8601e-04 102 2194e-03 109 2194e-03 109

120 4243e-04 101 4243e-04 101 1030e-03 106 1030e-03 106

140 2104e-04 101 2104e-04 101 4934e-04 104 4934e-04 104

180 1046e-04 1046e-04 2392e-04 2392e-04


h

α=02 α=05



180 8196e-04 196 8196e-04 196 6045e-04 197 6045e-04 197

1160 2112e-04 201 2112e-04 201 1547e-04 201 1547e-04 201

1320 5246e-05 200 5246e-05 200 3842e-05 200 3842e-05 200

1640 1316e-05 1316e-05 9649e-06 9649e-06

CPU(s) 3880 107191 4022 85058




4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)


4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)




h ) between the


h )






∆t

α=02 α=05



110 1089e-03 104 1089e-03 104 2961e-03 106 2961e-03 106

120 5304e-04 102 5304-04 102 1418e-03 104 1418e-03 104

140 2614e-04 101 2614e-04 101 6896e-04 103 6896e-04 103

180 1294e-04 1294e-04 3379e-04 3379e-04


h

α=02 α=05



180 9051e-04 196 9051e-04 196 6882e-04 198 6882e-04 198

1160 2319e-04 201 2319e-04 201 1746e-04 201 1746e-04 201

1320 5762e-05 200 5762e-05 200 4344e-05 200 4344e-05 200

1640 1447e-05 1447e-05 1091e-05 1091e-05

CPU(s) 3869 110658 4014 85587


h ) between the


h )




5 Conclusions





Acknowledgments




FC0 D

αt un=



+1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

=∆tminusα

n

Γ(1minusα)

(

un

1minusαminus(

α

1minusα+a0)u

nminus1minusnminus2

suml=1


1

nα)u0

)

(A1)

where

an =α∆tαNexp

sumj=1


j bn =α∆tαNexp

sumj=1


j



λ1j =

eminuss j∆t

s2j ∆t


eminuss j∆t

s2j ∆t



(FC0 D

αt gk)gk =

1

∆tαΓ(1minusα)

(

1

1minusα(gk)2minus

( α

1minusα+a0

)

gkminus1gk

minuskminus2

suml=1


bkminus2+1

kα

)

g0gk

)

ge 1

∆tαΓ(1minusα)

[

(

2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minus1

2

( α

1minusα+a0

)

(gkminus1)2minus 1

2

kminus2

suml=1


minus1

2

(

bkminus2+1

kα

)

(g0)2

)]

(A2)


∆tn

sumk=1

(FC0 D


Γ(1minusα)

[

n

sumk=2

(

1

1minusαminus α

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minusn

sumk=2

(

α

2(1minusα)+

a0

2

)

(gkminus1)2minus 1

2

n

sumk=2

kminus2

suml=1

(


)

(gl)2

+1



2

n

sumk=2

(

bkminus2+1

kα

)

(g0)2

]

=∆t1minusα

Γ(1minusα)

n

sumk=1

(

Ck(gk)2minusC0(g0)2

)

(A3)


Cnk =

1

2(1minusα)+

1

2

n

sumk=2

(

bkminus2+1

kα

)

k=0

1

2minus 1

2

nminus2

suml=0

(al+bl)+1

2bnminus2 k=1

1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1


2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα k=n

(A4)




1

t1+αminusεle

Nexp

sumj=1

ωjeminuss jtle 1

t1+α+ε (A5)


kminus2

suml=0


∆t

Nexp

sumj=1



(

1minus 1

kα

)


suml=0

(al+bl)le(

1minus 1

kα

)



Cn0 le

n1minusα

2(1minusα)+

α∆tαtnminus1ε

2

Cn1 ge

1

2minus 1

2

nminus2

suml=0

(al+bl)ge1


Cnk =1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1

2bnminuskminus1


Cnn ge

2minusα

2(1minusα)minus

nminus2

suml=0

(al+bl)minus1

2nαge 1


(A8)




FC0 D

αt ek

i =δ2xek


FC0 D

αt ek

0=2

h

[

δxek12minusFC

0 Dα2t ek

0

]

+Tk0 (B2)

FC0 D

αt ek

NS=

2

h

[

minusδxenNSminus 1

2minusFC

0 Dα2t ek

NS

]

+TkNS

(B3)





Tki =minus

[

C0 Dα

t u(xitk)minusFC0 D

αt Uk

i

]

+[


i

]


Tk0 =

uxx(x0tk)minus2

h

[


]

minus 2

h

[

C0 D

α2t u(x0tk)minusFC

0 D

α2t Uk

0

]

minus[

C0 Dα

t u(x0tk)minusFC0 D

αt Uk

0

]

1le kleNT

TkNS

=

uxx(xNStk)minus

2

h

[

ux(xNStk)minusδxUk

NSminus 12

]

minus 2

h

[

C0 D

α2t u(xNS

tk)minusFC0 D

α2t Uk

NS

]

minus[

C0 Dα

t u(xNStk)minusFC

0 Dαt Uk

NS

]

1le kleNT



∣

∣

∣Tk

i

∣

∣


∣

∣

∣Tk

0

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B6)

∣

∣

∣Tk

NS

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B7)


1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

le c21

2ν

(


)2+

2c21L

micro

(


le2c21

ν

(


+2c2

1L

micro

(


le4c21

(1

ν+

L

micro

)


(1

ν+

L

micro

)

ε2 (B8)


∆tn

sumk=1

ek2infin le ∆t(1+

radic

1+L2micro)

Lmicro

n

sumk=1

(

1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

)

(B9)




References












































































C0 Dα

t u(t)|t=tn =1

Γ(1minusα)

int tn

0

uprime(s)ds

(tnminuss)α



=1

Γ(1minusα)

int tn

tnminus1

uprime(s)ds

(tnminuss)α+

1

Γ(1minusα)

int tnminus1

0

uprime(s)ds

(tnminuss)α

=Cl(tn)+Ch(tn) (21)



We have


∆tnΓ(1minusα)

int tn

tnminus1

1

(tnminuss)αds=


∆tαnΓ(2minusα)

(22)


Ch(tn)=1

Γ(1minusα)

int tnminus1

0

uprime(s)ds

(tnminuss)α

=1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαint tnminus1

0

u(s)ds

(tnminuss)1+α

]

(23)



∣

∣

∣

∣

∣

1

tβminus

Nexp

sumi=1

ωieminussit

∣

∣

∣

∣

∣





tβ=

1

Γ(β)

int infin



function [49]







∣

∣

∣

∣

1

Γ(β)

int infin


∣

∣

∣

∣

le

eminusδp pβminus1

Γ(β)1δ 0ltβle1

eminusδp2βminus1(

pβ

Γ(β)+ 1

δβ

)

1ltβlt2(26)


∣

∣

∣

1

Γ(β)

int infin


∣

∣

∣

∣

=

∣

∣

∣

∣

1

Γ(β)eminustp

int infin


∣

∣

∣

∣


Γ(β)

int infin

0eminusδxdx


Γ(β)

1

δ (27)


∣

∣

∣

1

Γ(β)

int infin


∣

∣

∣

∣

=

∣

∣

∣

∣

1

Γ(β)eminustp

int infin


∣

∣

∣

∣

le eminusδp 1

Γ(β)

(

int p

0(2p)βminus1dx+

int infin


)


Γ(β)

(

pβ+int infin


)


(

pβ

Γ(β)+

1

δβ

)

(28)



eminusδp pβminus1

Γ(β)

1

δlt eminusδp pβ


when 1ltβlt2

eminusδp2βminus1

(

pβ

Γ(β)+

1

δβ

)


(

pβ

Γ(β)+pβ

)


Thus p=O(

log(

1εδ

)

δ)






∣

∣

∣

∣

∣

int b


n

sumk=1


∣

∣

∣

∣

∣

lt252 πaβ

(

e1e

4

)2n

(211)


∣

∣

∣

∣

∣

int b


n

sumk=1


∣

∣

∣

∣

∣

le (bminusa)2n+1

2n+1

(n)4

[(2n)]3max

sisin(ab)

∣

∣

∣D2n


∣

∣ (212)




we obtain(n)4

[(2n)]3lt2

radicπ( e

8

)2nradic

n

n2n (214)


∣

∣Dk

s sβminus1∣

∣

∣= sβminus1le2

radicπradic


∣

∣Dk

s sβminus1∣

∣

∣=∣

∣


∣

∣

∣le2

radicπradic


∣

∣Dk

s sβminus1∣

∣

∣=∣

∣


∣

∣


le2radic


πradic


(215)




∣

∣

∣D2n


∣

∣=

∣

∣

∣

∣

∣

2n

sumk=0

(

2nk

)

(

D2nminusks eminusst

)(

Dks sβminus1

)

∣

∣

∣

∣

∣

le2radic

πradic


sumk=0

(

2nk

)


=2radic

πradic

2nsβminus1eminusst

(

t+2n

es

)2n

(217)




∣

∣

∣

∣

∣

int b


n

sumk=1


∣

∣

∣

∣

∣

le maxalesleb

2radic


(

e(bminusa)t

8n+

bminusa

4s

)2n

le252 πaβeminusat

(

eat

8n+

1

4

)2n

(218)


maxxgt0

eminusx

(

ex

8n+

1

4

)2n

=

(

e1e

4

)2n

nge2 (219)




∣

∣

∣

∣

∣

int a


n

sumk=1

wkeminusskt

∣

∣

∣

∣

∣

lt4radic

πaβn32( e

8

)2n(

aT

nminus1

)2n

(220)


∣

∣

∣

∣

∣

int a


n

sumk=1

wkeminusskt

∣

∣

∣

∣

∣

le a2n+β

2n+β

(n)2[Γ(n+β)]2

(2n)[Γ(2n+β)]2max

sisin(0a)

∣

∣D2ns eminusst

∣

∣ (221)


∣

∣

∣

∣

∣

int a


n

sumk=1

wkeminusskt

∣

∣

∣

∣

∣

le max0leslea

a2n+β



le aβ

2n



le4radic

πaβn32( e

8

)2n(

aT

nminus1

)2n

(222)










log 1ε

)




log 1ε +log 1

δ

)


∣

∣

∣

∣

∣

1

tβminus(

no

sumk=1


sumj=minusM

ns

sumk=1


N

sumj=0

nl

sumk=1


)∣

∣

∣

∣

∣

le ε (223)









∆t=10minus3 α=02 α=05

ǫ

NexpT∆t 103 104 105 106 103 104 105 106

10minus3 25 29 33 37 30 35 40 44

10minus6 34 40 45 51 39 46 52 59

10minus9 43 51 58 66 49 57 65 74


T=1 α=02 α=05

ǫ


10minus3 25 34 43 55 30 41 52 66

10minus6 34 45 58 69 39 51 65 81

10minus8 40 51 64 79 43 57 72 84

Ch(tn)asymp1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωi

int tnminus1


]

=1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

(224)

Define





tnminus2





int tnminus1

tnminus2


s2i ∆tnminus1

[



(226)





23 Error analysis



C0 Dα


∆tαΓ(2minusα)+

1

Γ(1minusα)

[

unminus1

∆tαminus u0

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

FC0 D



FC0 D

αt u1=

∆tminusα




C0 D

αt un=

∆tminusα

Γ(2minusα)

[

a(α)0 unminus

nminus1

sumk=1


(α)nminusk)u


]

(229)




Rnu =C0 Dα

t u(t)∣

∣

t=tnminusC

0 Dαt un (230)

where 0ltαlt1 Then


Γ(2minusα)

(

1minusα

12+

22minusα


)

max0letletn



0 Dαt un in (227) and (228)




FRnu =C0 Dα

t u(t)∣

∣

t=tnminusFC

0 Dαt un (232)

where 0ltαlt1 Then


Γ(2minusα)

(

1minusα

12+

22minusα


)

max0letletn


Γ(1minusα)max

0letletnminus1

|u(t)| (233)


αt un and the L1-

approximation C0 D



∣

∣

∣

FC0 D

αt unminusC

0 Dαt un∣

∣

∣le αε

Γ(1minusα)

nminus1

suml=1

int tl

tlminus1

|Π1lu(s)|ds (234)


tminustlminus1



Γ(1minusα)

nminus1

suml=1

int tl

tlminus1

|Π1lu(s)|ds (235)

where

nminus1

suml=1

int tl

tlminus1






∆tn

sumk=1

(FC0 D



2Γ(1minusα)∆t

n

sumk=1


2Γ(2minusα)(g0)2 (237)






C0 Dα





C0 Dα



partu(xt)

partx=

1

Γ(1minus α2 )

int t

0

us(xs)

(tminuss)α2

ds =C0 D

α2

t u(xt) x= xl (36)

partu(xt)

partx=minus 1

Γ(1minus α2 )

int t

0

us(xs)

(tminuss)α2

ds =minusC0 D

α2

t u(xt) x= xr (37)





j=0


δtuni =

uni minusunminus1

i

∆t δxun

i+ 12=

uni+1minusun

i

h

δ2xun

i =δxui+ 1

2minusδxuiminus 1

2

h δxun

i =un

i+1minusuniminus1

2h




uxx(x0)minus2

h

[

δxu 12minusux(x0)

]

=minush

3uxxx(x0+θ1h) θ1 isin (01) (38)

uxx(xNS)minus 2

h

[


2

]

=h

3uxxx(xNS



C0 D

αt un

i =δ2xun


C0 D

αt un

0 =2

h

[

δxun12minusC

0 Dα2t un

0

]


C0 D

αt un

NS=

2

h

[

minusδxunNSminus 1

2minusC

0 Dα2t un

NS

]

+ f nNS

1lenleNT (312)




FC0 D

αt un

i =δ2xun


FC0 D

αt un

0 =2

h

[

δxun12minusFC

0 Dα2t un

0

]


FC0 D

αt un

NS=

2

h

[

minusδxunNSminus 1

2minusFC

0 Dα2t un

NS

]

+ f nNS

1lenleNT (316)





u2infin le θδxu2+

(1

θ+

1

L

)

u2 forallθgt0





∆tn

sumk=1

∥

∥

∥uk∥

∥

∥

2

infinle2(

1+radic

1+L2micro)

Lmicro

(

ρ∥

∥u0∥

∥

2+[

(u00)

2+(u0NS)2]

+∆t

8ν

n

sumk=1

[

(h f k0 )

2+(h f kNS)2]

+∆t

micro

n

sumk=1

hNSminus1

sumi=1

( f ki )

2

)

(318)



where

ρ=T1minusα


Tminusαminus2αεT

Γ(1minusα)

=T1minus α

2

2Γ(2minus α2 )

ν=Tminus α

2 minusαεT

Γ(1minus α2 )

(319)


have

hNSminus1

sumi=1

(FC0 D

αt uk

i )uki minush

NSminus1

sumi=1

(δ2xuk

i )uki =h

NSminus1

sumi=1

f ki uk

i

Multiplying h2 uk

0 and h2 uk



(FC0 D

αt ukuk)+

[

minus(δxvk12)uk

0minushNSminus1

sumi=1

(δ2xuk

i )uki +(δxuk

NSminus 12)uk

NS

]

+(FC0 D

α2t uk

0)uk0+(FC

0 D

α2t uk

NS)uk

NS=

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS (320)


minus(δxvk12)uk

0minushNSminus1

sumi=1

(δ2xuk

i )uki +(δxuk

NSminus 12)uk

NS=∥

∥

∥δxuk

∥

∥

∥

2 (321)



2Γ(1minusα)

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

tminus α


2Γ(1minus α2 )

n

sumk=1

[

(uk0)

2+(ukNS)2]

+∆tn

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2


2Γ(2minusα)

∥

∥u0∥

∥

2+

t1minus α

2n + α


2Γ(2minus α2 )

[

(u00)

2+(u0NS)2]

+∆tn

sumk=1

[

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS

]

(322)


1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS



le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]

+hNSminus1

sumi=1

[


4Γ(1minusα)(uk

i )2+

Γ(1minusα)


( f ki )

2

]

le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]


4Γ(1minusα)

∥

∥

∥uk∥

∥

∥

2+h

NSminus1

sumi=1

Γ(1minusα)


( f ki )

2 (323)


micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2leρn

∥

∥u0∥

∥

2+n

[

(u00)

2+(u0NS)2]

+∆t

8νn

n

sumk=1

[

(h f k0 )

2+(h f kNS)2]

+∆t

micron

n

sumk=1

hNSminus1

sumi=1

( f ki )

2 (324)

where




Γ(1minusα)

n =t1minus α

2n + α


2Γ(2minus α2 )

νn =tminus α


Γ(1minus α2 )

(325)


θn = micron

4

(

ie θn = 2(

1+radic

1+L2micron)

(Lmicron))


∆tn

sumk=1

∥

∥

∥uk∥

∥

∥

2

infinle 2(

1+radic

1+L2micron)

Lmicron

(

micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2)

(326)












radic

∆tn

sumk=1

ek2infin le c2(h


where c22 =

4c21T(

1+radic

1+L2micro)

Lmicro

(

1ν +

Lmicro

)


defined in (319)







exp(minusx)t3+α+1]



f (xt)=






E(h∆t)=

radic

radic

radic

radic∆tNT

sumk=1

ek2infin rt = log2

E(h∆t)

E(h∆t2) rs = log2

E(h∆t)

E(h2∆t)




∆t

α=02 α=05



110 1570e-02 170 1570e-02 170 8151e-02 148 8151e-02 148

120 4846e-03 170 4844e-03 170 2922e-02 147 2923e-02 147

140 1489e-03 172 1488e-03 171 1052e-02 148 1052e-02 148

180 4524e-04 170 4541e-04 172 3781e-03 148 3784e-03 148

1160 1395e-04 1375e-04 1357e-03 1357e-03


h

α=02 α=05



π10 8862e-01 206 8632e-01 203 8652e-01 205 8035e-01 199

π20 2122e-01 201 2107e-01 201 2087e-01 201 2016e-01 199

π40 5258e-02 200 5244e-02 200 5177e-02 200 5064e-02 199

π80 1312e-02 200 1311e-02 200 1292e-02 200 1272e-02 199

π160 3280e-03 3277e-03 3229e-03 3195e-03

CPU(s) 4337 330466 4365 222606




30000











42 44 46 48 5 5205

1

15

2

25

3

35

4

Log10

(NT)

Log 10

(cpu

time)





C0 Dα

t u(xt)=uxx+ f (u)

u(x0)=u0(x)(41)





C0 Dα



(partx+3sα20 )

C0 Dα

t u+(3sα0partx+s

3α2

0 )u=(partx+3sα20 ) f (u) x= xr (44)

(partxminus3sα20 )

C0 Dα


3α2




C0 D

αt Un

i =δ2xUn


(δx+3sα20 )

C0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

C0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (48)


0 Dαt by our



FC0 D

αt Un

i =δ2xUn


(δx+3sα20 )

FC0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

FC0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (411)













∆t

α=02 α=05



110 8601e-04 102 8601e-04 102 2194e-03 109 2194e-03 109

120 4243e-04 101 4243e-04 101 1030e-03 106 1030e-03 106

140 2104e-04 101 2104e-04 101 4934e-04 104 4934e-04 104

180 1046e-04 1046e-04 2392e-04 2392e-04


h

α=02 α=05



180 8196e-04 196 8196e-04 196 6045e-04 197 6045e-04 197

1160 2112e-04 201 2112e-04 201 1547e-04 201 1547e-04 201

1320 5246e-05 200 5246e-05 200 3842e-05 200 3842e-05 200

1640 1316e-05 1316e-05 9649e-06 9649e-06

CPU(s) 3880 107191 4022 85058




4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)


4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)




h ) between the


h )






∆t

α=02 α=05



110 1089e-03 104 1089e-03 104 2961e-03 106 2961e-03 106

120 5304e-04 102 5304-04 102 1418e-03 104 1418e-03 104

140 2614e-04 101 2614e-04 101 6896e-04 103 6896e-04 103

180 1294e-04 1294e-04 3379e-04 3379e-04


h

α=02 α=05



180 9051e-04 196 9051e-04 196 6882e-04 198 6882e-04 198

1160 2319e-04 201 2319e-04 201 1746e-04 201 1746e-04 201

1320 5762e-05 200 5762e-05 200 4344e-05 200 4344e-05 200

1640 1447e-05 1447e-05 1091e-05 1091e-05

CPU(s) 3869 110658 4014 85587


h ) between the


h )




5 Conclusions





Acknowledgments




FC0 D

αt un=



+1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

=∆tminusα

n

Γ(1minusα)

(

un

1minusαminus(

α

1minusα+a0)u

nminus1minusnminus2

suml=1


1

nα)u0

)

(A1)

where

an =α∆tαNexp

sumj=1


j bn =α∆tαNexp

sumj=1


j



λ1j =

eminuss j∆t

s2j ∆t


eminuss j∆t

s2j ∆t



(FC0 D

αt gk)gk =

1

∆tαΓ(1minusα)

(

1

1minusα(gk)2minus

( α

1minusα+a0

)

gkminus1gk

minuskminus2

suml=1


bkminus2+1

kα

)

g0gk

)

ge 1

∆tαΓ(1minusα)

[

(

2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minus1

2

( α

1minusα+a0

)

(gkminus1)2minus 1

2

kminus2

suml=1


minus1

2

(

bkminus2+1

kα

)

(g0)2

)]

(A2)


∆tn

sumk=1

(FC0 D


Γ(1minusα)

[

n

sumk=2

(

1

1minusαminus α

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minusn

sumk=2

(

α

2(1minusα)+

a0

2

)

(gkminus1)2minus 1

2

n

sumk=2

kminus2

suml=1

(


)

(gl)2

+1



2

n

sumk=2

(

bkminus2+1

kα

)

(g0)2

]

=∆t1minusα

Γ(1minusα)

n

sumk=1

(

Ck(gk)2minusC0(g0)2

)

(A3)


Cnk =

1

2(1minusα)+

1

2

n

sumk=2

(

bkminus2+1

kα

)

k=0

1

2minus 1

2

nminus2

suml=0

(al+bl)+1

2bnminus2 k=1

1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1


2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα k=n

(A4)




1

t1+αminusεle

Nexp

sumj=1

ωjeminuss jtle 1

t1+α+ε (A5)


kminus2

suml=0


∆t

Nexp

sumj=1



(

1minus 1

kα

)


suml=0

(al+bl)le(

1minus 1

kα

)



Cn0 le

n1minusα

2(1minusα)+

α∆tαtnminus1ε

2

Cn1 ge

1

2minus 1

2

nminus2

suml=0

(al+bl)ge1


Cnk =1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1

2bnminuskminus1


Cnn ge

2minusα

2(1minusα)minus

nminus2

suml=0

(al+bl)minus1

2nαge 1


(A8)




FC0 D

αt ek

i =δ2xek


FC0 D

αt ek

0=2

h

[

δxek12minusFC

0 Dα2t ek

0

]

+Tk0 (B2)

FC0 D

αt ek

NS=

2

h

[

minusδxenNSminus 1

2minusFC

0 Dα2t ek

NS

]

+TkNS

(B3)





Tki =minus

[

C0 Dα

t u(xitk)minusFC0 D

αt Uk

i

]

+[


i

]


Tk0 =

uxx(x0tk)minus2

h

[


]

minus 2

h

[

C0 D

α2t u(x0tk)minusFC

0 D

α2t Uk

0

]

minus[

C0 Dα

t u(x0tk)minusFC0 D

αt Uk

0

]

1le kleNT

TkNS

=

uxx(xNStk)minus

2

h

[

ux(xNStk)minusδxUk

NSminus 12

]

minus 2

h

[

C0 D

α2t u(xNS

tk)minusFC0 D

α2t Uk

NS

]

minus[

C0 Dα

t u(xNStk)minusFC

0 Dαt Uk

NS

]

1le kleNT



∣

∣

∣Tk

i

∣

∣


∣

∣

∣Tk

0

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B6)

∣

∣

∣Tk

NS

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B7)


1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

le c21

2ν

(


)2+

2c21L

micro

(


le2c21

ν

(


+2c2

1L

micro

(


le4c21

(1

ν+

L

micro

)


(1

ν+

L

micro

)

ε2 (B8)


∆tn

sumk=1

ek2infin le ∆t(1+

radic

1+L2micro)

Lmicro

n

sumk=1

(

1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

)

(B9)




References


































































=1

Γ(1minusα)

int tn

tnminus1

uprime(s)ds

(tnminuss)α+

1

Γ(1minusα)

int tnminus1

0

uprime(s)ds

(tnminuss)α

=Cl(tn)+Ch(tn) (21)



We have


∆tnΓ(1minusα)

int tn

tnminus1

1

(tnminuss)αds=


∆tαnΓ(2minusα)

(22)


Ch(tn)=1

Γ(1minusα)

int tnminus1

0

uprime(s)ds

(tnminuss)α

=1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαint tnminus1

0

u(s)ds

(tnminuss)1+α

]

(23)



∣

∣

∣

∣

∣

1

tβminus

Nexp

sumi=1

ωieminussit

∣

∣

∣

∣

∣





tβ=

1

Γ(β)

int infin



function [49]







∣

∣

∣

∣

1

Γ(β)

int infin


∣

∣

∣

∣

le

eminusδp pβminus1

Γ(β)1δ 0ltβle1

eminusδp2βminus1(

pβ

Γ(β)+ 1

δβ

)

1ltβlt2(26)


∣

∣

∣

1

Γ(β)

int infin


∣

∣

∣

∣

=

∣

∣

∣

∣

1

Γ(β)eminustp

int infin


∣

∣

∣

∣


Γ(β)

int infin

0eminusδxdx


Γ(β)

1

δ (27)


∣

∣

∣

1

Γ(β)

int infin


∣

∣

∣

∣

=

∣

∣

∣

∣

1

Γ(β)eminustp

int infin


∣

∣

∣

∣

le eminusδp 1

Γ(β)

(

int p

0(2p)βminus1dx+

int infin


)


Γ(β)

(

pβ+int infin


)


(

pβ

Γ(β)+

1

δβ

)

(28)



eminusδp pβminus1

Γ(β)

1

δlt eminusδp pβ


when 1ltβlt2

eminusδp2βminus1

(

pβ

Γ(β)+

1

δβ

)


(

pβ

Γ(β)+pβ

)


Thus p=O(

log(

1εδ

)

δ)






∣

∣

∣

∣

∣

int b


n

sumk=1


∣

∣

∣

∣

∣

lt252 πaβ

(

e1e

4

)2n

(211)


∣

∣

∣

∣

∣

int b


n

sumk=1


∣

∣

∣

∣

∣

le (bminusa)2n+1

2n+1

(n)4

[(2n)]3max

sisin(ab)

∣

∣

∣D2n


∣

∣ (212)




we obtain(n)4

[(2n)]3lt2

radicπ( e

8

)2nradic

n

n2n (214)


∣

∣Dk

s sβminus1∣

∣

∣= sβminus1le2

radicπradic


∣

∣Dk

s sβminus1∣

∣

∣=∣

∣


∣

∣

∣le2

radicπradic


∣

∣Dk

s sβminus1∣

∣

∣=∣

∣


∣

∣


le2radic


πradic


(215)




∣

∣

∣D2n


∣

∣=

∣

∣

∣

∣

∣

2n

sumk=0

(

2nk

)

(

D2nminusks eminusst

)(

Dks sβminus1

)

∣

∣

∣

∣

∣

le2radic

πradic


sumk=0

(

2nk

)


=2radic

πradic

2nsβminus1eminusst

(

t+2n

es

)2n

(217)




∣

∣

∣

∣

∣

int b


n

sumk=1


∣

∣

∣

∣

∣

le maxalesleb

2radic


(

e(bminusa)t

8n+

bminusa

4s

)2n

le252 πaβeminusat

(

eat

8n+

1

4

)2n

(218)


maxxgt0

eminusx

(

ex

8n+

1

4

)2n

=

(

e1e

4

)2n

nge2 (219)




∣

∣

∣

∣

∣

int a


n

sumk=1

wkeminusskt

∣

∣

∣

∣

∣

lt4radic

πaβn32( e

8

)2n(

aT

nminus1

)2n

(220)


∣

∣

∣

∣

∣

int a


n

sumk=1

wkeminusskt

∣

∣

∣

∣

∣

le a2n+β

2n+β

(n)2[Γ(n+β)]2

(2n)[Γ(2n+β)]2max

sisin(0a)

∣

∣D2ns eminusst

∣

∣ (221)


∣

∣

∣

∣

∣

int a


n

sumk=1

wkeminusskt

∣

∣

∣

∣

∣

le max0leslea

a2n+β



le aβ

2n



le4radic

πaβn32( e

8

)2n(

aT

nminus1

)2n

(222)










log 1ε

)




log 1ε +log 1

δ

)


∣

∣

∣

∣

∣

1

tβminus(

no

sumk=1


sumj=minusM

ns

sumk=1


N

sumj=0

nl

sumk=1


)∣

∣

∣

∣

∣

le ε (223)









∆t=10minus3 α=02 α=05

ǫ

NexpT∆t 103 104 105 106 103 104 105 106

10minus3 25 29 33 37 30 35 40 44

10minus6 34 40 45 51 39 46 52 59

10minus9 43 51 58 66 49 57 65 74


T=1 α=02 α=05

ǫ


10minus3 25 34 43 55 30 41 52 66

10minus6 34 45 58 69 39 51 65 81

10minus8 40 51 64 79 43 57 72 84

Ch(tn)asymp1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωi

int tnminus1


]

=1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

(224)

Define





tnminus2





int tnminus1

tnminus2


s2i ∆tnminus1

[



(226)





23 Error analysis



C0 Dα


∆tαΓ(2minusα)+

1

Γ(1minusα)

[

unminus1

∆tαminus u0

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

FC0 D



FC0 D

αt u1=

∆tminusα




C0 D

αt un=

∆tminusα

Γ(2minusα)

[

a(α)0 unminus

nminus1

sumk=1


(α)nminusk)u


]

(229)




Rnu =C0 Dα

t u(t)∣

∣

t=tnminusC

0 Dαt un (230)

where 0ltαlt1 Then


Γ(2minusα)

(

1minusα

12+

22minusα


)

max0letletn



0 Dαt un in (227) and (228)




FRnu =C0 Dα

t u(t)∣

∣

t=tnminusFC

0 Dαt un (232)

where 0ltαlt1 Then


Γ(2minusα)

(

1minusα

12+

22minusα


)

max0letletn


Γ(1minusα)max

0letletnminus1

|u(t)| (233)


αt un and the L1-

approximation C0 D



∣

∣

∣

FC0 D

αt unminusC

0 Dαt un∣

∣

∣le αε

Γ(1minusα)

nminus1

suml=1

int tl

tlminus1

|Π1lu(s)|ds (234)


tminustlminus1



Γ(1minusα)

nminus1

suml=1

int tl

tlminus1

|Π1lu(s)|ds (235)

where

nminus1

suml=1

int tl

tlminus1






∆tn

sumk=1

(FC0 D



2Γ(1minusα)∆t

n

sumk=1


2Γ(2minusα)(g0)2 (237)






C0 Dα





C0 Dα



partu(xt)

partx=

1

Γ(1minus α2 )

int t

0

us(xs)

(tminuss)α2

ds =C0 D

α2

t u(xt) x= xl (36)

partu(xt)

partx=minus 1

Γ(1minus α2 )

int t

0

us(xs)

(tminuss)α2

ds =minusC0 D

α2

t u(xt) x= xr (37)





j=0


δtuni =

uni minusunminus1

i

∆t δxun

i+ 12=

uni+1minusun

i

h

δ2xun

i =δxui+ 1

2minusδxuiminus 1

2

h δxun

i =un

i+1minusuniminus1

2h




uxx(x0)minus2

h

[

δxu 12minusux(x0)

]

=minush

3uxxx(x0+θ1h) θ1 isin (01) (38)

uxx(xNS)minus 2

h

[


2

]

=h

3uxxx(xNS



C0 D

αt un

i =δ2xun


C0 D

αt un

0 =2

h

[

δxun12minusC

0 Dα2t un

0

]


C0 D

αt un

NS=

2

h

[

minusδxunNSminus 1

2minusC

0 Dα2t un

NS

]

+ f nNS

1lenleNT (312)




FC0 D

αt un

i =δ2xun


FC0 D

αt un

0 =2

h

[

δxun12minusFC

0 Dα2t un

0

]


FC0 D

αt un

NS=

2

h

[

minusδxunNSminus 1

2minusFC

0 Dα2t un

NS

]

+ f nNS

1lenleNT (316)





u2infin le θδxu2+

(1

θ+

1

L

)

u2 forallθgt0





∆tn

sumk=1

∥

∥

∥uk∥

∥

∥

2

infinle2(

1+radic

1+L2micro)

Lmicro

(

ρ∥

∥u0∥

∥

2+[

(u00)

2+(u0NS)2]

+∆t

8ν

n

sumk=1

[

(h f k0 )

2+(h f kNS)2]

+∆t

micro

n

sumk=1

hNSminus1

sumi=1

( f ki )

2

)

(318)



where

ρ=T1minusα


Tminusαminus2αεT

Γ(1minusα)

=T1minus α

2

2Γ(2minus α2 )

ν=Tminus α

2 minusαεT

Γ(1minus α2 )

(319)


have

hNSminus1

sumi=1

(FC0 D

αt uk

i )uki minush

NSminus1

sumi=1

(δ2xuk

i )uki =h

NSminus1

sumi=1

f ki uk

i

Multiplying h2 uk

0 and h2 uk



(FC0 D

αt ukuk)+

[

minus(δxvk12)uk

0minushNSminus1

sumi=1

(δ2xuk

i )uki +(δxuk

NSminus 12)uk

NS

]

+(FC0 D

α2t uk

0)uk0+(FC

0 D

α2t uk

NS)uk

NS=

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS (320)


minus(δxvk12)uk

0minushNSminus1

sumi=1

(δ2xuk

i )uki +(δxuk

NSminus 12)uk

NS=∥

∥

∥δxuk

∥

∥

∥

2 (321)



2Γ(1minusα)

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

tminus α


2Γ(1minus α2 )

n

sumk=1

[

(uk0)

2+(ukNS)2]

+∆tn

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2


2Γ(2minusα)

∥

∥u0∥

∥

2+

t1minus α

2n + α


2Γ(2minus α2 )

[

(u00)

2+(u0NS)2]

+∆tn

sumk=1

[

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS

]

(322)


1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS



le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]

+hNSminus1

sumi=1

[


4Γ(1minusα)(uk

i )2+

Γ(1minusα)


( f ki )

2

]

le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]


4Γ(1minusα)

∥

∥

∥uk∥

∥

∥

2+h

NSminus1

sumi=1

Γ(1minusα)


( f ki )

2 (323)


micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2leρn

∥

∥u0∥

∥

2+n

[

(u00)

2+(u0NS)2]

+∆t

8νn

n

sumk=1

[

(h f k0 )

2+(h f kNS)2]

+∆t

micron

n

sumk=1

hNSminus1

sumi=1

( f ki )

2 (324)

where




Γ(1minusα)

n =t1minus α

2n + α


2Γ(2minus α2 )

νn =tminus α


Γ(1minus α2 )

(325)


θn = micron

4

(

ie θn = 2(

1+radic

1+L2micron)

(Lmicron))


∆tn

sumk=1

∥

∥

∥uk∥

∥

∥

2

infinle 2(

1+radic

1+L2micron)

Lmicron

(

micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2)

(326)












radic

∆tn

sumk=1

ek2infin le c2(h


where c22 =

4c21T(

1+radic

1+L2micro)

Lmicro

(

1ν +

Lmicro

)


defined in (319)







exp(minusx)t3+α+1]



f (xt)=






E(h∆t)=

radic

radic

radic

radic∆tNT

sumk=1

ek2infin rt = log2

E(h∆t)

E(h∆t2) rs = log2

E(h∆t)

E(h2∆t)




∆t

α=02 α=05



110 1570e-02 170 1570e-02 170 8151e-02 148 8151e-02 148

120 4846e-03 170 4844e-03 170 2922e-02 147 2923e-02 147

140 1489e-03 172 1488e-03 171 1052e-02 148 1052e-02 148

180 4524e-04 170 4541e-04 172 3781e-03 148 3784e-03 148

1160 1395e-04 1375e-04 1357e-03 1357e-03


h

α=02 α=05



π10 8862e-01 206 8632e-01 203 8652e-01 205 8035e-01 199

π20 2122e-01 201 2107e-01 201 2087e-01 201 2016e-01 199

π40 5258e-02 200 5244e-02 200 5177e-02 200 5064e-02 199

π80 1312e-02 200 1311e-02 200 1292e-02 200 1272e-02 199

π160 3280e-03 3277e-03 3229e-03 3195e-03

CPU(s) 4337 330466 4365 222606




30000











42 44 46 48 5 5205

1

15

2

25

3

35

4

Log10

(NT)

Log 10

(cpu

time)





C0 Dα

t u(xt)=uxx+ f (u)

u(x0)=u0(x)(41)





C0 Dα



(partx+3sα20 )

C0 Dα

t u+(3sα0partx+s

3α2

0 )u=(partx+3sα20 ) f (u) x= xr (44)

(partxminus3sα20 )

C0 Dα


3α2




C0 D

αt Un

i =δ2xUn


(δx+3sα20 )

C0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

C0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (48)


0 Dαt by our



FC0 D

αt Un

i =δ2xUn


(δx+3sα20 )

FC0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

FC0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (411)













∆t

α=02 α=05



110 8601e-04 102 8601e-04 102 2194e-03 109 2194e-03 109

120 4243e-04 101 4243e-04 101 1030e-03 106 1030e-03 106

140 2104e-04 101 2104e-04 101 4934e-04 104 4934e-04 104

180 1046e-04 1046e-04 2392e-04 2392e-04


h

α=02 α=05



180 8196e-04 196 8196e-04 196 6045e-04 197 6045e-04 197

1160 2112e-04 201 2112e-04 201 1547e-04 201 1547e-04 201

1320 5246e-05 200 5246e-05 200 3842e-05 200 3842e-05 200

1640 1316e-05 1316e-05 9649e-06 9649e-06

CPU(s) 3880 107191 4022 85058




4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)


4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)




h ) between the


h )






∆t

α=02 α=05



110 1089e-03 104 1089e-03 104 2961e-03 106 2961e-03 106

120 5304e-04 102 5304-04 102 1418e-03 104 1418e-03 104

140 2614e-04 101 2614e-04 101 6896e-04 103 6896e-04 103

180 1294e-04 1294e-04 3379e-04 3379e-04


h

α=02 α=05



180 9051e-04 196 9051e-04 196 6882e-04 198 6882e-04 198

1160 2319e-04 201 2319e-04 201 1746e-04 201 1746e-04 201

1320 5762e-05 200 5762e-05 200 4344e-05 200 4344e-05 200

1640 1447e-05 1447e-05 1091e-05 1091e-05

CPU(s) 3869 110658 4014 85587


h ) between the


h )




5 Conclusions





Acknowledgments




FC0 D

αt un=



+1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

=∆tminusα

n

Γ(1minusα)

(

un

1minusαminus(

α

1minusα+a0)u

nminus1minusnminus2

suml=1


1

nα)u0

)

(A1)

where

an =α∆tαNexp

sumj=1


j bn =α∆tαNexp

sumj=1


j



λ1j =

eminuss j∆t

s2j ∆t


eminuss j∆t

s2j ∆t



(FC0 D

αt gk)gk =

1

∆tαΓ(1minusα)

(

1

1minusα(gk)2minus

( α

1minusα+a0

)

gkminus1gk

minuskminus2

suml=1


bkminus2+1

kα

)

g0gk

)

ge 1

∆tαΓ(1minusα)

[

(

2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minus1

2

( α

1minusα+a0

)

(gkminus1)2minus 1

2

kminus2

suml=1


minus1

2

(

bkminus2+1

kα

)

(g0)2

)]

(A2)


∆tn

sumk=1

(FC0 D


Γ(1minusα)

[

n

sumk=2

(

1

1minusαminus α

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minusn

sumk=2

(

α

2(1minusα)+

a0

2

)

(gkminus1)2minus 1

2

n

sumk=2

kminus2

suml=1

(


)

(gl)2

+1



2

n

sumk=2

(

bkminus2+1

kα

)

(g0)2

]

=∆t1minusα

Γ(1minusα)

n

sumk=1

(

Ck(gk)2minusC0(g0)2

)

(A3)


Cnk =

1

2(1minusα)+

1

2

n

sumk=2

(

bkminus2+1

kα

)

k=0

1

2minus 1

2

nminus2

suml=0

(al+bl)+1

2bnminus2 k=1

1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1


2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα k=n

(A4)




1

t1+αminusεle

Nexp

sumj=1

ωjeminuss jtle 1

t1+α+ε (A5)


kminus2

suml=0


∆t

Nexp

sumj=1



(

1minus 1

kα

)


suml=0

(al+bl)le(

1minus 1

kα

)



Cn0 le

n1minusα

2(1minusα)+

α∆tαtnminus1ε

2

Cn1 ge

1

2minus 1

2

nminus2

suml=0

(al+bl)ge1


Cnk =1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1

2bnminuskminus1


Cnn ge

2minusα

2(1minusα)minus

nminus2

suml=0

(al+bl)minus1

2nαge 1


(A8)




FC0 D

αt ek

i =δ2xek


FC0 D

αt ek

0=2

h

[

δxek12minusFC

0 Dα2t ek

0

]

+Tk0 (B2)

FC0 D

αt ek

NS=

2

h

[

minusδxenNSminus 1

2minusFC

0 Dα2t ek

NS

]

+TkNS

(B3)





Tki =minus

[

C0 Dα

t u(xitk)minusFC0 D

αt Uk

i

]

+[


i

]


Tk0 =

uxx(x0tk)minus2

h

[


]

minus 2

h

[

C0 D

α2t u(x0tk)minusFC

0 D

α2t Uk

0

]

minus[

C0 Dα

t u(x0tk)minusFC0 D

αt Uk

0

]

1le kleNT

TkNS

=

uxx(xNStk)minus

2

h

[

ux(xNStk)minusδxUk

NSminus 12

]

minus 2

h

[

C0 D

α2t u(xNS

tk)minusFC0 D

α2t Uk

NS

]

minus[

C0 Dα

t u(xNStk)minusFC

0 Dαt Uk

NS

]

1le kleNT



∣

∣

∣Tk

i

∣

∣


∣

∣

∣Tk

0

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B6)

∣

∣

∣Tk

NS

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B7)


1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

le c21

2ν

(


)2+

2c21L

micro

(


le2c21

ν

(


+2c2

1L

micro

(


le4c21

(1

ν+

L

micro

)


(1

ν+

L

micro

)

ε2 (B8)


∆tn

sumk=1

ek2infin le ∆t(1+

radic

1+L2micro)

Lmicro

n

sumk=1

(

1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

)

(B9)




References





































































∣

∣

∣

∣

1

Γ(β)

int infin


∣

∣

∣

∣

le

eminusδp pβminus1

Γ(β)1δ 0ltβle1

eminusδp2βminus1(

pβ

Γ(β)+ 1

δβ

)

1ltβlt2(26)


∣

∣

∣

1

Γ(β)

int infin


∣

∣

∣

∣

=

∣

∣

∣

∣

1

Γ(β)eminustp

int infin


∣

∣

∣

∣


Γ(β)

int infin

0eminusδxdx


Γ(β)

1

δ (27)


∣

∣

∣

1

Γ(β)

int infin


∣

∣

∣

∣

=

∣

∣

∣

∣

1

Γ(β)eminustp

int infin


∣

∣

∣

∣

le eminusδp 1

Γ(β)

(

int p

0(2p)βminus1dx+

int infin


)


Γ(β)

(

pβ+int infin


)


(

pβ

Γ(β)+

1

δβ

)

(28)



eminusδp pβminus1

Γ(β)

1

δlt eminusδp pβ


when 1ltβlt2

eminusδp2βminus1

(

pβ

Γ(β)+

1

δβ

)


(

pβ

Γ(β)+pβ

)


Thus p=O(

log(

1εδ

)

δ)






∣

∣

∣

∣

∣

int b


n

sumk=1


∣

∣

∣

∣

∣

lt252 πaβ

(

e1e

4

)2n

(211)


∣

∣

∣

∣

∣

int b


n

sumk=1


∣

∣

∣

∣

∣

le (bminusa)2n+1

2n+1

(n)4

[(2n)]3max

sisin(ab)

∣

∣

∣D2n


∣

∣ (212)




we obtain(n)4

[(2n)]3lt2

radicπ( e

8

)2nradic

n

n2n (214)


∣

∣Dk

s sβminus1∣

∣

∣= sβminus1le2

radicπradic


∣

∣Dk

s sβminus1∣

∣

∣=∣

∣


∣

∣

∣le2

radicπradic


∣

∣Dk

s sβminus1∣

∣

∣=∣

∣


∣

∣


le2radic


πradic


(215)




∣

∣

∣D2n


∣

∣=

∣

∣

∣

∣

∣

2n

sumk=0

(

2nk

)

(

D2nminusks eminusst

)(

Dks sβminus1

)

∣

∣

∣

∣

∣

le2radic

πradic


sumk=0

(

2nk

)


=2radic

πradic

2nsβminus1eminusst

(

t+2n

es

)2n

(217)




∣

∣

∣

∣

∣

int b


n

sumk=1


∣

∣

∣

∣

∣

le maxalesleb

2radic


(

e(bminusa)t

8n+

bminusa

4s

)2n

le252 πaβeminusat

(

eat

8n+

1

4

)2n

(218)


maxxgt0

eminusx

(

ex

8n+

1

4

)2n

=

(

e1e

4

)2n

nge2 (219)




∣

∣

∣

∣

∣

int a


n

sumk=1

wkeminusskt

∣

∣

∣

∣

∣

lt4radic

πaβn32( e

8

)2n(

aT

nminus1

)2n

(220)


∣

∣

∣

∣

∣

int a


n

sumk=1

wkeminusskt

∣

∣

∣

∣

∣

le a2n+β

2n+β

(n)2[Γ(n+β)]2

(2n)[Γ(2n+β)]2max

sisin(0a)

∣

∣D2ns eminusst

∣

∣ (221)


∣

∣

∣

∣

∣

int a


n

sumk=1

wkeminusskt

∣

∣

∣

∣

∣

le max0leslea

a2n+β



le aβ

2n



le4radic

πaβn32( e

8

)2n(

aT

nminus1

)2n

(222)










log 1ε

)




log 1ε +log 1

δ

)


∣

∣

∣

∣

∣

1

tβminus(

no

sumk=1


sumj=minusM

ns

sumk=1


N

sumj=0

nl

sumk=1


)∣

∣

∣

∣

∣

le ε (223)









∆t=10minus3 α=02 α=05

ǫ

NexpT∆t 103 104 105 106 103 104 105 106

10minus3 25 29 33 37 30 35 40 44

10minus6 34 40 45 51 39 46 52 59

10minus9 43 51 58 66 49 57 65 74


T=1 α=02 α=05

ǫ


10minus3 25 34 43 55 30 41 52 66

10minus6 34 45 58 69 39 51 65 81

10minus8 40 51 64 79 43 57 72 84

Ch(tn)asymp1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωi

int tnminus1


]

=1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

(224)

Define





tnminus2





int tnminus1

tnminus2


s2i ∆tnminus1

[



(226)





23 Error analysis



C0 Dα


∆tαΓ(2minusα)+

1

Γ(1minusα)

[

unminus1

∆tαminus u0

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

FC0 D



FC0 D

αt u1=

∆tminusα




C0 D

αt un=

∆tminusα

Γ(2minusα)

[

a(α)0 unminus

nminus1

sumk=1


(α)nminusk)u


]

(229)




Rnu =C0 Dα

t u(t)∣

∣

t=tnminusC

0 Dαt un (230)

where 0ltαlt1 Then


Γ(2minusα)

(

1minusα

12+

22minusα


)

max0letletn



0 Dαt un in (227) and (228)




FRnu =C0 Dα

t u(t)∣

∣

t=tnminusFC

0 Dαt un (232)

where 0ltαlt1 Then


Γ(2minusα)

(

1minusα

12+

22minusα


)

max0letletn


Γ(1minusα)max

0letletnminus1

|u(t)| (233)


αt un and the L1-

approximation C0 D



∣

∣

∣

FC0 D

αt unminusC

0 Dαt un∣

∣

∣le αε

Γ(1minusα)

nminus1

suml=1

int tl

tlminus1

|Π1lu(s)|ds (234)


tminustlminus1



Γ(1minusα)

nminus1

suml=1

int tl

tlminus1

|Π1lu(s)|ds (235)

where

nminus1

suml=1

int tl

tlminus1






∆tn

sumk=1

(FC0 D



2Γ(1minusα)∆t

n

sumk=1


2Γ(2minusα)(g0)2 (237)






C0 Dα





C0 Dα



partu(xt)

partx=

1

Γ(1minus α2 )

int t

0

us(xs)

(tminuss)α2

ds =C0 D

α2

t u(xt) x= xl (36)

partu(xt)

partx=minus 1

Γ(1minus α2 )

int t

0

us(xs)

(tminuss)α2

ds =minusC0 D

α2

t u(xt) x= xr (37)





j=0


δtuni =

uni minusunminus1

i

∆t δxun

i+ 12=

uni+1minusun

i

h

δ2xun

i =δxui+ 1

2minusδxuiminus 1

2

h δxun

i =un

i+1minusuniminus1

2h




uxx(x0)minus2

h

[

δxu 12minusux(x0)

]

=minush

3uxxx(x0+θ1h) θ1 isin (01) (38)

uxx(xNS)minus 2

h

[


2

]

=h

3uxxx(xNS



C0 D

αt un

i =δ2xun


C0 D

αt un

0 =2

h

[

δxun12minusC

0 Dα2t un

0

]


C0 D

αt un

NS=

2

h

[

minusδxunNSminus 1

2minusC

0 Dα2t un

NS

]

+ f nNS

1lenleNT (312)




FC0 D

αt un

i =δ2xun


FC0 D

αt un

0 =2

h

[

δxun12minusFC

0 Dα2t un

0

]


FC0 D

αt un

NS=

2

h

[

minusδxunNSminus 1

2minusFC

0 Dα2t un

NS

]

+ f nNS

1lenleNT (316)





u2infin le θδxu2+

(1

θ+

1

L

)

u2 forallθgt0





∆tn

sumk=1

∥

∥

∥uk∥

∥

∥

2

infinle2(

1+radic

1+L2micro)

Lmicro

(

ρ∥

∥u0∥

∥

2+[

(u00)

2+(u0NS)2]

+∆t

8ν

n

sumk=1

[

(h f k0 )

2+(h f kNS)2]

+∆t

micro

n

sumk=1

hNSminus1

sumi=1

( f ki )

2

)

(318)



where

ρ=T1minusα


Tminusαminus2αεT

Γ(1minusα)

=T1minus α

2

2Γ(2minus α2 )

ν=Tminus α

2 minusαεT

Γ(1minus α2 )

(319)


have

hNSminus1

sumi=1

(FC0 D

αt uk

i )uki minush

NSminus1

sumi=1

(δ2xuk

i )uki =h

NSminus1

sumi=1

f ki uk

i

Multiplying h2 uk

0 and h2 uk



(FC0 D

αt ukuk)+

[

minus(δxvk12)uk

0minushNSminus1

sumi=1

(δ2xuk

i )uki +(δxuk

NSminus 12)uk

NS

]

+(FC0 D

α2t uk

0)uk0+(FC

0 D

α2t uk

NS)uk

NS=

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS (320)


minus(δxvk12)uk

0minushNSminus1

sumi=1

(δ2xuk

i )uki +(δxuk

NSminus 12)uk

NS=∥

∥

∥δxuk

∥

∥

∥

2 (321)



2Γ(1minusα)

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

tminus α


2Γ(1minus α2 )

n

sumk=1

[

(uk0)

2+(ukNS)2]

+∆tn

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2


2Γ(2minusα)

∥

∥u0∥

∥

2+

t1minus α

2n + α


2Γ(2minus α2 )

[

(u00)

2+(u0NS)2]

+∆tn

sumk=1

[

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS

]

(322)


1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS



le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]

+hNSminus1

sumi=1

[


4Γ(1minusα)(uk

i )2+

Γ(1minusα)


( f ki )

2

]

le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]


4Γ(1minusα)

∥

∥

∥uk∥

∥

∥

2+h

NSminus1

sumi=1

Γ(1minusα)


( f ki )

2 (323)


micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2leρn

∥

∥u0∥

∥

2+n

[

(u00)

2+(u0NS)2]

+∆t

8νn

n

sumk=1

[

(h f k0 )

2+(h f kNS)2]

+∆t

micron

n

sumk=1

hNSminus1

sumi=1

( f ki )

2 (324)

where




Γ(1minusα)

n =t1minus α

2n + α


2Γ(2minus α2 )

νn =tminus α


Γ(1minus α2 )

(325)


θn = micron

4

(

ie θn = 2(

1+radic

1+L2micron)

(Lmicron))


∆tn

sumk=1

∥

∥

∥uk∥

∥

∥

2

infinle 2(

1+radic

1+L2micron)

Lmicron

(

micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2)

(326)












radic

∆tn

sumk=1

ek2infin le c2(h


where c22 =

4c21T(

1+radic

1+L2micro)

Lmicro

(

1ν +

Lmicro

)


defined in (319)







exp(minusx)t3+α+1]



f (xt)=






E(h∆t)=

radic

radic

radic

radic∆tNT

sumk=1

ek2infin rt = log2

E(h∆t)

E(h∆t2) rs = log2

E(h∆t)

E(h2∆t)




∆t

α=02 α=05



110 1570e-02 170 1570e-02 170 8151e-02 148 8151e-02 148

120 4846e-03 170 4844e-03 170 2922e-02 147 2923e-02 147

140 1489e-03 172 1488e-03 171 1052e-02 148 1052e-02 148

180 4524e-04 170 4541e-04 172 3781e-03 148 3784e-03 148

1160 1395e-04 1375e-04 1357e-03 1357e-03


h

α=02 α=05



π10 8862e-01 206 8632e-01 203 8652e-01 205 8035e-01 199

π20 2122e-01 201 2107e-01 201 2087e-01 201 2016e-01 199

π40 5258e-02 200 5244e-02 200 5177e-02 200 5064e-02 199

π80 1312e-02 200 1311e-02 200 1292e-02 200 1272e-02 199

π160 3280e-03 3277e-03 3229e-03 3195e-03

CPU(s) 4337 330466 4365 222606




30000











42 44 46 48 5 5205

1

15

2

25

3

35

4

Log10

(NT)

Log 10

(cpu

time)





C0 Dα

t u(xt)=uxx+ f (u)

u(x0)=u0(x)(41)





C0 Dα



(partx+3sα20 )

C0 Dα

t u+(3sα0partx+s

3α2

0 )u=(partx+3sα20 ) f (u) x= xr (44)

(partxminus3sα20 )

C0 Dα


3α2




C0 D

αt Un

i =δ2xUn


(δx+3sα20 )

C0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

C0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (48)


0 Dαt by our



FC0 D

αt Un

i =δ2xUn


(δx+3sα20 )

FC0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

FC0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (411)













∆t

α=02 α=05



110 8601e-04 102 8601e-04 102 2194e-03 109 2194e-03 109

120 4243e-04 101 4243e-04 101 1030e-03 106 1030e-03 106

140 2104e-04 101 2104e-04 101 4934e-04 104 4934e-04 104

180 1046e-04 1046e-04 2392e-04 2392e-04


h

α=02 α=05



180 8196e-04 196 8196e-04 196 6045e-04 197 6045e-04 197

1160 2112e-04 201 2112e-04 201 1547e-04 201 1547e-04 201

1320 5246e-05 200 5246e-05 200 3842e-05 200 3842e-05 200

1640 1316e-05 1316e-05 9649e-06 9649e-06

CPU(s) 3880 107191 4022 85058




4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)


4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)




h ) between the


h )






∆t

α=02 α=05



110 1089e-03 104 1089e-03 104 2961e-03 106 2961e-03 106

120 5304e-04 102 5304-04 102 1418e-03 104 1418e-03 104

140 2614e-04 101 2614e-04 101 6896e-04 103 6896e-04 103

180 1294e-04 1294e-04 3379e-04 3379e-04


h

α=02 α=05



180 9051e-04 196 9051e-04 196 6882e-04 198 6882e-04 198

1160 2319e-04 201 2319e-04 201 1746e-04 201 1746e-04 201

1320 5762e-05 200 5762e-05 200 4344e-05 200 4344e-05 200

1640 1447e-05 1447e-05 1091e-05 1091e-05

CPU(s) 3869 110658 4014 85587


h ) between the


h )




5 Conclusions





Acknowledgments




FC0 D

αt un=



+1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

=∆tminusα

n

Γ(1minusα)

(

un

1minusαminus(

α

1minusα+a0)u

nminus1minusnminus2

suml=1


1

nα)u0

)

(A1)

where

an =α∆tαNexp

sumj=1


j bn =α∆tαNexp

sumj=1


j



λ1j =

eminuss j∆t

s2j ∆t


eminuss j∆t

s2j ∆t



(FC0 D

αt gk)gk =

1

∆tαΓ(1minusα)

(

1

1minusα(gk)2minus

( α

1minusα+a0

)

gkminus1gk

minuskminus2

suml=1


bkminus2+1

kα

)

g0gk

)

ge 1

∆tαΓ(1minusα)

[

(

2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minus1

2

( α

1minusα+a0

)

(gkminus1)2minus 1

2

kminus2

suml=1


minus1

2

(

bkminus2+1

kα

)

(g0)2

)]

(A2)


∆tn

sumk=1

(FC0 D


Γ(1minusα)

[

n

sumk=2

(

1

1minusαminus α

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minusn

sumk=2

(

α

2(1minusα)+

a0

2

)

(gkminus1)2minus 1

2

n

sumk=2

kminus2

suml=1

(


)

(gl)2

+1



2

n

sumk=2

(

bkminus2+1

kα

)

(g0)2

]

=∆t1minusα

Γ(1minusα)

n

sumk=1

(

Ck(gk)2minusC0(g0)2

)

(A3)


Cnk =

1

2(1minusα)+

1

2

n

sumk=2

(

bkminus2+1

kα

)

k=0

1

2minus 1

2

nminus2

suml=0

(al+bl)+1

2bnminus2 k=1

1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1


2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα k=n

(A4)




1

t1+αminusεle

Nexp

sumj=1

ωjeminuss jtle 1

t1+α+ε (A5)


kminus2

suml=0


∆t

Nexp

sumj=1



(

1minus 1

kα

)


suml=0

(al+bl)le(

1minus 1

kα

)



Cn0 le

n1minusα

2(1minusα)+

α∆tαtnminus1ε

2

Cn1 ge

1

2minus 1

2

nminus2

suml=0

(al+bl)ge1


Cnk =1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1

2bnminuskminus1


Cnn ge

2minusα

2(1minusα)minus

nminus2

suml=0

(al+bl)minus1

2nαge 1


(A8)




FC0 D

αt ek

i =δ2xek


FC0 D

αt ek

0=2

h

[

δxek12minusFC

0 Dα2t ek

0

]

+Tk0 (B2)

FC0 D

αt ek

NS=

2

h

[

minusδxenNSminus 1

2minusFC

0 Dα2t ek

NS

]

+TkNS

(B3)





Tki =minus

[

C0 Dα

t u(xitk)minusFC0 D

αt Uk

i

]

+[


i

]


Tk0 =

uxx(x0tk)minus2

h

[


]

minus 2

h

[

C0 D

α2t u(x0tk)minusFC

0 D

α2t Uk

0

]

minus[

C0 Dα

t u(x0tk)minusFC0 D

αt Uk

0

]

1le kleNT

TkNS

=

uxx(xNStk)minus

2

h

[

ux(xNStk)minusδxUk

NSminus 12

]

minus 2

h

[

C0 D

α2t u(xNS

tk)minusFC0 D

α2t Uk

NS

]

minus[

C0 Dα

t u(xNStk)minusFC

0 Dαt Uk

NS

]

1le kleNT



∣

∣

∣Tk

i

∣

∣


∣

∣

∣Tk

0

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B6)

∣

∣

∣Tk

NS

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B7)


1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

le c21

2ν

(


)2+

2c21L

micro

(


le2c21

ν

(


+2c2

1L

micro

(


le4c21

(1

ν+

L

micro

)


(1

ν+

L

micro

)

ε2 (B8)


∆tn

sumk=1

ek2infin le ∆t(1+

radic

1+L2micro)

Lmicro

n

sumk=1

(

1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

)

(B9)




References




































































∣

∣

∣

∣

∣

int b


n

sumk=1


∣

∣

∣

∣

∣

lt252 πaβ

(

e1e

4

)2n

(211)


∣

∣

∣

∣

∣

int b


n

sumk=1


∣

∣

∣

∣

∣

le (bminusa)2n+1

2n+1

(n)4

[(2n)]3max

sisin(ab)

∣

∣

∣D2n


∣

∣ (212)




we obtain(n)4

[(2n)]3lt2

radicπ( e

8

)2nradic

n

n2n (214)


∣

∣Dk

s sβminus1∣

∣

∣= sβminus1le2

radicπradic


∣

∣Dk

s sβminus1∣

∣

∣=∣

∣


∣

∣

∣le2

radicπradic


∣

∣Dk

s sβminus1∣

∣

∣=∣

∣


∣

∣


le2radic


πradic


(215)




∣

∣

∣D2n


∣

∣=

∣

∣

∣

∣

∣

2n

sumk=0

(

2nk

)

(

D2nminusks eminusst

)(

Dks sβminus1

)

∣

∣

∣

∣

∣

le2radic

πradic


sumk=0

(

2nk

)


=2radic

πradic

2nsβminus1eminusst

(

t+2n

es

)2n

(217)




∣

∣

∣

∣

∣

int b


n

sumk=1


∣

∣

∣

∣

∣

le maxalesleb

2radic


(

e(bminusa)t

8n+

bminusa

4s

)2n

le252 πaβeminusat

(

eat

8n+

1

4

)2n

(218)


maxxgt0

eminusx

(

ex

8n+

1

4

)2n

=

(

e1e

4

)2n

nge2 (219)




∣

∣

∣

∣

∣

int a


n

sumk=1

wkeminusskt

∣

∣

∣

∣

∣

lt4radic

πaβn32( e

8

)2n(

aT

nminus1

)2n

(220)


∣

∣

∣

∣

∣

int a


n

sumk=1

wkeminusskt

∣

∣

∣

∣

∣

le a2n+β

2n+β

(n)2[Γ(n+β)]2

(2n)[Γ(2n+β)]2max

sisin(0a)

∣

∣D2ns eminusst

∣

∣ (221)


∣

∣

∣

∣

∣

int a


n

sumk=1

wkeminusskt

∣

∣

∣

∣

∣

le max0leslea

a2n+β



le aβ

2n



le4radic

πaβn32( e

8

)2n(

aT

nminus1

)2n

(222)










log 1ε

)




log 1ε +log 1

δ

)


∣

∣

∣

∣

∣

1

tβminus(

no

sumk=1


sumj=minusM

ns

sumk=1


N

sumj=0

nl

sumk=1


)∣

∣

∣

∣

∣

le ε (223)









∆t=10minus3 α=02 α=05

ǫ

NexpT∆t 103 104 105 106 103 104 105 106

10minus3 25 29 33 37 30 35 40 44

10minus6 34 40 45 51 39 46 52 59

10minus9 43 51 58 66 49 57 65 74


T=1 α=02 α=05

ǫ


10minus3 25 34 43 55 30 41 52 66

10minus6 34 45 58 69 39 51 65 81

10minus8 40 51 64 79 43 57 72 84

Ch(tn)asymp1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωi

int tnminus1


]

=1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

(224)

Define





tnminus2





int tnminus1

tnminus2


s2i ∆tnminus1

[



(226)





23 Error analysis



C0 Dα


∆tαΓ(2minusα)+

1

Γ(1minusα)

[

unminus1

∆tαminus u0

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

FC0 D



FC0 D

αt u1=

∆tminusα




C0 D

αt un=

∆tminusα

Γ(2minusα)

[

a(α)0 unminus

nminus1

sumk=1


(α)nminusk)u


]

(229)




Rnu =C0 Dα

t u(t)∣

∣

t=tnminusC

0 Dαt un (230)

where 0ltαlt1 Then


Γ(2minusα)

(

1minusα

12+

22minusα


)

max0letletn



0 Dαt un in (227) and (228)




FRnu =C0 Dα

t u(t)∣

∣

t=tnminusFC

0 Dαt un (232)

where 0ltαlt1 Then


Γ(2minusα)

(

1minusα

12+

22minusα


)

max0letletn


Γ(1minusα)max

0letletnminus1

|u(t)| (233)


αt un and the L1-

approximation C0 D



∣

∣

∣

FC0 D

αt unminusC

0 Dαt un∣

∣

∣le αε

Γ(1minusα)

nminus1

suml=1

int tl

tlminus1

|Π1lu(s)|ds (234)


tminustlminus1



Γ(1minusα)

nminus1

suml=1

int tl

tlminus1

|Π1lu(s)|ds (235)

where

nminus1

suml=1

int tl

tlminus1






∆tn

sumk=1

(FC0 D



2Γ(1minusα)∆t

n

sumk=1


2Γ(2minusα)(g0)2 (237)






C0 Dα





C0 Dα



partu(xt)

partx=

1

Γ(1minus α2 )

int t

0

us(xs)

(tminuss)α2

ds =C0 D

α2

t u(xt) x= xl (36)

partu(xt)

partx=minus 1

Γ(1minus α2 )

int t

0

us(xs)

(tminuss)α2

ds =minusC0 D

α2

t u(xt) x= xr (37)





j=0


δtuni =

uni minusunminus1

i

∆t δxun

i+ 12=

uni+1minusun

i

h

δ2xun

i =δxui+ 1

2minusδxuiminus 1

2

h δxun

i =un

i+1minusuniminus1

2h




uxx(x0)minus2

h

[

δxu 12minusux(x0)

]

=minush

3uxxx(x0+θ1h) θ1 isin (01) (38)

uxx(xNS)minus 2

h

[


2

]

=h

3uxxx(xNS



C0 D

αt un

i =δ2xun


C0 D

αt un

0 =2

h

[

δxun12minusC

0 Dα2t un

0

]


C0 D

αt un

NS=

2

h

[

minusδxunNSminus 1

2minusC

0 Dα2t un

NS

]

+ f nNS

1lenleNT (312)




FC0 D

αt un

i =δ2xun


FC0 D

αt un

0 =2

h

[

δxun12minusFC

0 Dα2t un

0

]


FC0 D

αt un

NS=

2

h

[

minusδxunNSminus 1

2minusFC

0 Dα2t un

NS

]

+ f nNS

1lenleNT (316)





u2infin le θδxu2+

(1

θ+

1

L

)

u2 forallθgt0





∆tn

sumk=1

∥

∥

∥uk∥

∥

∥

2

infinle2(

1+radic

1+L2micro)

Lmicro

(

ρ∥

∥u0∥

∥

2+[

(u00)

2+(u0NS)2]

+∆t

8ν

n

sumk=1

[

(h f k0 )

2+(h f kNS)2]

+∆t

micro

n

sumk=1

hNSminus1

sumi=1

( f ki )

2

)

(318)



where

ρ=T1minusα


Tminusαminus2αεT

Γ(1minusα)

=T1minus α

2

2Γ(2minus α2 )

ν=Tminus α

2 minusαεT

Γ(1minus α2 )

(319)


have

hNSminus1

sumi=1

(FC0 D

αt uk

i )uki minush

NSminus1

sumi=1

(δ2xuk

i )uki =h

NSminus1

sumi=1

f ki uk

i

Multiplying h2 uk

0 and h2 uk



(FC0 D

αt ukuk)+

[

minus(δxvk12)uk

0minushNSminus1

sumi=1

(δ2xuk

i )uki +(δxuk

NSminus 12)uk

NS

]

+(FC0 D

α2t uk

0)uk0+(FC

0 D

α2t uk

NS)uk

NS=

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS (320)


minus(δxvk12)uk

0minushNSminus1

sumi=1

(δ2xuk

i )uki +(δxuk

NSminus 12)uk

NS=∥

∥

∥δxuk

∥

∥

∥

2 (321)



2Γ(1minusα)

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

tminus α


2Γ(1minus α2 )

n

sumk=1

[

(uk0)

2+(ukNS)2]

+∆tn

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2


2Γ(2minusα)

∥

∥u0∥

∥

2+

t1minus α

2n + α


2Γ(2minus α2 )

[

(u00)

2+(u0NS)2]

+∆tn

sumk=1

[

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS

]

(322)


1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS



le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]

+hNSminus1

sumi=1

[


4Γ(1minusα)(uk

i )2+

Γ(1minusα)


( f ki )

2

]

le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]


4Γ(1minusα)

∥

∥

∥uk∥

∥

∥

2+h

NSminus1

sumi=1

Γ(1minusα)


( f ki )

2 (323)


micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2leρn

∥

∥u0∥

∥

2+n

[

(u00)

2+(u0NS)2]

+∆t

8νn

n

sumk=1

[

(h f k0 )

2+(h f kNS)2]

+∆t

micron

n

sumk=1

hNSminus1

sumi=1

( f ki )

2 (324)

where




Γ(1minusα)

n =t1minus α

2n + α


2Γ(2minus α2 )

νn =tminus α


Γ(1minus α2 )

(325)


θn = micron

4

(

ie θn = 2(

1+radic

1+L2micron)

(Lmicron))


∆tn

sumk=1

∥

∥

∥uk∥

∥

∥

2

infinle 2(

1+radic

1+L2micron)

Lmicron

(

micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2)

(326)












radic

∆tn

sumk=1

ek2infin le c2(h


where c22 =

4c21T(

1+radic

1+L2micro)

Lmicro

(

1ν +

Lmicro

)


defined in (319)







exp(minusx)t3+α+1]



f (xt)=






E(h∆t)=

radic

radic

radic

radic∆tNT

sumk=1

ek2infin rt = log2

E(h∆t)

E(h∆t2) rs = log2

E(h∆t)

E(h2∆t)




∆t

α=02 α=05



110 1570e-02 170 1570e-02 170 8151e-02 148 8151e-02 148

120 4846e-03 170 4844e-03 170 2922e-02 147 2923e-02 147

140 1489e-03 172 1488e-03 171 1052e-02 148 1052e-02 148

180 4524e-04 170 4541e-04 172 3781e-03 148 3784e-03 148

1160 1395e-04 1375e-04 1357e-03 1357e-03


h

α=02 α=05



π10 8862e-01 206 8632e-01 203 8652e-01 205 8035e-01 199

π20 2122e-01 201 2107e-01 201 2087e-01 201 2016e-01 199

π40 5258e-02 200 5244e-02 200 5177e-02 200 5064e-02 199

π80 1312e-02 200 1311e-02 200 1292e-02 200 1272e-02 199

π160 3280e-03 3277e-03 3229e-03 3195e-03

CPU(s) 4337 330466 4365 222606




30000











42 44 46 48 5 5205

1

15

2

25

3

35

4

Log10

(NT)

Log 10

(cpu

time)





C0 Dα

t u(xt)=uxx+ f (u)

u(x0)=u0(x)(41)





C0 Dα



(partx+3sα20 )

C0 Dα

t u+(3sα0partx+s

3α2

0 )u=(partx+3sα20 ) f (u) x= xr (44)

(partxminus3sα20 )

C0 Dα


3α2




C0 D

αt Un

i =δ2xUn


(δx+3sα20 )

C0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

C0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (48)


0 Dαt by our



FC0 D

αt Un

i =δ2xUn


(δx+3sα20 )

FC0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

FC0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (411)













∆t

α=02 α=05



110 8601e-04 102 8601e-04 102 2194e-03 109 2194e-03 109

120 4243e-04 101 4243e-04 101 1030e-03 106 1030e-03 106

140 2104e-04 101 2104e-04 101 4934e-04 104 4934e-04 104

180 1046e-04 1046e-04 2392e-04 2392e-04


h

α=02 α=05



180 8196e-04 196 8196e-04 196 6045e-04 197 6045e-04 197

1160 2112e-04 201 2112e-04 201 1547e-04 201 1547e-04 201

1320 5246e-05 200 5246e-05 200 3842e-05 200 3842e-05 200

1640 1316e-05 1316e-05 9649e-06 9649e-06

CPU(s) 3880 107191 4022 85058




4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)


4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)




h ) between the


h )






∆t

α=02 α=05



110 1089e-03 104 1089e-03 104 2961e-03 106 2961e-03 106

120 5304e-04 102 5304-04 102 1418e-03 104 1418e-03 104

140 2614e-04 101 2614e-04 101 6896e-04 103 6896e-04 103

180 1294e-04 1294e-04 3379e-04 3379e-04


h

α=02 α=05



180 9051e-04 196 9051e-04 196 6882e-04 198 6882e-04 198

1160 2319e-04 201 2319e-04 201 1746e-04 201 1746e-04 201

1320 5762e-05 200 5762e-05 200 4344e-05 200 4344e-05 200

1640 1447e-05 1447e-05 1091e-05 1091e-05

CPU(s) 3869 110658 4014 85587


h ) between the


h )




5 Conclusions





Acknowledgments




FC0 D

αt un=



+1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

=∆tminusα

n

Γ(1minusα)

(

un

1minusαminus(

α

1minusα+a0)u

nminus1minusnminus2

suml=1


1

nα)u0

)

(A1)

where

an =α∆tαNexp

sumj=1


j bn =α∆tαNexp

sumj=1


j



λ1j =

eminuss j∆t

s2j ∆t


eminuss j∆t

s2j ∆t



(FC0 D

αt gk)gk =

1

∆tαΓ(1minusα)

(

1

1minusα(gk)2minus

( α

1minusα+a0

)

gkminus1gk

minuskminus2

suml=1


bkminus2+1

kα

)

g0gk

)

ge 1

∆tαΓ(1minusα)

[

(

2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minus1

2

( α

1minusα+a0

)

(gkminus1)2minus 1

2

kminus2

suml=1


minus1

2

(

bkminus2+1

kα

)

(g0)2

)]

(A2)


∆tn

sumk=1

(FC0 D


Γ(1minusα)

[

n

sumk=2

(

1

1minusαminus α

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minusn

sumk=2

(

α

2(1minusα)+

a0

2

)

(gkminus1)2minus 1

2

n

sumk=2

kminus2

suml=1

(


)

(gl)2

+1



2

n

sumk=2

(

bkminus2+1

kα

)

(g0)2

]

=∆t1minusα

Γ(1minusα)

n

sumk=1

(

Ck(gk)2minusC0(g0)2

)

(A3)


Cnk =

1

2(1minusα)+

1

2

n

sumk=2

(

bkminus2+1

kα

)

k=0

1

2minus 1

2

nminus2

suml=0

(al+bl)+1

2bnminus2 k=1

1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1


2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα k=n

(A4)




1

t1+αminusεle

Nexp

sumj=1

ωjeminuss jtle 1

t1+α+ε (A5)


kminus2

suml=0


∆t

Nexp

sumj=1



(

1minus 1

kα

)


suml=0

(al+bl)le(

1minus 1

kα

)



Cn0 le

n1minusα

2(1minusα)+

α∆tαtnminus1ε

2

Cn1 ge

1

2minus 1

2

nminus2

suml=0

(al+bl)ge1


Cnk =1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1

2bnminuskminus1


Cnn ge

2minusα

2(1minusα)minus

nminus2

suml=0

(al+bl)minus1

2nαge 1


(A8)




FC0 D

αt ek

i =δ2xek


FC0 D

αt ek

0=2

h

[

δxek12minusFC

0 Dα2t ek

0

]

+Tk0 (B2)

FC0 D

αt ek

NS=

2

h

[

minusδxenNSminus 1

2minusFC

0 Dα2t ek

NS

]

+TkNS

(B3)





Tki =minus

[

C0 Dα

t u(xitk)minusFC0 D

αt Uk

i

]

+[


i

]


Tk0 =

uxx(x0tk)minus2

h

[


]

minus 2

h

[

C0 D

α2t u(x0tk)minusFC

0 D

α2t Uk

0

]

minus[

C0 Dα

t u(x0tk)minusFC0 D

αt Uk

0

]

1le kleNT

TkNS

=

uxx(xNStk)minus

2

h

[

ux(xNStk)minusδxUk

NSminus 12

]

minus 2

h

[

C0 D

α2t u(xNS

tk)minusFC0 D

α2t Uk

NS

]

minus[

C0 Dα

t u(xNStk)minusFC

0 Dαt Uk

NS

]

1le kleNT



∣

∣

∣Tk

i

∣

∣


∣

∣

∣Tk

0

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B6)

∣

∣

∣Tk

NS

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B7)


1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

le c21

2ν

(


)2+

2c21L

micro

(


le2c21

ν

(


+2c2

1L

micro

(


le4c21

(1

ν+

L

micro

)


(1

ν+

L

micro

)

ε2 (B8)


∆tn

sumk=1

ek2infin le ∆t(1+

radic

1+L2micro)

Lmicro

n

sumk=1

(

1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

)

(B9)




References



































































∣

∣

∣

∣

∣

int b


n

sumk=1


∣

∣

∣

∣

∣

le maxalesleb

2radic


(

e(bminusa)t

8n+

bminusa

4s

)2n

le252 πaβeminusat

(

eat

8n+

1

4

)2n

(218)


maxxgt0

eminusx

(

ex

8n+

1

4

)2n

=

(

e1e

4

)2n

nge2 (219)




∣

∣

∣

∣

∣

int a


n

sumk=1

wkeminusskt

∣

∣

∣

∣

∣

lt4radic

πaβn32( e

8

)2n(

aT

nminus1

)2n

(220)


∣

∣

∣

∣

∣

int a


n

sumk=1

wkeminusskt

∣

∣

∣

∣

∣

le a2n+β

2n+β

(n)2[Γ(n+β)]2

(2n)[Γ(2n+β)]2max

sisin(0a)

∣

∣D2ns eminusst

∣

∣ (221)


∣

∣

∣

∣

∣

int a


n

sumk=1

wkeminusskt

∣

∣

∣

∣

∣

le max0leslea

a2n+β



le aβ

2n



le4radic

πaβn32( e

8

)2n(

aT

nminus1

)2n

(222)










log 1ε

)




log 1ε +log 1

δ

)


∣

∣

∣

∣

∣

1

tβminus(

no

sumk=1


sumj=minusM

ns

sumk=1


N

sumj=0

nl

sumk=1


)∣

∣

∣

∣

∣

le ε (223)









∆t=10minus3 α=02 α=05

ǫ

NexpT∆t 103 104 105 106 103 104 105 106

10minus3 25 29 33 37 30 35 40 44

10minus6 34 40 45 51 39 46 52 59

10minus9 43 51 58 66 49 57 65 74


T=1 α=02 α=05

ǫ


10minus3 25 34 43 55 30 41 52 66

10minus6 34 45 58 69 39 51 65 81

10minus8 40 51 64 79 43 57 72 84

Ch(tn)asymp1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωi

int tnminus1


]

=1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

(224)

Define





tnminus2





int tnminus1

tnminus2


s2i ∆tnminus1

[



(226)





23 Error analysis



C0 Dα


∆tαΓ(2minusα)+

1

Γ(1minusα)

[

unminus1

∆tαminus u0

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

FC0 D



FC0 D

αt u1=

∆tminusα




C0 D

αt un=

∆tminusα

Γ(2minusα)

[

a(α)0 unminus

nminus1

sumk=1


(α)nminusk)u


]

(229)




Rnu =C0 Dα

t u(t)∣

∣

t=tnminusC

0 Dαt un (230)

where 0ltαlt1 Then


Γ(2minusα)

(

1minusα

12+

22minusα


)

max0letletn



0 Dαt un in (227) and (228)




FRnu =C0 Dα

t u(t)∣

∣

t=tnminusFC

0 Dαt un (232)

where 0ltαlt1 Then


Γ(2minusα)

(

1minusα

12+

22minusα


)

max0letletn


Γ(1minusα)max

0letletnminus1

|u(t)| (233)


αt un and the L1-

approximation C0 D



∣

∣

∣

FC0 D

αt unminusC

0 Dαt un∣

∣

∣le αε

Γ(1minusα)

nminus1

suml=1

int tl

tlminus1

|Π1lu(s)|ds (234)


tminustlminus1



Γ(1minusα)

nminus1

suml=1

int tl

tlminus1

|Π1lu(s)|ds (235)

where

nminus1

suml=1

int tl

tlminus1






∆tn

sumk=1

(FC0 D



2Γ(1minusα)∆t

n

sumk=1


2Γ(2minusα)(g0)2 (237)






C0 Dα





C0 Dα



partu(xt)

partx=

1

Γ(1minus α2 )

int t

0

us(xs)

(tminuss)α2

ds =C0 D

α2

t u(xt) x= xl (36)

partu(xt)

partx=minus 1

Γ(1minus α2 )

int t

0

us(xs)

(tminuss)α2

ds =minusC0 D

α2

t u(xt) x= xr (37)





j=0


δtuni =

uni minusunminus1

i

∆t δxun

i+ 12=

uni+1minusun

i

h

δ2xun

i =δxui+ 1

2minusδxuiminus 1

2

h δxun

i =un

i+1minusuniminus1

2h




uxx(x0)minus2

h

[

δxu 12minusux(x0)

]

=minush

3uxxx(x0+θ1h) θ1 isin (01) (38)

uxx(xNS)minus 2

h

[


2

]

=h

3uxxx(xNS



C0 D

αt un

i =δ2xun


C0 D

αt un

0 =2

h

[

δxun12minusC

0 Dα2t un

0

]


C0 D

αt un

NS=

2

h

[

minusδxunNSminus 1

2minusC

0 Dα2t un

NS

]

+ f nNS

1lenleNT (312)




FC0 D

αt un

i =δ2xun


FC0 D

αt un

0 =2

h

[

δxun12minusFC

0 Dα2t un

0

]


FC0 D

αt un

NS=

2

h

[

minusδxunNSminus 1

2minusFC

0 Dα2t un

NS

]

+ f nNS

1lenleNT (316)





u2infin le θδxu2+

(1

θ+

1

L

)

u2 forallθgt0





∆tn

sumk=1

∥

∥

∥uk∥

∥

∥

2

infinle2(

1+radic

1+L2micro)

Lmicro

(

ρ∥

∥u0∥

∥

2+[

(u00)

2+(u0NS)2]

+∆t

8ν

n

sumk=1

[

(h f k0 )

2+(h f kNS)2]

+∆t

micro

n

sumk=1

hNSminus1

sumi=1

( f ki )

2

)

(318)



where

ρ=T1minusα


Tminusαminus2αεT

Γ(1minusα)

=T1minus α

2

2Γ(2minus α2 )

ν=Tminus α

2 minusαεT

Γ(1minus α2 )

(319)


have

hNSminus1

sumi=1

(FC0 D

αt uk

i )uki minush

NSminus1

sumi=1

(δ2xuk

i )uki =h

NSminus1

sumi=1

f ki uk

i

Multiplying h2 uk

0 and h2 uk



(FC0 D

αt ukuk)+

[

minus(δxvk12)uk

0minushNSminus1

sumi=1

(δ2xuk

i )uki +(δxuk

NSminus 12)uk

NS

]

+(FC0 D

α2t uk

0)uk0+(FC

0 D

α2t uk

NS)uk

NS=

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS (320)


minus(δxvk12)uk

0minushNSminus1

sumi=1

(δ2xuk

i )uki +(δxuk

NSminus 12)uk

NS=∥

∥

∥δxuk

∥

∥

∥

2 (321)



2Γ(1minusα)

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

tminus α


2Γ(1minus α2 )

n

sumk=1

[

(uk0)

2+(ukNS)2]

+∆tn

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2


2Γ(2minusα)

∥

∥u0∥

∥

2+

t1minus α

2n + α


2Γ(2minus α2 )

[

(u00)

2+(u0NS)2]

+∆tn

sumk=1

[

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS

]

(322)


1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS



le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]

+hNSminus1

sumi=1

[


4Γ(1minusα)(uk

i )2+

Γ(1minusα)


( f ki )

2

]

le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]


4Γ(1minusα)

∥

∥

∥uk∥

∥

∥

2+h

NSminus1

sumi=1

Γ(1minusα)


( f ki )

2 (323)


micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2leρn

∥

∥u0∥

∥

2+n

[

(u00)

2+(u0NS)2]

+∆t

8νn

n

sumk=1

[

(h f k0 )

2+(h f kNS)2]

+∆t

micron

n

sumk=1

hNSminus1

sumi=1

( f ki )

2 (324)

where




Γ(1minusα)

n =t1minus α

2n + α


2Γ(2minus α2 )

νn =tminus α


Γ(1minus α2 )

(325)


θn = micron

4

(

ie θn = 2(

1+radic

1+L2micron)

(Lmicron))


∆tn

sumk=1

∥

∥

∥uk∥

∥

∥

2

infinle 2(

1+radic

1+L2micron)

Lmicron

(

micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2)

(326)












radic

∆tn

sumk=1

ek2infin le c2(h


where c22 =

4c21T(

1+radic

1+L2micro)

Lmicro

(

1ν +

Lmicro

)


defined in (319)







exp(minusx)t3+α+1]



f (xt)=






E(h∆t)=

radic

radic

radic

radic∆tNT

sumk=1

ek2infin rt = log2

E(h∆t)

E(h∆t2) rs = log2

E(h∆t)

E(h2∆t)




∆t

α=02 α=05



110 1570e-02 170 1570e-02 170 8151e-02 148 8151e-02 148

120 4846e-03 170 4844e-03 170 2922e-02 147 2923e-02 147

140 1489e-03 172 1488e-03 171 1052e-02 148 1052e-02 148

180 4524e-04 170 4541e-04 172 3781e-03 148 3784e-03 148

1160 1395e-04 1375e-04 1357e-03 1357e-03


h

α=02 α=05



π10 8862e-01 206 8632e-01 203 8652e-01 205 8035e-01 199

π20 2122e-01 201 2107e-01 201 2087e-01 201 2016e-01 199

π40 5258e-02 200 5244e-02 200 5177e-02 200 5064e-02 199

π80 1312e-02 200 1311e-02 200 1292e-02 200 1272e-02 199

π160 3280e-03 3277e-03 3229e-03 3195e-03

CPU(s) 4337 330466 4365 222606




30000











42 44 46 48 5 5205

1

15

2

25

3

35

4

Log10

(NT)

Log 10

(cpu

time)





C0 Dα

t u(xt)=uxx+ f (u)

u(x0)=u0(x)(41)





C0 Dα



(partx+3sα20 )

C0 Dα

t u+(3sα0partx+s

3α2

0 )u=(partx+3sα20 ) f (u) x= xr (44)

(partxminus3sα20 )

C0 Dα


3α2




C0 D

αt Un

i =δ2xUn


(δx+3sα20 )

C0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

C0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (48)


0 Dαt by our



FC0 D

αt Un

i =δ2xUn


(δx+3sα20 )

FC0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

FC0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (411)













∆t

α=02 α=05



110 8601e-04 102 8601e-04 102 2194e-03 109 2194e-03 109

120 4243e-04 101 4243e-04 101 1030e-03 106 1030e-03 106

140 2104e-04 101 2104e-04 101 4934e-04 104 4934e-04 104

180 1046e-04 1046e-04 2392e-04 2392e-04


h

α=02 α=05



180 8196e-04 196 8196e-04 196 6045e-04 197 6045e-04 197

1160 2112e-04 201 2112e-04 201 1547e-04 201 1547e-04 201

1320 5246e-05 200 5246e-05 200 3842e-05 200 3842e-05 200

1640 1316e-05 1316e-05 9649e-06 9649e-06

CPU(s) 3880 107191 4022 85058




4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)


4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)




h ) between the


h )






∆t

α=02 α=05



110 1089e-03 104 1089e-03 104 2961e-03 106 2961e-03 106

120 5304e-04 102 5304-04 102 1418e-03 104 1418e-03 104

140 2614e-04 101 2614e-04 101 6896e-04 103 6896e-04 103

180 1294e-04 1294e-04 3379e-04 3379e-04


h

α=02 α=05



180 9051e-04 196 9051e-04 196 6882e-04 198 6882e-04 198

1160 2319e-04 201 2319e-04 201 1746e-04 201 1746e-04 201

1320 5762e-05 200 5762e-05 200 4344e-05 200 4344e-05 200

1640 1447e-05 1447e-05 1091e-05 1091e-05

CPU(s) 3869 110658 4014 85587


h ) between the


h )




5 Conclusions





Acknowledgments




FC0 D

αt un=



+1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

=∆tminusα

n

Γ(1minusα)

(

un

1minusαminus(

α

1minusα+a0)u

nminus1minusnminus2

suml=1


1

nα)u0

)

(A1)

where

an =α∆tαNexp

sumj=1


j bn =α∆tαNexp

sumj=1


j



λ1j =

eminuss j∆t

s2j ∆t


eminuss j∆t

s2j ∆t



(FC0 D

αt gk)gk =

1

∆tαΓ(1minusα)

(

1

1minusα(gk)2minus

( α

1minusα+a0

)

gkminus1gk

minuskminus2

suml=1


bkminus2+1

kα

)

g0gk

)

ge 1

∆tαΓ(1minusα)

[

(

2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minus1

2

( α

1minusα+a0

)

(gkminus1)2minus 1

2

kminus2

suml=1


minus1

2

(

bkminus2+1

kα

)

(g0)2

)]

(A2)


∆tn

sumk=1

(FC0 D


Γ(1minusα)

[

n

sumk=2

(

1

1minusαminus α

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minusn

sumk=2

(

α

2(1minusα)+

a0

2

)

(gkminus1)2minus 1

2

n

sumk=2

kminus2

suml=1

(


)

(gl)2

+1



2

n

sumk=2

(

bkminus2+1

kα

)

(g0)2

]

=∆t1minusα

Γ(1minusα)

n

sumk=1

(

Ck(gk)2minusC0(g0)2

)

(A3)


Cnk =

1

2(1minusα)+

1

2

n

sumk=2

(

bkminus2+1

kα

)

k=0

1

2minus 1

2

nminus2

suml=0

(al+bl)+1

2bnminus2 k=1

1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1


2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα k=n

(A4)




1

t1+αminusεle

Nexp

sumj=1

ωjeminuss jtle 1

t1+α+ε (A5)


kminus2

suml=0


∆t

Nexp

sumj=1



(

1minus 1

kα

)


suml=0

(al+bl)le(

1minus 1

kα

)



Cn0 le

n1minusα

2(1minusα)+

α∆tαtnminus1ε

2

Cn1 ge

1

2minus 1

2

nminus2

suml=0

(al+bl)ge1


Cnk =1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1

2bnminuskminus1


Cnn ge

2minusα

2(1minusα)minus

nminus2

suml=0

(al+bl)minus1

2nαge 1


(A8)




FC0 D

αt ek

i =δ2xek


FC0 D

αt ek

0=2

h

[

δxek12minusFC

0 Dα2t ek

0

]

+Tk0 (B2)

FC0 D

αt ek

NS=

2

h

[

minusδxenNSminus 1

2minusFC

0 Dα2t ek

NS

]

+TkNS

(B3)





Tki =minus

[

C0 Dα

t u(xitk)minusFC0 D

αt Uk

i

]

+[


i

]


Tk0 =

uxx(x0tk)minus2

h

[


]

minus 2

h

[

C0 D

α2t u(x0tk)minusFC

0 D

α2t Uk

0

]

minus[

C0 Dα

t u(x0tk)minusFC0 D

αt Uk

0

]

1le kleNT

TkNS

=

uxx(xNStk)minus

2

h

[

ux(xNStk)minusδxUk

NSminus 12

]

minus 2

h

[

C0 D

α2t u(xNS

tk)minusFC0 D

α2t Uk

NS

]

minus[

C0 Dα

t u(xNStk)minusFC

0 Dαt Uk

NS

]

1le kleNT



∣

∣

∣Tk

i

∣

∣


∣

∣

∣Tk

0

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B6)

∣

∣

∣Tk

NS

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B7)


1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

le c21

2ν

(


)2+

2c21L

micro

(


le2c21

ν

(


+2c2

1L

micro

(


le4c21

(1

ν+

L

micro

)


(1

ν+

L

micro

)

ε2 (B8)


∆tn

sumk=1

ek2infin le ∆t(1+

radic

1+L2micro)

Lmicro

n

sumk=1

(

1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

)

(B9)




References








































































log 1ε

)




log 1ε +log 1

δ

)


∣

∣

∣

∣

∣

1

tβminus(

no

sumk=1


sumj=minusM

ns

sumk=1


N

sumj=0

nl

sumk=1


)∣

∣

∣

∣

∣

le ε (223)









∆t=10minus3 α=02 α=05

ǫ

NexpT∆t 103 104 105 106 103 104 105 106

10minus3 25 29 33 37 30 35 40 44

10minus6 34 40 45 51 39 46 52 59

10minus9 43 51 58 66 49 57 65 74


T=1 α=02 α=05

ǫ


10minus3 25 34 43 55 30 41 52 66

10minus6 34 45 58 69 39 51 65 81

10minus8 40 51 64 79 43 57 72 84

Ch(tn)asymp1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωi

int tnminus1


]

=1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

(224)

Define





tnminus2





int tnminus1

tnminus2


s2i ∆tnminus1

[



(226)





23 Error analysis



C0 Dα


∆tαΓ(2minusα)+

1

Γ(1minusα)

[

unminus1

∆tαminus u0

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

FC0 D



FC0 D

αt u1=

∆tminusα




C0 D

αt un=

∆tminusα

Γ(2minusα)

[

a(α)0 unminus

nminus1

sumk=1


(α)nminusk)u


]

(229)




Rnu =C0 Dα

t u(t)∣

∣

t=tnminusC

0 Dαt un (230)

where 0ltαlt1 Then


Γ(2minusα)

(

1minusα

12+

22minusα


)

max0letletn



0 Dαt un in (227) and (228)




FRnu =C0 Dα

t u(t)∣

∣

t=tnminusFC

0 Dαt un (232)

where 0ltαlt1 Then


Γ(2minusα)

(

1minusα

12+

22minusα


)

max0letletn


Γ(1minusα)max

0letletnminus1

|u(t)| (233)


αt un and the L1-

approximation C0 D



∣

∣

∣

FC0 D

αt unminusC

0 Dαt un∣

∣

∣le αε

Γ(1minusα)

nminus1

suml=1

int tl

tlminus1

|Π1lu(s)|ds (234)


tminustlminus1



Γ(1minusα)

nminus1

suml=1

int tl

tlminus1

|Π1lu(s)|ds (235)

where

nminus1

suml=1

int tl

tlminus1






∆tn

sumk=1

(FC0 D



2Γ(1minusα)∆t

n

sumk=1


2Γ(2minusα)(g0)2 (237)






C0 Dα





C0 Dα



partu(xt)

partx=

1

Γ(1minus α2 )

int t

0

us(xs)

(tminuss)α2

ds =C0 D

α2

t u(xt) x= xl (36)

partu(xt)

partx=minus 1

Γ(1minus α2 )

int t

0

us(xs)

(tminuss)α2

ds =minusC0 D

α2

t u(xt) x= xr (37)





j=0


δtuni =

uni minusunminus1

i

∆t δxun

i+ 12=

uni+1minusun

i

h

δ2xun

i =δxui+ 1

2minusδxuiminus 1

2

h δxun

i =un

i+1minusuniminus1

2h




uxx(x0)minus2

h

[

δxu 12minusux(x0)

]

=minush

3uxxx(x0+θ1h) θ1 isin (01) (38)

uxx(xNS)minus 2

h

[


2

]

=h

3uxxx(xNS



C0 D

αt un

i =δ2xun


C0 D

αt un

0 =2

h

[

δxun12minusC

0 Dα2t un

0

]


C0 D

αt un

NS=

2

h

[

minusδxunNSminus 1

2minusC

0 Dα2t un

NS

]

+ f nNS

1lenleNT (312)




FC0 D

αt un

i =δ2xun


FC0 D

αt un

0 =2

h

[

δxun12minusFC

0 Dα2t un

0

]


FC0 D

αt un

NS=

2

h

[

minusδxunNSminus 1

2minusFC

0 Dα2t un

NS

]

+ f nNS

1lenleNT (316)





u2infin le θδxu2+

(1

θ+

1

L

)

u2 forallθgt0





∆tn

sumk=1

∥

∥

∥uk∥

∥

∥

2

infinle2(

1+radic

1+L2micro)

Lmicro

(

ρ∥

∥u0∥

∥

2+[

(u00)

2+(u0NS)2]

+∆t

8ν

n

sumk=1

[

(h f k0 )

2+(h f kNS)2]

+∆t

micro

n

sumk=1

hNSminus1

sumi=1

( f ki )

2

)

(318)



where

ρ=T1minusα


Tminusαminus2αεT

Γ(1minusα)

=T1minus α

2

2Γ(2minus α2 )

ν=Tminus α

2 minusαεT

Γ(1minus α2 )

(319)


have

hNSminus1

sumi=1

(FC0 D

αt uk

i )uki minush

NSminus1

sumi=1

(δ2xuk

i )uki =h

NSminus1

sumi=1

f ki uk

i

Multiplying h2 uk

0 and h2 uk



(FC0 D

αt ukuk)+

[

minus(δxvk12)uk

0minushNSminus1

sumi=1

(δ2xuk

i )uki +(δxuk

NSminus 12)uk

NS

]

+(FC0 D

α2t uk

0)uk0+(FC

0 D

α2t uk

NS)uk

NS=

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS (320)


minus(δxvk12)uk

0minushNSminus1

sumi=1

(δ2xuk

i )uki +(δxuk

NSminus 12)uk

NS=∥

∥

∥δxuk

∥

∥

∥

2 (321)



2Γ(1minusα)

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

tminus α


2Γ(1minus α2 )

n

sumk=1

[

(uk0)

2+(ukNS)2]

+∆tn

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2


2Γ(2minusα)

∥

∥u0∥

∥

2+

t1minus α

2n + α


2Γ(2minus α2 )

[

(u00)

2+(u0NS)2]

+∆tn

sumk=1

[

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS

]

(322)


1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS



le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]

+hNSminus1

sumi=1

[


4Γ(1minusα)(uk

i )2+

Γ(1minusα)


( f ki )

2

]

le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]


4Γ(1minusα)

∥

∥

∥uk∥

∥

∥

2+h

NSminus1

sumi=1

Γ(1minusα)


( f ki )

2 (323)


micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2leρn

∥

∥u0∥

∥

2+n

[

(u00)

2+(u0NS)2]

+∆t

8νn

n

sumk=1

[

(h f k0 )

2+(h f kNS)2]

+∆t

micron

n

sumk=1

hNSminus1

sumi=1

( f ki )

2 (324)

where




Γ(1minusα)

n =t1minus α

2n + α


2Γ(2minus α2 )

νn =tminus α


Γ(1minus α2 )

(325)


θn = micron

4

(

ie θn = 2(

1+radic

1+L2micron)

(Lmicron))


∆tn

sumk=1

∥

∥

∥uk∥

∥

∥

2

infinle 2(

1+radic

1+L2micron)

Lmicron

(

micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2)

(326)












radic

∆tn

sumk=1

ek2infin le c2(h


where c22 =

4c21T(

1+radic

1+L2micro)

Lmicro

(

1ν +

Lmicro

)


defined in (319)







exp(minusx)t3+α+1]



f (xt)=






E(h∆t)=

radic

radic

radic

radic∆tNT

sumk=1

ek2infin rt = log2

E(h∆t)

E(h∆t2) rs = log2

E(h∆t)

E(h2∆t)




∆t

α=02 α=05



110 1570e-02 170 1570e-02 170 8151e-02 148 8151e-02 148

120 4846e-03 170 4844e-03 170 2922e-02 147 2923e-02 147

140 1489e-03 172 1488e-03 171 1052e-02 148 1052e-02 148

180 4524e-04 170 4541e-04 172 3781e-03 148 3784e-03 148

1160 1395e-04 1375e-04 1357e-03 1357e-03


h

α=02 α=05



π10 8862e-01 206 8632e-01 203 8652e-01 205 8035e-01 199

π20 2122e-01 201 2107e-01 201 2087e-01 201 2016e-01 199

π40 5258e-02 200 5244e-02 200 5177e-02 200 5064e-02 199

π80 1312e-02 200 1311e-02 200 1292e-02 200 1272e-02 199

π160 3280e-03 3277e-03 3229e-03 3195e-03

CPU(s) 4337 330466 4365 222606




30000











42 44 46 48 5 5205

1

15

2

25

3

35

4

Log10

(NT)

Log 10

(cpu

time)





C0 Dα

t u(xt)=uxx+ f (u)

u(x0)=u0(x)(41)





C0 Dα



(partx+3sα20 )

C0 Dα

t u+(3sα0partx+s

3α2

0 )u=(partx+3sα20 ) f (u) x= xr (44)

(partxminus3sα20 )

C0 Dα


3α2




C0 D

αt Un

i =δ2xUn


(δx+3sα20 )

C0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

C0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (48)


0 Dαt by our



FC0 D

αt Un

i =δ2xUn


(δx+3sα20 )

FC0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

FC0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (411)













∆t

α=02 α=05



110 8601e-04 102 8601e-04 102 2194e-03 109 2194e-03 109

120 4243e-04 101 4243e-04 101 1030e-03 106 1030e-03 106

140 2104e-04 101 2104e-04 101 4934e-04 104 4934e-04 104

180 1046e-04 1046e-04 2392e-04 2392e-04


h

α=02 α=05



180 8196e-04 196 8196e-04 196 6045e-04 197 6045e-04 197

1160 2112e-04 201 2112e-04 201 1547e-04 201 1547e-04 201

1320 5246e-05 200 5246e-05 200 3842e-05 200 3842e-05 200

1640 1316e-05 1316e-05 9649e-06 9649e-06

CPU(s) 3880 107191 4022 85058




4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)


4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)




h ) between the


h )






∆t

α=02 α=05



110 1089e-03 104 1089e-03 104 2961e-03 106 2961e-03 106

120 5304e-04 102 5304-04 102 1418e-03 104 1418e-03 104

140 2614e-04 101 2614e-04 101 6896e-04 103 6896e-04 103

180 1294e-04 1294e-04 3379e-04 3379e-04


h

α=02 α=05



180 9051e-04 196 9051e-04 196 6882e-04 198 6882e-04 198

1160 2319e-04 201 2319e-04 201 1746e-04 201 1746e-04 201

1320 5762e-05 200 5762e-05 200 4344e-05 200 4344e-05 200

1640 1447e-05 1447e-05 1091e-05 1091e-05

CPU(s) 3869 110658 4014 85587


h ) between the


h )




5 Conclusions





Acknowledgments




FC0 D

αt un=



+1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

=∆tminusα

n

Γ(1minusα)

(

un

1minusαminus(

α

1minusα+a0)u

nminus1minusnminus2

suml=1


1

nα)u0

)

(A1)

where

an =α∆tαNexp

sumj=1


j bn =α∆tαNexp

sumj=1


j



λ1j =

eminuss j∆t

s2j ∆t


eminuss j∆t

s2j ∆t



(FC0 D

αt gk)gk =

1

∆tαΓ(1minusα)

(

1

1minusα(gk)2minus

( α

1minusα+a0

)

gkminus1gk

minuskminus2

suml=1


bkminus2+1

kα

)

g0gk

)

ge 1

∆tαΓ(1minusα)

[

(

2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minus1

2

( α

1minusα+a0

)

(gkminus1)2minus 1

2

kminus2

suml=1


minus1

2

(

bkminus2+1

kα

)

(g0)2

)]

(A2)


∆tn

sumk=1

(FC0 D


Γ(1minusα)

[

n

sumk=2

(

1

1minusαminus α

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minusn

sumk=2

(

α

2(1minusα)+

a0

2

)

(gkminus1)2minus 1

2

n

sumk=2

kminus2

suml=1

(


)

(gl)2

+1



2

n

sumk=2

(

bkminus2+1

kα

)

(g0)2

]

=∆t1minusα

Γ(1minusα)

n

sumk=1

(

Ck(gk)2minusC0(g0)2

)

(A3)


Cnk =

1

2(1minusα)+

1

2

n

sumk=2

(

bkminus2+1

kα

)

k=0

1

2minus 1

2

nminus2

suml=0

(al+bl)+1

2bnminus2 k=1

1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1


2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα k=n

(A4)




1

t1+αminusεle

Nexp

sumj=1

ωjeminuss jtle 1

t1+α+ε (A5)


kminus2

suml=0


∆t

Nexp

sumj=1



(

1minus 1

kα

)


suml=0

(al+bl)le(

1minus 1

kα

)



Cn0 le

n1minusα

2(1minusα)+

α∆tαtnminus1ε

2

Cn1 ge

1

2minus 1

2

nminus2

suml=0

(al+bl)ge1


Cnk =1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1

2bnminuskminus1


Cnn ge

2minusα

2(1minusα)minus

nminus2

suml=0

(al+bl)minus1

2nαge 1


(A8)




FC0 D

αt ek

i =δ2xek


FC0 D

αt ek

0=2

h

[

δxek12minusFC

0 Dα2t ek

0

]

+Tk0 (B2)

FC0 D

αt ek

NS=

2

h

[

minusδxenNSminus 1

2minusFC

0 Dα2t ek

NS

]

+TkNS

(B3)





Tki =minus

[

C0 Dα

t u(xitk)minusFC0 D

αt Uk

i

]

+[


i

]


Tk0 =

uxx(x0tk)minus2

h

[


]

minus 2

h

[

C0 D

α2t u(x0tk)minusFC

0 D

α2t Uk

0

]

minus[

C0 Dα

t u(x0tk)minusFC0 D

αt Uk

0

]

1le kleNT

TkNS

=

uxx(xNStk)minus

2

h

[

ux(xNStk)minusδxUk

NSminus 12

]

minus 2

h

[

C0 D

α2t u(xNS

tk)minusFC0 D

α2t Uk

NS

]

minus[

C0 Dα

t u(xNStk)minusFC

0 Dαt Uk

NS

]

1le kleNT



∣

∣

∣Tk

i

∣

∣


∣

∣

∣Tk

0

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B6)

∣

∣

∣Tk

NS

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B7)


1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

le c21

2ν

(


)2+

2c21L

micro

(


le2c21

ν

(


+2c2

1L

micro

(


le4c21

(1

ν+

L

micro

)


(1

ν+

L

micro

)

ε2 (B8)


∆tn

sumk=1

ek2infin le ∆t(1+

radic

1+L2micro)

Lmicro

n

sumk=1

(

1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

)

(B9)




References



































































∆t=10minus3 α=02 α=05

ǫ

NexpT∆t 103 104 105 106 103 104 105 106

10minus3 25 29 33 37 30 35 40 44

10minus6 34 40 45 51 39 46 52 59

10minus9 43 51 58 66 49 57 65 74


T=1 α=02 α=05

ǫ


10minus3 25 34 43 55 30 41 52 66

10minus6 34 45 58 69 39 51 65 81

10minus8 40 51 64 79 43 57 72 84

Ch(tn)asymp1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωi

int tnminus1


]

=1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

(224)

Define





tnminus2





int tnminus1

tnminus2


s2i ∆tnminus1

[



(226)





23 Error analysis



C0 Dα


∆tαΓ(2minusα)+

1

Γ(1minusα)

[

unminus1

∆tαminus u0

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

FC0 D



FC0 D

αt u1=

∆tminusα




C0 D

αt un=

∆tminusα

Γ(2minusα)

[

a(α)0 unminus

nminus1

sumk=1


(α)nminusk)u


]

(229)




Rnu =C0 Dα

t u(t)∣

∣

t=tnminusC

0 Dαt un (230)

where 0ltαlt1 Then


Γ(2minusα)

(

1minusα

12+

22minusα


)

max0letletn



0 Dαt un in (227) and (228)




FRnu =C0 Dα

t u(t)∣

∣

t=tnminusFC

0 Dαt un (232)

where 0ltαlt1 Then


Γ(2minusα)

(

1minusα

12+

22minusα


)

max0letletn


Γ(1minusα)max

0letletnminus1

|u(t)| (233)


αt un and the L1-

approximation C0 D



∣

∣

∣

FC0 D

αt unminusC

0 Dαt un∣

∣

∣le αε

Γ(1minusα)

nminus1

suml=1

int tl

tlminus1

|Π1lu(s)|ds (234)


tminustlminus1



Γ(1minusα)

nminus1

suml=1

int tl

tlminus1

|Π1lu(s)|ds (235)

where

nminus1

suml=1

int tl

tlminus1






∆tn

sumk=1

(FC0 D



2Γ(1minusα)∆t

n

sumk=1


2Γ(2minusα)(g0)2 (237)






C0 Dα





C0 Dα



partu(xt)

partx=

1

Γ(1minus α2 )

int t

0

us(xs)

(tminuss)α2

ds =C0 D

α2

t u(xt) x= xl (36)

partu(xt)

partx=minus 1

Γ(1minus α2 )

int t

0

us(xs)

(tminuss)α2

ds =minusC0 D

α2

t u(xt) x= xr (37)





j=0


δtuni =

uni minusunminus1

i

∆t δxun

i+ 12=

uni+1minusun

i

h

δ2xun

i =δxui+ 1

2minusδxuiminus 1

2

h δxun

i =un

i+1minusuniminus1

2h




uxx(x0)minus2

h

[

δxu 12minusux(x0)

]

=minush

3uxxx(x0+θ1h) θ1 isin (01) (38)

uxx(xNS)minus 2

h

[


2

]

=h

3uxxx(xNS



C0 D

αt un

i =δ2xun


C0 D

αt un

0 =2

h

[

δxun12minusC

0 Dα2t un

0

]


C0 D

αt un

NS=

2

h

[

minusδxunNSminus 1

2minusC

0 Dα2t un

NS

]

+ f nNS

1lenleNT (312)




FC0 D

αt un

i =δ2xun


FC0 D

αt un

0 =2

h

[

δxun12minusFC

0 Dα2t un

0

]


FC0 D

αt un

NS=

2

h

[

minusδxunNSminus 1

2minusFC

0 Dα2t un

NS

]

+ f nNS

1lenleNT (316)





u2infin le θδxu2+

(1

θ+

1

L

)

u2 forallθgt0





∆tn

sumk=1

∥

∥

∥uk∥

∥

∥

2

infinle2(

1+radic

1+L2micro)

Lmicro

(

ρ∥

∥u0∥

∥

2+[

(u00)

2+(u0NS)2]

+∆t

8ν

n

sumk=1

[

(h f k0 )

2+(h f kNS)2]

+∆t

micro

n

sumk=1

hNSminus1

sumi=1

( f ki )

2

)

(318)



where

ρ=T1minusα


Tminusαminus2αεT

Γ(1minusα)

=T1minus α

2

2Γ(2minus α2 )

ν=Tminus α

2 minusαεT

Γ(1minus α2 )

(319)


have

hNSminus1

sumi=1

(FC0 D

αt uk

i )uki minush

NSminus1

sumi=1

(δ2xuk

i )uki =h

NSminus1

sumi=1

f ki uk

i

Multiplying h2 uk

0 and h2 uk



(FC0 D

αt ukuk)+

[

minus(δxvk12)uk

0minushNSminus1

sumi=1

(δ2xuk

i )uki +(δxuk

NSminus 12)uk

NS

]

+(FC0 D

α2t uk

0)uk0+(FC

0 D

α2t uk

NS)uk

NS=

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS (320)


minus(δxvk12)uk

0minushNSminus1

sumi=1

(δ2xuk

i )uki +(δxuk

NSminus 12)uk

NS=∥

∥

∥δxuk

∥

∥

∥

2 (321)



2Γ(1minusα)

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

tminus α


2Γ(1minus α2 )

n

sumk=1

[

(uk0)

2+(ukNS)2]

+∆tn

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2


2Γ(2minusα)

∥

∥u0∥

∥

2+

t1minus α

2n + α


2Γ(2minus α2 )

[

(u00)

2+(u0NS)2]

+∆tn

sumk=1

[

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS

]

(322)


1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS



le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]

+hNSminus1

sumi=1

[


4Γ(1minusα)(uk

i )2+

Γ(1minusα)


( f ki )

2

]

le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]


4Γ(1minusα)

∥

∥

∥uk∥

∥

∥

2+h

NSminus1

sumi=1

Γ(1minusα)


( f ki )

2 (323)


micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2leρn

∥

∥u0∥

∥

2+n

[

(u00)

2+(u0NS)2]

+∆t

8νn

n

sumk=1

[

(h f k0 )

2+(h f kNS)2]

+∆t

micron

n

sumk=1

hNSminus1

sumi=1

( f ki )

2 (324)

where




Γ(1minusα)

n =t1minus α

2n + α


2Γ(2minus α2 )

νn =tminus α


Γ(1minus α2 )

(325)


θn = micron

4

(

ie θn = 2(

1+radic

1+L2micron)

(Lmicron))


∆tn

sumk=1

∥

∥

∥uk∥

∥

∥

2

infinle 2(

1+radic

1+L2micron)

Lmicron

(

micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2)

(326)












radic

∆tn

sumk=1

ek2infin le c2(h


where c22 =

4c21T(

1+radic

1+L2micro)

Lmicro

(

1ν +

Lmicro

)


defined in (319)







exp(minusx)t3+α+1]



f (xt)=






E(h∆t)=

radic

radic

radic

radic∆tNT

sumk=1

ek2infin rt = log2

E(h∆t)

E(h∆t2) rs = log2

E(h∆t)

E(h2∆t)




∆t

α=02 α=05



110 1570e-02 170 1570e-02 170 8151e-02 148 8151e-02 148

120 4846e-03 170 4844e-03 170 2922e-02 147 2923e-02 147

140 1489e-03 172 1488e-03 171 1052e-02 148 1052e-02 148

180 4524e-04 170 4541e-04 172 3781e-03 148 3784e-03 148

1160 1395e-04 1375e-04 1357e-03 1357e-03


h

α=02 α=05



π10 8862e-01 206 8632e-01 203 8652e-01 205 8035e-01 199

π20 2122e-01 201 2107e-01 201 2087e-01 201 2016e-01 199

π40 5258e-02 200 5244e-02 200 5177e-02 200 5064e-02 199

π80 1312e-02 200 1311e-02 200 1292e-02 200 1272e-02 199

π160 3280e-03 3277e-03 3229e-03 3195e-03

CPU(s) 4337 330466 4365 222606




30000











42 44 46 48 5 5205

1

15

2

25

3

35

4

Log10

(NT)

Log 10

(cpu

time)





C0 Dα

t u(xt)=uxx+ f (u)

u(x0)=u0(x)(41)





C0 Dα



(partx+3sα20 )

C0 Dα

t u+(3sα0partx+s

3α2

0 )u=(partx+3sα20 ) f (u) x= xr (44)

(partxminus3sα20 )

C0 Dα


3α2




C0 D

αt Un

i =δ2xUn


(δx+3sα20 )

C0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

C0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (48)


0 Dαt by our



FC0 D

αt Un

i =δ2xUn


(δx+3sα20 )

FC0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

FC0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (411)













∆t

α=02 α=05



110 8601e-04 102 8601e-04 102 2194e-03 109 2194e-03 109

120 4243e-04 101 4243e-04 101 1030e-03 106 1030e-03 106

140 2104e-04 101 2104e-04 101 4934e-04 104 4934e-04 104

180 1046e-04 1046e-04 2392e-04 2392e-04


h

α=02 α=05



180 8196e-04 196 8196e-04 196 6045e-04 197 6045e-04 197

1160 2112e-04 201 2112e-04 201 1547e-04 201 1547e-04 201

1320 5246e-05 200 5246e-05 200 3842e-05 200 3842e-05 200

1640 1316e-05 1316e-05 9649e-06 9649e-06

CPU(s) 3880 107191 4022 85058




4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)


4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)




h ) between the


h )






∆t

α=02 α=05



110 1089e-03 104 1089e-03 104 2961e-03 106 2961e-03 106

120 5304e-04 102 5304-04 102 1418e-03 104 1418e-03 104

140 2614e-04 101 2614e-04 101 6896e-04 103 6896e-04 103

180 1294e-04 1294e-04 3379e-04 3379e-04


h

α=02 α=05



180 9051e-04 196 9051e-04 196 6882e-04 198 6882e-04 198

1160 2319e-04 201 2319e-04 201 1746e-04 201 1746e-04 201

1320 5762e-05 200 5762e-05 200 4344e-05 200 4344e-05 200

1640 1447e-05 1447e-05 1091e-05 1091e-05

CPU(s) 3869 110658 4014 85587


h ) between the


h )




5 Conclusions





Acknowledgments




FC0 D

αt un=



+1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

=∆tminusα

n

Γ(1minusα)

(

un

1minusαminus(

α

1minusα+a0)u

nminus1minusnminus2

suml=1


1

nα)u0

)

(A1)

where

an =α∆tαNexp

sumj=1


j bn =α∆tαNexp

sumj=1


j



λ1j =

eminuss j∆t

s2j ∆t


eminuss j∆t

s2j ∆t



(FC0 D

αt gk)gk =

1

∆tαΓ(1minusα)

(

1

1minusα(gk)2minus

( α

1minusα+a0

)

gkminus1gk

minuskminus2

suml=1


bkminus2+1

kα

)

g0gk

)

ge 1

∆tαΓ(1minusα)

[

(

2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minus1

2

( α

1minusα+a0

)

(gkminus1)2minus 1

2

kminus2

suml=1


minus1

2

(

bkminus2+1

kα

)

(g0)2

)]

(A2)


∆tn

sumk=1

(FC0 D


Γ(1minusα)

[

n

sumk=2

(

1

1minusαminus α

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minusn

sumk=2

(

α

2(1minusα)+

a0

2

)

(gkminus1)2minus 1

2

n

sumk=2

kminus2

suml=1

(


)

(gl)2

+1



2

n

sumk=2

(

bkminus2+1

kα

)

(g0)2

]

=∆t1minusα

Γ(1minusα)

n

sumk=1

(

Ck(gk)2minusC0(g0)2

)

(A3)


Cnk =

1

2(1minusα)+

1

2

n

sumk=2

(

bkminus2+1

kα

)

k=0

1

2minus 1

2

nminus2

suml=0

(al+bl)+1

2bnminus2 k=1

1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1


2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα k=n

(A4)




1

t1+αminusεle

Nexp

sumj=1

ωjeminuss jtle 1

t1+α+ε (A5)


kminus2

suml=0


∆t

Nexp

sumj=1



(

1minus 1

kα

)


suml=0

(al+bl)le(

1minus 1

kα

)



Cn0 le

n1minusα

2(1minusα)+

α∆tαtnminus1ε

2

Cn1 ge

1

2minus 1

2

nminus2

suml=0

(al+bl)ge1


Cnk =1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1

2bnminuskminus1


Cnn ge

2minusα

2(1minusα)minus

nminus2

suml=0

(al+bl)minus1

2nαge 1


(A8)




FC0 D

αt ek

i =δ2xek


FC0 D

αt ek

0=2

h

[

δxek12minusFC

0 Dα2t ek

0

]

+Tk0 (B2)

FC0 D

αt ek

NS=

2

h

[

minusδxenNSminus 1

2minusFC

0 Dα2t ek

NS

]

+TkNS

(B3)





Tki =minus

[

C0 Dα

t u(xitk)minusFC0 D

αt Uk

i

]

+[


i

]


Tk0 =

uxx(x0tk)minus2

h

[


]

minus 2

h

[

C0 D

α2t u(x0tk)minusFC

0 D

α2t Uk

0

]

minus[

C0 Dα

t u(x0tk)minusFC0 D

αt Uk

0

]

1le kleNT

TkNS

=

uxx(xNStk)minus

2

h

[

ux(xNStk)minusδxUk

NSminus 12

]

minus 2

h

[

C0 D

α2t u(xNS

tk)minusFC0 D

α2t Uk

NS

]

minus[

C0 Dα

t u(xNStk)minusFC

0 Dαt Uk

NS

]

1le kleNT



∣

∣

∣Tk

i

∣

∣


∣

∣

∣Tk

0

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B6)

∣

∣

∣Tk

NS

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B7)


1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

le c21

2ν

(


)2+

2c21L

micro

(


le2c21

ν

(


+2c2

1L

micro

(


le4c21

(1

ν+

L

micro

)


(1

ν+

L

micro

)

ε2 (B8)


∆tn

sumk=1

ek2infin le ∆t(1+

radic

1+L2micro)

Lmicro

n

sumk=1

(

1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

)

(B9)




References




































































23 Error analysis



C0 Dα


∆tαΓ(2minusα)+

1

Γ(1minusα)

[

unminus1

∆tαminus u0

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

FC0 D



FC0 D

αt u1=

∆tminusα




C0 D

αt un=

∆tminusα

Γ(2minusα)

[

a(α)0 unminus

nminus1

sumk=1


(α)nminusk)u


]

(229)




Rnu =C0 Dα

t u(t)∣

∣

t=tnminusC

0 Dαt un (230)

where 0ltαlt1 Then


Γ(2minusα)

(

1minusα

12+

22minusα


)

max0letletn



0 Dαt un in (227) and (228)




FRnu =C0 Dα

t u(t)∣

∣

t=tnminusFC

0 Dαt un (232)

where 0ltαlt1 Then


Γ(2minusα)

(

1minusα

12+

22minusα


)

max0letletn


Γ(1minusα)max

0letletnminus1

|u(t)| (233)


αt un and the L1-

approximation C0 D



∣

∣

∣

FC0 D

αt unminusC

0 Dαt un∣

∣

∣le αε

Γ(1minusα)

nminus1

suml=1

int tl

tlminus1

|Π1lu(s)|ds (234)


tminustlminus1



Γ(1minusα)

nminus1

suml=1

int tl

tlminus1

|Π1lu(s)|ds (235)

where

nminus1

suml=1

int tl

tlminus1






∆tn

sumk=1

(FC0 D



2Γ(1minusα)∆t

n

sumk=1


2Γ(2minusα)(g0)2 (237)






C0 Dα





C0 Dα



partu(xt)

partx=

1

Γ(1minus α2 )

int t

0

us(xs)

(tminuss)α2

ds =C0 D

α2

t u(xt) x= xl (36)

partu(xt)

partx=minus 1

Γ(1minus α2 )

int t

0

us(xs)

(tminuss)α2

ds =minusC0 D

α2

t u(xt) x= xr (37)





j=0


δtuni =

uni minusunminus1

i

∆t δxun

i+ 12=

uni+1minusun

i

h

δ2xun

i =δxui+ 1

2minusδxuiminus 1

2

h δxun

i =un

i+1minusuniminus1

2h




uxx(x0)minus2

h

[

δxu 12minusux(x0)

]

=minush

3uxxx(x0+θ1h) θ1 isin (01) (38)

uxx(xNS)minus 2

h

[


2

]

=h

3uxxx(xNS



C0 D

αt un

i =δ2xun


C0 D

αt un

0 =2

h

[

δxun12minusC

0 Dα2t un

0

]


C0 D

αt un

NS=

2

h

[

minusδxunNSminus 1

2minusC

0 Dα2t un

NS

]

+ f nNS

1lenleNT (312)




FC0 D

αt un

i =δ2xun


FC0 D

αt un

0 =2

h

[

δxun12minusFC

0 Dα2t un

0

]


FC0 D

αt un

NS=

2

h

[

minusδxunNSminus 1

2minusFC

0 Dα2t un

NS

]

+ f nNS

1lenleNT (316)





u2infin le θδxu2+

(1

θ+

1

L

)

u2 forallθgt0





∆tn

sumk=1

∥

∥

∥uk∥

∥

∥

2

infinle2(

1+radic

1+L2micro)

Lmicro

(

ρ∥

∥u0∥

∥

2+[

(u00)

2+(u0NS)2]

+∆t

8ν

n

sumk=1

[

(h f k0 )

2+(h f kNS)2]

+∆t

micro

n

sumk=1

hNSminus1

sumi=1

( f ki )

2

)

(318)



where

ρ=T1minusα


Tminusαminus2αεT

Γ(1minusα)

=T1minus α

2

2Γ(2minus α2 )

ν=Tminus α

2 minusαεT

Γ(1minus α2 )

(319)


have

hNSminus1

sumi=1

(FC0 D

αt uk

i )uki minush

NSminus1

sumi=1

(δ2xuk

i )uki =h

NSminus1

sumi=1

f ki uk

i

Multiplying h2 uk

0 and h2 uk



(FC0 D

αt ukuk)+

[

minus(δxvk12)uk

0minushNSminus1

sumi=1

(δ2xuk

i )uki +(δxuk

NSminus 12)uk

NS

]

+(FC0 D

α2t uk

0)uk0+(FC

0 D

α2t uk

NS)uk

NS=

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS (320)


minus(δxvk12)uk

0minushNSminus1

sumi=1

(δ2xuk

i )uki +(δxuk

NSminus 12)uk

NS=∥

∥

∥δxuk

∥

∥

∥

2 (321)



2Γ(1minusα)

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

tminus α


2Γ(1minus α2 )

n

sumk=1

[

(uk0)

2+(ukNS)2]

+∆tn

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2


2Γ(2minusα)

∥

∥u0∥

∥

2+

t1minus α

2n + α


2Γ(2minus α2 )

[

(u00)

2+(u0NS)2]

+∆tn

sumk=1

[

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS

]

(322)


1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS



le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]

+hNSminus1

sumi=1

[


4Γ(1minusα)(uk

i )2+

Γ(1minusα)


( f ki )

2

]

le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]


4Γ(1minusα)

∥

∥

∥uk∥

∥

∥

2+h

NSminus1

sumi=1

Γ(1minusα)


( f ki )

2 (323)


micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2leρn

∥

∥u0∥

∥

2+n

[

(u00)

2+(u0NS)2]

+∆t

8νn

n

sumk=1

[

(h f k0 )

2+(h f kNS)2]

+∆t

micron

n

sumk=1

hNSminus1

sumi=1

( f ki )

2 (324)

where




Γ(1minusα)

n =t1minus α

2n + α


2Γ(2minus α2 )

νn =tminus α


Γ(1minus α2 )

(325)


θn = micron

4

(

ie θn = 2(

1+radic

1+L2micron)

(Lmicron))


∆tn

sumk=1

∥

∥

∥uk∥

∥

∥

2

infinle 2(

1+radic

1+L2micron)

Lmicron

(

micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2)

(326)












radic

∆tn

sumk=1

ek2infin le c2(h


where c22 =

4c21T(

1+radic

1+L2micro)

Lmicro

(

1ν +

Lmicro

)


defined in (319)







exp(minusx)t3+α+1]



f (xt)=






E(h∆t)=

radic

radic

radic

radic∆tNT

sumk=1

ek2infin rt = log2

E(h∆t)

E(h∆t2) rs = log2

E(h∆t)

E(h2∆t)




∆t

α=02 α=05



110 1570e-02 170 1570e-02 170 8151e-02 148 8151e-02 148

120 4846e-03 170 4844e-03 170 2922e-02 147 2923e-02 147

140 1489e-03 172 1488e-03 171 1052e-02 148 1052e-02 148

180 4524e-04 170 4541e-04 172 3781e-03 148 3784e-03 148

1160 1395e-04 1375e-04 1357e-03 1357e-03


h

α=02 α=05



π10 8862e-01 206 8632e-01 203 8652e-01 205 8035e-01 199

π20 2122e-01 201 2107e-01 201 2087e-01 201 2016e-01 199

π40 5258e-02 200 5244e-02 200 5177e-02 200 5064e-02 199

π80 1312e-02 200 1311e-02 200 1292e-02 200 1272e-02 199

π160 3280e-03 3277e-03 3229e-03 3195e-03

CPU(s) 4337 330466 4365 222606




30000











42 44 46 48 5 5205

1

15

2

25

3

35

4

Log10

(NT)

Log 10

(cpu

time)





C0 Dα

t u(xt)=uxx+ f (u)

u(x0)=u0(x)(41)





C0 Dα



(partx+3sα20 )

C0 Dα

t u+(3sα0partx+s

3α2

0 )u=(partx+3sα20 ) f (u) x= xr (44)

(partxminus3sα20 )

C0 Dα


3α2




C0 D

αt Un

i =δ2xUn


(δx+3sα20 )

C0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

C0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (48)


0 Dαt by our



FC0 D

αt Un

i =δ2xUn


(δx+3sα20 )

FC0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

FC0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (411)













∆t

α=02 α=05



110 8601e-04 102 8601e-04 102 2194e-03 109 2194e-03 109

120 4243e-04 101 4243e-04 101 1030e-03 106 1030e-03 106

140 2104e-04 101 2104e-04 101 4934e-04 104 4934e-04 104

180 1046e-04 1046e-04 2392e-04 2392e-04


h

α=02 α=05



180 8196e-04 196 8196e-04 196 6045e-04 197 6045e-04 197

1160 2112e-04 201 2112e-04 201 1547e-04 201 1547e-04 201

1320 5246e-05 200 5246e-05 200 3842e-05 200 3842e-05 200

1640 1316e-05 1316e-05 9649e-06 9649e-06

CPU(s) 3880 107191 4022 85058




4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)


4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)




h ) between the


h )






∆t

α=02 α=05



110 1089e-03 104 1089e-03 104 2961e-03 106 2961e-03 106

120 5304e-04 102 5304-04 102 1418e-03 104 1418e-03 104

140 2614e-04 101 2614e-04 101 6896e-04 103 6896e-04 103

180 1294e-04 1294e-04 3379e-04 3379e-04


h

α=02 α=05



180 9051e-04 196 9051e-04 196 6882e-04 198 6882e-04 198

1160 2319e-04 201 2319e-04 201 1746e-04 201 1746e-04 201

1320 5762e-05 200 5762e-05 200 4344e-05 200 4344e-05 200

1640 1447e-05 1447e-05 1091e-05 1091e-05

CPU(s) 3869 110658 4014 85587


h ) between the


h )




5 Conclusions





Acknowledgments




FC0 D

αt un=



+1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

=∆tminusα

n

Γ(1minusα)

(

un

1minusαminus(

α

1minusα+a0)u

nminus1minusnminus2

suml=1


1

nα)u0

)

(A1)

where

an =α∆tαNexp

sumj=1


j bn =α∆tαNexp

sumj=1


j



λ1j =

eminuss j∆t

s2j ∆t


eminuss j∆t

s2j ∆t



(FC0 D

αt gk)gk =

1

∆tαΓ(1minusα)

(

1

1minusα(gk)2minus

( α

1minusα+a0

)

gkminus1gk

minuskminus2

suml=1


bkminus2+1

kα

)

g0gk

)

ge 1

∆tαΓ(1minusα)

[

(

2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minus1

2

( α

1minusα+a0

)

(gkminus1)2minus 1

2

kminus2

suml=1


minus1

2

(

bkminus2+1

kα

)

(g0)2

)]

(A2)


∆tn

sumk=1

(FC0 D


Γ(1minusα)

[

n

sumk=2

(

1

1minusαminus α

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minusn

sumk=2

(

α

2(1minusα)+

a0

2

)

(gkminus1)2minus 1

2

n

sumk=2

kminus2

suml=1

(


)

(gl)2

+1



2

n

sumk=2

(

bkminus2+1

kα

)

(g0)2

]

=∆t1minusα

Γ(1minusα)

n

sumk=1

(

Ck(gk)2minusC0(g0)2

)

(A3)


Cnk =

1

2(1minusα)+

1

2

n

sumk=2

(

bkminus2+1

kα

)

k=0

1

2minus 1

2

nminus2

suml=0

(al+bl)+1

2bnminus2 k=1

1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1


2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα k=n

(A4)




1

t1+αminusεle

Nexp

sumj=1

ωjeminuss jtle 1

t1+α+ε (A5)


kminus2

suml=0


∆t

Nexp

sumj=1



(

1minus 1

kα

)


suml=0

(al+bl)le(

1minus 1

kα

)



Cn0 le

n1minusα

2(1minusα)+

α∆tαtnminus1ε

2

Cn1 ge

1

2minus 1

2

nminus2

suml=0

(al+bl)ge1


Cnk =1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1

2bnminuskminus1


Cnn ge

2minusα

2(1minusα)minus

nminus2

suml=0

(al+bl)minus1

2nαge 1


(A8)




FC0 D

αt ek

i =δ2xek


FC0 D

αt ek

0=2

h

[

δxek12minusFC

0 Dα2t ek

0

]

+Tk0 (B2)

FC0 D

αt ek

NS=

2

h

[

minusδxenNSminus 1

2minusFC

0 Dα2t ek

NS

]

+TkNS

(B3)





Tki =minus

[

C0 Dα

t u(xitk)minusFC0 D

αt Uk

i

]

+[


i

]


Tk0 =

uxx(x0tk)minus2

h

[


]

minus 2

h

[

C0 D

α2t u(x0tk)minusFC

0 D

α2t Uk

0

]

minus[

C0 Dα

t u(x0tk)minusFC0 D

αt Uk

0

]

1le kleNT

TkNS

=

uxx(xNStk)minus

2

h

[

ux(xNStk)minusδxUk

NSminus 12

]

minus 2

h

[

C0 D

α2t u(xNS

tk)minusFC0 D

α2t Uk

NS

]

minus[

C0 Dα

t u(xNStk)minusFC

0 Dαt Uk

NS

]

1le kleNT



∣

∣

∣Tk

i

∣

∣


∣

∣

∣Tk

0

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B6)

∣

∣

∣Tk

NS

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B7)


1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

le c21

2ν

(


)2+

2c21L

micro

(


le2c21

ν

(


+2c2

1L

micro

(


le4c21

(1

ν+

L

micro

)


(1

ν+

L

micro

)

ε2 (B8)


∆tn

sumk=1

ek2infin le ∆t(1+

radic

1+L2micro)

Lmicro

n

sumk=1

(

1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

)

(B9)




References



































































FRnu =C0 Dα

t u(t)∣

∣

t=tnminusFC

0 Dαt un (232)

where 0ltαlt1 Then


Γ(2minusα)

(

1minusα

12+

22minusα


)

max0letletn


Γ(1minusα)max

0letletnminus1

|u(t)| (233)


αt un and the L1-

approximation C0 D



∣

∣

∣

FC0 D

αt unminusC

0 Dαt un∣

∣

∣le αε

Γ(1minusα)

nminus1

suml=1

int tl

tlminus1

|Π1lu(s)|ds (234)


tminustlminus1



Γ(1minusα)

nminus1

suml=1

int tl

tlminus1

|Π1lu(s)|ds (235)

where

nminus1

suml=1

int tl

tlminus1






∆tn

sumk=1

(FC0 D



2Γ(1minusα)∆t

n

sumk=1


2Γ(2minusα)(g0)2 (237)






C0 Dα





C0 Dα



partu(xt)

partx=

1

Γ(1minus α2 )

int t

0

us(xs)

(tminuss)α2

ds =C0 D

α2

t u(xt) x= xl (36)

partu(xt)

partx=minus 1

Γ(1minus α2 )

int t

0

us(xs)

(tminuss)α2

ds =minusC0 D

α2

t u(xt) x= xr (37)





j=0


δtuni =

uni minusunminus1

i

∆t δxun

i+ 12=

uni+1minusun

i

h

δ2xun

i =δxui+ 1

2minusδxuiminus 1

2

h δxun

i =un

i+1minusuniminus1

2h




uxx(x0)minus2

h

[

δxu 12minusux(x0)

]

=minush

3uxxx(x0+θ1h) θ1 isin (01) (38)

uxx(xNS)minus 2

h

[


2

]

=h

3uxxx(xNS



C0 D

αt un

i =δ2xun


C0 D

αt un

0 =2

h

[

δxun12minusC

0 Dα2t un

0

]


C0 D

αt un

NS=

2

h

[

minusδxunNSminus 1

2minusC

0 Dα2t un

NS

]

+ f nNS

1lenleNT (312)




FC0 D

αt un

i =δ2xun


FC0 D

αt un

0 =2

h

[

δxun12minusFC

0 Dα2t un

0

]


FC0 D

αt un

NS=

2

h

[

minusδxunNSminus 1

2minusFC

0 Dα2t un

NS

]

+ f nNS

1lenleNT (316)





u2infin le θδxu2+

(1

θ+

1

L

)

u2 forallθgt0





∆tn

sumk=1

∥

∥

∥uk∥

∥

∥

2

infinle2(

1+radic

1+L2micro)

Lmicro

(

ρ∥

∥u0∥

∥

2+[

(u00)

2+(u0NS)2]

+∆t

8ν

n

sumk=1

[

(h f k0 )

2+(h f kNS)2]

+∆t

micro

n

sumk=1

hNSminus1

sumi=1

( f ki )

2

)

(318)



where

ρ=T1minusα


Tminusαminus2αεT

Γ(1minusα)

=T1minus α

2

2Γ(2minus α2 )

ν=Tminus α

2 minusαεT

Γ(1minus α2 )

(319)


have

hNSminus1

sumi=1

(FC0 D

αt uk

i )uki minush

NSminus1

sumi=1

(δ2xuk

i )uki =h

NSminus1

sumi=1

f ki uk

i

Multiplying h2 uk

0 and h2 uk



(FC0 D

αt ukuk)+

[

minus(δxvk12)uk

0minushNSminus1

sumi=1

(δ2xuk

i )uki +(δxuk

NSminus 12)uk

NS

]

+(FC0 D

α2t uk

0)uk0+(FC

0 D

α2t uk

NS)uk

NS=

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS (320)


minus(δxvk12)uk

0minushNSminus1

sumi=1

(δ2xuk

i )uki +(δxuk

NSminus 12)uk

NS=∥

∥

∥δxuk

∥

∥

∥

2 (321)



2Γ(1minusα)

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

tminus α


2Γ(1minus α2 )

n

sumk=1

[

(uk0)

2+(ukNS)2]

+∆tn

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2


2Γ(2minusα)

∥

∥u0∥

∥

2+

t1minus α

2n + α


2Γ(2minus α2 )

[

(u00)

2+(u0NS)2]

+∆tn

sumk=1

[

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS

]

(322)


1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS



le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]

+hNSminus1

sumi=1

[


4Γ(1minusα)(uk

i )2+

Γ(1minusα)


( f ki )

2

]

le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]


4Γ(1minusα)

∥

∥

∥uk∥

∥

∥

2+h

NSminus1

sumi=1

Γ(1minusα)


( f ki )

2 (323)


micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2leρn

∥

∥u0∥

∥

2+n

[

(u00)

2+(u0NS)2]

+∆t

8νn

n

sumk=1

[

(h f k0 )

2+(h f kNS)2]

+∆t

micron

n

sumk=1

hNSminus1

sumi=1

( f ki )

2 (324)

where




Γ(1minusα)

n =t1minus α

2n + α


2Γ(2minus α2 )

νn =tminus α


Γ(1minus α2 )

(325)


θn = micron

4

(

ie θn = 2(

1+radic

1+L2micron)

(Lmicron))


∆tn

sumk=1

∥

∥

∥uk∥

∥

∥

2

infinle 2(

1+radic

1+L2micron)

Lmicron

(

micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2)

(326)












radic

∆tn

sumk=1

ek2infin le c2(h


where c22 =

4c21T(

1+radic

1+L2micro)

Lmicro

(

1ν +

Lmicro

)


defined in (319)







exp(minusx)t3+α+1]



f (xt)=






E(h∆t)=

radic

radic

radic

radic∆tNT

sumk=1

ek2infin rt = log2

E(h∆t)

E(h∆t2) rs = log2

E(h∆t)

E(h2∆t)




∆t

α=02 α=05



110 1570e-02 170 1570e-02 170 8151e-02 148 8151e-02 148

120 4846e-03 170 4844e-03 170 2922e-02 147 2923e-02 147

140 1489e-03 172 1488e-03 171 1052e-02 148 1052e-02 148

180 4524e-04 170 4541e-04 172 3781e-03 148 3784e-03 148

1160 1395e-04 1375e-04 1357e-03 1357e-03


h

α=02 α=05



π10 8862e-01 206 8632e-01 203 8652e-01 205 8035e-01 199

π20 2122e-01 201 2107e-01 201 2087e-01 201 2016e-01 199

π40 5258e-02 200 5244e-02 200 5177e-02 200 5064e-02 199

π80 1312e-02 200 1311e-02 200 1292e-02 200 1272e-02 199

π160 3280e-03 3277e-03 3229e-03 3195e-03

CPU(s) 4337 330466 4365 222606




30000











42 44 46 48 5 5205

1

15

2

25

3

35

4

Log10

(NT)

Log 10

(cpu

time)





C0 Dα

t u(xt)=uxx+ f (u)

u(x0)=u0(x)(41)





C0 Dα



(partx+3sα20 )

C0 Dα

t u+(3sα0partx+s

3α2

0 )u=(partx+3sα20 ) f (u) x= xr (44)

(partxminus3sα20 )

C0 Dα


3α2




C0 D

αt Un

i =δ2xUn


(δx+3sα20 )

C0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

C0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (48)


0 Dαt by our



FC0 D

αt Un

i =δ2xUn


(δx+3sα20 )

FC0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

FC0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (411)













∆t

α=02 α=05



110 8601e-04 102 8601e-04 102 2194e-03 109 2194e-03 109

120 4243e-04 101 4243e-04 101 1030e-03 106 1030e-03 106

140 2104e-04 101 2104e-04 101 4934e-04 104 4934e-04 104

180 1046e-04 1046e-04 2392e-04 2392e-04


h

α=02 α=05



180 8196e-04 196 8196e-04 196 6045e-04 197 6045e-04 197

1160 2112e-04 201 2112e-04 201 1547e-04 201 1547e-04 201

1320 5246e-05 200 5246e-05 200 3842e-05 200 3842e-05 200

1640 1316e-05 1316e-05 9649e-06 9649e-06

CPU(s) 3880 107191 4022 85058




4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)


4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)




h ) between the


h )






∆t

α=02 α=05



110 1089e-03 104 1089e-03 104 2961e-03 106 2961e-03 106

120 5304e-04 102 5304-04 102 1418e-03 104 1418e-03 104

140 2614e-04 101 2614e-04 101 6896e-04 103 6896e-04 103

180 1294e-04 1294e-04 3379e-04 3379e-04


h

α=02 α=05



180 9051e-04 196 9051e-04 196 6882e-04 198 6882e-04 198

1160 2319e-04 201 2319e-04 201 1746e-04 201 1746e-04 201

1320 5762e-05 200 5762e-05 200 4344e-05 200 4344e-05 200

1640 1447e-05 1447e-05 1091e-05 1091e-05

CPU(s) 3869 110658 4014 85587


h ) between the


h )




5 Conclusions





Acknowledgments




FC0 D

αt un=



+1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

=∆tminusα

n

Γ(1minusα)

(

un

1minusαminus(

α

1minusα+a0)u

nminus1minusnminus2

suml=1


1

nα)u0

)

(A1)

where

an =α∆tαNexp

sumj=1


j bn =α∆tαNexp

sumj=1


j



λ1j =

eminuss j∆t

s2j ∆t


eminuss j∆t

s2j ∆t



(FC0 D

αt gk)gk =

1

∆tαΓ(1minusα)

(

1

1minusα(gk)2minus

( α

1minusα+a0

)

gkminus1gk

minuskminus2

suml=1


bkminus2+1

kα

)

g0gk

)

ge 1

∆tαΓ(1minusα)

[

(

2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minus1

2

( α

1minusα+a0

)

(gkminus1)2minus 1

2

kminus2

suml=1


minus1

2

(

bkminus2+1

kα

)

(g0)2

)]

(A2)


∆tn

sumk=1

(FC0 D


Γ(1minusα)

[

n

sumk=2

(

1

1minusαminus α

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minusn

sumk=2

(

α

2(1minusα)+

a0

2

)

(gkminus1)2minus 1

2

n

sumk=2

kminus2

suml=1

(


)

(gl)2

+1



2

n

sumk=2

(

bkminus2+1

kα

)

(g0)2

]

=∆t1minusα

Γ(1minusα)

n

sumk=1

(

Ck(gk)2minusC0(g0)2

)

(A3)


Cnk =

1

2(1minusα)+

1

2

n

sumk=2

(

bkminus2+1

kα

)

k=0

1

2minus 1

2

nminus2

suml=0

(al+bl)+1

2bnminus2 k=1

1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1


2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα k=n

(A4)




1

t1+αminusεle

Nexp

sumj=1

ωjeminuss jtle 1

t1+α+ε (A5)


kminus2

suml=0


∆t

Nexp

sumj=1



(

1minus 1

kα

)


suml=0

(al+bl)le(

1minus 1

kα

)



Cn0 le

n1minusα

2(1minusα)+

α∆tαtnminus1ε

2

Cn1 ge

1

2minus 1

2

nminus2

suml=0

(al+bl)ge1


Cnk =1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1

2bnminuskminus1


Cnn ge

2minusα

2(1minusα)minus

nminus2

suml=0

(al+bl)minus1

2nαge 1


(A8)




FC0 D

αt ek

i =δ2xek


FC0 D

αt ek

0=2

h

[

δxek12minusFC

0 Dα2t ek

0

]

+Tk0 (B2)

FC0 D

αt ek

NS=

2

h

[

minusδxenNSminus 1

2minusFC

0 Dα2t ek

NS

]

+TkNS

(B3)





Tki =minus

[

C0 Dα

t u(xitk)minusFC0 D

αt Uk

i

]

+[


i

]


Tk0 =

uxx(x0tk)minus2

h

[


]

minus 2

h

[

C0 D

α2t u(x0tk)minusFC

0 D

α2t Uk

0

]

minus[

C0 Dα

t u(x0tk)minusFC0 D

αt Uk

0

]

1le kleNT

TkNS

=

uxx(xNStk)minus

2

h

[

ux(xNStk)minusδxUk

NSminus 12

]

minus 2

h

[

C0 D

α2t u(xNS

tk)minusFC0 D

α2t Uk

NS

]

minus[

C0 Dα

t u(xNStk)minusFC

0 Dαt Uk

NS

]

1le kleNT



∣

∣

∣Tk

i

∣

∣


∣

∣

∣Tk

0

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B6)

∣

∣

∣Tk

NS

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B7)


1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

le c21

2ν

(


)2+

2c21L

micro

(


le2c21

ν

(


+2c2

1L

micro

(


le4c21

(1

ν+

L

micro

)


(1

ν+

L

micro

)

ε2 (B8)


∆tn

sumk=1

ek2infin le ∆t(1+

radic

1+L2micro)

Lmicro

n

sumk=1

(

1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

)

(B9)




References




































































C0 Dα





C0 Dα



partu(xt)

partx=

1

Γ(1minus α2 )

int t

0

us(xs)

(tminuss)α2

ds =C0 D

α2

t u(xt) x= xl (36)

partu(xt)

partx=minus 1

Γ(1minus α2 )

int t

0

us(xs)

(tminuss)α2

ds =minusC0 D

α2

t u(xt) x= xr (37)





j=0


δtuni =

uni minusunminus1

i

∆t δxun

i+ 12=

uni+1minusun

i

h

δ2xun

i =δxui+ 1

2minusδxuiminus 1

2

h δxun

i =un

i+1minusuniminus1

2h




uxx(x0)minus2

h

[

δxu 12minusux(x0)

]

=minush

3uxxx(x0+θ1h) θ1 isin (01) (38)

uxx(xNS)minus 2

h

[


2

]

=h

3uxxx(xNS



C0 D

αt un

i =δ2xun


C0 D

αt un

0 =2

h

[

δxun12minusC

0 Dα2t un

0

]


C0 D

αt un

NS=

2

h

[

minusδxunNSminus 1

2minusC

0 Dα2t un

NS

]

+ f nNS

1lenleNT (312)




FC0 D

αt un

i =δ2xun


FC0 D

αt un

0 =2

h

[

δxun12minusFC

0 Dα2t un

0

]


FC0 D

αt un

NS=

2

h

[

minusδxunNSminus 1

2minusFC

0 Dα2t un

NS

]

+ f nNS

1lenleNT (316)





u2infin le θδxu2+

(1

θ+

1

L

)

u2 forallθgt0





∆tn

sumk=1

∥

∥

∥uk∥

∥

∥

2

infinle2(

1+radic

1+L2micro)

Lmicro

(

ρ∥

∥u0∥

∥

2+[

(u00)

2+(u0NS)2]

+∆t

8ν

n

sumk=1

[

(h f k0 )

2+(h f kNS)2]

+∆t

micro

n

sumk=1

hNSminus1

sumi=1

( f ki )

2

)

(318)



where

ρ=T1minusα


Tminusαminus2αεT

Γ(1minusα)

=T1minus α

2

2Γ(2minus α2 )

ν=Tminus α

2 minusαεT

Γ(1minus α2 )

(319)


have

hNSminus1

sumi=1

(FC0 D

αt uk

i )uki minush

NSminus1

sumi=1

(δ2xuk

i )uki =h

NSminus1

sumi=1

f ki uk

i

Multiplying h2 uk

0 and h2 uk



(FC0 D

αt ukuk)+

[

minus(δxvk12)uk

0minushNSminus1

sumi=1

(δ2xuk

i )uki +(δxuk

NSminus 12)uk

NS

]

+(FC0 D

α2t uk

0)uk0+(FC

0 D

α2t uk

NS)uk

NS=

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS (320)


minus(δxvk12)uk

0minushNSminus1

sumi=1

(δ2xuk

i )uki +(δxuk

NSminus 12)uk

NS=∥

∥

∥δxuk

∥

∥

∥

2 (321)



2Γ(1minusα)

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

tminus α


2Γ(1minus α2 )

n

sumk=1

[

(uk0)

2+(ukNS)2]

+∆tn

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2


2Γ(2minusα)

∥

∥u0∥

∥

2+

t1minus α

2n + α


2Γ(2minus α2 )

[

(u00)

2+(u0NS)2]

+∆tn

sumk=1

[

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS

]

(322)


1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS



le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]

+hNSminus1

sumi=1

[


4Γ(1minusα)(uk

i )2+

Γ(1minusα)


( f ki )

2

]

le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]


4Γ(1minusα)

∥

∥

∥uk∥

∥

∥

2+h

NSminus1

sumi=1

Γ(1minusα)


( f ki )

2 (323)


micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2leρn

∥

∥u0∥

∥

2+n

[

(u00)

2+(u0NS)2]

+∆t

8νn

n

sumk=1

[

(h f k0 )

2+(h f kNS)2]

+∆t

micron

n

sumk=1

hNSminus1

sumi=1

( f ki )

2 (324)

where




Γ(1minusα)

n =t1minus α

2n + α


2Γ(2minus α2 )

νn =tminus α


Γ(1minus α2 )

(325)


θn = micron

4

(

ie θn = 2(

1+radic

1+L2micron)

(Lmicron))


∆tn

sumk=1

∥

∥

∥uk∥

∥

∥

2

infinle 2(

1+radic

1+L2micron)

Lmicron

(

micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2)

(326)












radic

∆tn

sumk=1

ek2infin le c2(h


where c22 =

4c21T(

1+radic

1+L2micro)

Lmicro

(

1ν +

Lmicro

)


defined in (319)







exp(minusx)t3+α+1]



f (xt)=






E(h∆t)=

radic

radic

radic

radic∆tNT

sumk=1

ek2infin rt = log2

E(h∆t)

E(h∆t2) rs = log2

E(h∆t)

E(h2∆t)




∆t

α=02 α=05



110 1570e-02 170 1570e-02 170 8151e-02 148 8151e-02 148

120 4846e-03 170 4844e-03 170 2922e-02 147 2923e-02 147

140 1489e-03 172 1488e-03 171 1052e-02 148 1052e-02 148

180 4524e-04 170 4541e-04 172 3781e-03 148 3784e-03 148

1160 1395e-04 1375e-04 1357e-03 1357e-03


h

α=02 α=05



π10 8862e-01 206 8632e-01 203 8652e-01 205 8035e-01 199

π20 2122e-01 201 2107e-01 201 2087e-01 201 2016e-01 199

π40 5258e-02 200 5244e-02 200 5177e-02 200 5064e-02 199

π80 1312e-02 200 1311e-02 200 1292e-02 200 1272e-02 199

π160 3280e-03 3277e-03 3229e-03 3195e-03

CPU(s) 4337 330466 4365 222606




30000











42 44 46 48 5 5205

1

15

2

25

3

35

4

Log10

(NT)

Log 10

(cpu

time)





C0 Dα

t u(xt)=uxx+ f (u)

u(x0)=u0(x)(41)





C0 Dα



(partx+3sα20 )

C0 Dα

t u+(3sα0partx+s

3α2

0 )u=(partx+3sα20 ) f (u) x= xr (44)

(partxminus3sα20 )

C0 Dα


3α2




C0 D

αt Un

i =δ2xUn


(δx+3sα20 )

C0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

C0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (48)


0 Dαt by our



FC0 D

αt Un

i =δ2xUn


(δx+3sα20 )

FC0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

FC0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (411)













∆t

α=02 α=05



110 8601e-04 102 8601e-04 102 2194e-03 109 2194e-03 109

120 4243e-04 101 4243e-04 101 1030e-03 106 1030e-03 106

140 2104e-04 101 2104e-04 101 4934e-04 104 4934e-04 104

180 1046e-04 1046e-04 2392e-04 2392e-04


h

α=02 α=05



180 8196e-04 196 8196e-04 196 6045e-04 197 6045e-04 197

1160 2112e-04 201 2112e-04 201 1547e-04 201 1547e-04 201

1320 5246e-05 200 5246e-05 200 3842e-05 200 3842e-05 200

1640 1316e-05 1316e-05 9649e-06 9649e-06

CPU(s) 3880 107191 4022 85058




4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)


4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)




h ) between the


h )






∆t

α=02 α=05



110 1089e-03 104 1089e-03 104 2961e-03 106 2961e-03 106

120 5304e-04 102 5304-04 102 1418e-03 104 1418e-03 104

140 2614e-04 101 2614e-04 101 6896e-04 103 6896e-04 103

180 1294e-04 1294e-04 3379e-04 3379e-04


h

α=02 α=05



180 9051e-04 196 9051e-04 196 6882e-04 198 6882e-04 198

1160 2319e-04 201 2319e-04 201 1746e-04 201 1746e-04 201

1320 5762e-05 200 5762e-05 200 4344e-05 200 4344e-05 200

1640 1447e-05 1447e-05 1091e-05 1091e-05

CPU(s) 3869 110658 4014 85587


h ) between the


h )




5 Conclusions





Acknowledgments




FC0 D

αt un=



+1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

=∆tminusα

n

Γ(1minusα)

(

un

1minusαminus(

α

1minusα+a0)u

nminus1minusnminus2

suml=1


1

nα)u0

)

(A1)

where

an =α∆tαNexp

sumj=1


j bn =α∆tαNexp

sumj=1


j



λ1j =

eminuss j∆t

s2j ∆t


eminuss j∆t

s2j ∆t



(FC0 D

αt gk)gk =

1

∆tαΓ(1minusα)

(

1

1minusα(gk)2minus

( α

1minusα+a0

)

gkminus1gk

minuskminus2

suml=1


bkminus2+1

kα

)

g0gk

)

ge 1

∆tαΓ(1minusα)

[

(

2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minus1

2

( α

1minusα+a0

)

(gkminus1)2minus 1

2

kminus2

suml=1


minus1

2

(

bkminus2+1

kα

)

(g0)2

)]

(A2)


∆tn

sumk=1

(FC0 D


Γ(1minusα)

[

n

sumk=2

(

1

1minusαminus α

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minusn

sumk=2

(

α

2(1minusα)+

a0

2

)

(gkminus1)2minus 1

2

n

sumk=2

kminus2

suml=1

(


)

(gl)2

+1



2

n

sumk=2

(

bkminus2+1

kα

)

(g0)2

]

=∆t1minusα

Γ(1minusα)

n

sumk=1

(

Ck(gk)2minusC0(g0)2

)

(A3)


Cnk =

1

2(1minusα)+

1

2

n

sumk=2

(

bkminus2+1

kα

)

k=0

1

2minus 1

2

nminus2

suml=0

(al+bl)+1

2bnminus2 k=1

1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1


2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα k=n

(A4)




1

t1+αminusεle

Nexp

sumj=1

ωjeminuss jtle 1

t1+α+ε (A5)


kminus2

suml=0


∆t

Nexp

sumj=1



(

1minus 1

kα

)


suml=0

(al+bl)le(

1minus 1

kα

)



Cn0 le

n1minusα

2(1minusα)+

α∆tαtnminus1ε

2

Cn1 ge

1

2minus 1

2

nminus2

suml=0

(al+bl)ge1


Cnk =1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1

2bnminuskminus1


Cnn ge

2minusα

2(1minusα)minus

nminus2

suml=0

(al+bl)minus1

2nαge 1


(A8)




FC0 D

αt ek

i =δ2xek


FC0 D

αt ek

0=2

h

[

δxek12minusFC

0 Dα2t ek

0

]

+Tk0 (B2)

FC0 D

αt ek

NS=

2

h

[

minusδxenNSminus 1

2minusFC

0 Dα2t ek

NS

]

+TkNS

(B3)





Tki =minus

[

C0 Dα

t u(xitk)minusFC0 D

αt Uk

i

]

+[


i

]


Tk0 =

uxx(x0tk)minus2

h

[


]

minus 2

h

[

C0 D

α2t u(x0tk)minusFC

0 D

α2t Uk

0

]

minus[

C0 Dα

t u(x0tk)minusFC0 D

αt Uk

0

]

1le kleNT

TkNS

=

uxx(xNStk)minus

2

h

[

ux(xNStk)minusδxUk

NSminus 12

]

minus 2

h

[

C0 D

α2t u(xNS

tk)minusFC0 D

α2t Uk

NS

]

minus[

C0 Dα

t u(xNStk)minusFC

0 Dαt Uk

NS

]

1le kleNT



∣

∣

∣Tk

i

∣

∣


∣

∣

∣Tk

0

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B6)

∣

∣

∣Tk

NS

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B7)


1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

le c21

2ν

(


)2+

2c21L

micro

(


le2c21

ν

(


+2c2

1L

micro

(


le4c21

(1

ν+

L

micro

)


(1

ν+

L

micro

)

ε2 (B8)


∆tn

sumk=1

ek2infin le ∆t(1+

radic

1+L2micro)

Lmicro

n

sumk=1

(

1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

)

(B9)




References



































































uxx(x0)minus2

h

[

δxu 12minusux(x0)

]

=minush

3uxxx(x0+θ1h) θ1 isin (01) (38)

uxx(xNS)minus 2

h

[


2

]

=h

3uxxx(xNS



C0 D

αt un

i =δ2xun


C0 D

αt un

0 =2

h

[

δxun12minusC

0 Dα2t un

0

]


C0 D

αt un

NS=

2

h

[

minusδxunNSminus 1

2minusC

0 Dα2t un

NS

]

+ f nNS

1lenleNT (312)




FC0 D

αt un

i =δ2xun


FC0 D

αt un

0 =2

h

[

δxun12minusFC

0 Dα2t un

0

]


FC0 D

αt un

NS=

2

h

[

minusδxunNSminus 1

2minusFC

0 Dα2t un

NS

]

+ f nNS

1lenleNT (316)





u2infin le θδxu2+

(1

θ+

1

L

)

u2 forallθgt0





∆tn

sumk=1

∥

∥

∥uk∥

∥

∥

2

infinle2(

1+radic

1+L2micro)

Lmicro

(

ρ∥

∥u0∥

∥

2+[

(u00)

2+(u0NS)2]

+∆t

8ν

n

sumk=1

[

(h f k0 )

2+(h f kNS)2]

+∆t

micro

n

sumk=1

hNSminus1

sumi=1

( f ki )

2

)

(318)



where

ρ=T1minusα


Tminusαminus2αεT

Γ(1minusα)

=T1minus α

2

2Γ(2minus α2 )

ν=Tminus α

2 minusαεT

Γ(1minus α2 )

(319)


have

hNSminus1

sumi=1

(FC0 D

αt uk

i )uki minush

NSminus1

sumi=1

(δ2xuk

i )uki =h

NSminus1

sumi=1

f ki uk

i

Multiplying h2 uk

0 and h2 uk



(FC0 D

αt ukuk)+

[

minus(δxvk12)uk

0minushNSminus1

sumi=1

(δ2xuk

i )uki +(δxuk

NSminus 12)uk

NS

]

+(FC0 D

α2t uk

0)uk0+(FC

0 D

α2t uk

NS)uk

NS=

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS (320)


minus(δxvk12)uk

0minushNSminus1

sumi=1

(δ2xuk

i )uki +(δxuk

NSminus 12)uk

NS=∥

∥

∥δxuk

∥

∥

∥

2 (321)



2Γ(1minusα)

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

tminus α


2Γ(1minus α2 )

n

sumk=1

[

(uk0)

2+(ukNS)2]

+∆tn

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2


2Γ(2minusα)

∥

∥u0∥

∥

2+

t1minus α

2n + α


2Γ(2minus α2 )

[

(u00)

2+(u0NS)2]

+∆tn

sumk=1

[

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS

]

(322)


1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS



le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]

+hNSminus1

sumi=1

[


4Γ(1minusα)(uk

i )2+

Γ(1minusα)


( f ki )

2

]

le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]


4Γ(1minusα)

∥

∥

∥uk∥

∥

∥

2+h

NSminus1

sumi=1

Γ(1minusα)


( f ki )

2 (323)


micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2leρn

∥

∥u0∥

∥

2+n

[

(u00)

2+(u0NS)2]

+∆t

8νn

n

sumk=1

[

(h f k0 )

2+(h f kNS)2]

+∆t

micron

n

sumk=1

hNSminus1

sumi=1

( f ki )

2 (324)

where




Γ(1minusα)

n =t1minus α

2n + α


2Γ(2minus α2 )

νn =tminus α


Γ(1minus α2 )

(325)


θn = micron

4

(

ie θn = 2(

1+radic

1+L2micron)

(Lmicron))


∆tn

sumk=1

∥

∥

∥uk∥

∥

∥

2

infinle 2(

1+radic

1+L2micron)

Lmicron

(

micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2)

(326)












radic

∆tn

sumk=1

ek2infin le c2(h


where c22 =

4c21T(

1+radic

1+L2micro)

Lmicro

(

1ν +

Lmicro

)


defined in (319)







exp(minusx)t3+α+1]



f (xt)=






E(h∆t)=

radic

radic

radic

radic∆tNT

sumk=1

ek2infin rt = log2

E(h∆t)

E(h∆t2) rs = log2

E(h∆t)

E(h2∆t)




∆t

α=02 α=05



110 1570e-02 170 1570e-02 170 8151e-02 148 8151e-02 148

120 4846e-03 170 4844e-03 170 2922e-02 147 2923e-02 147

140 1489e-03 172 1488e-03 171 1052e-02 148 1052e-02 148

180 4524e-04 170 4541e-04 172 3781e-03 148 3784e-03 148

1160 1395e-04 1375e-04 1357e-03 1357e-03


h

α=02 α=05



π10 8862e-01 206 8632e-01 203 8652e-01 205 8035e-01 199

π20 2122e-01 201 2107e-01 201 2087e-01 201 2016e-01 199

π40 5258e-02 200 5244e-02 200 5177e-02 200 5064e-02 199

π80 1312e-02 200 1311e-02 200 1292e-02 200 1272e-02 199

π160 3280e-03 3277e-03 3229e-03 3195e-03

CPU(s) 4337 330466 4365 222606




30000











42 44 46 48 5 5205

1

15

2

25

3

35

4

Log10

(NT)

Log 10

(cpu

time)





C0 Dα

t u(xt)=uxx+ f (u)

u(x0)=u0(x)(41)





C0 Dα



(partx+3sα20 )

C0 Dα

t u+(3sα0partx+s

3α2

0 )u=(partx+3sα20 ) f (u) x= xr (44)

(partxminus3sα20 )

C0 Dα


3α2




C0 D

αt Un

i =δ2xUn


(δx+3sα20 )

C0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

C0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (48)


0 Dαt by our



FC0 D

αt Un

i =δ2xUn


(δx+3sα20 )

FC0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

FC0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (411)













∆t

α=02 α=05



110 8601e-04 102 8601e-04 102 2194e-03 109 2194e-03 109

120 4243e-04 101 4243e-04 101 1030e-03 106 1030e-03 106

140 2104e-04 101 2104e-04 101 4934e-04 104 4934e-04 104

180 1046e-04 1046e-04 2392e-04 2392e-04


h

α=02 α=05



180 8196e-04 196 8196e-04 196 6045e-04 197 6045e-04 197

1160 2112e-04 201 2112e-04 201 1547e-04 201 1547e-04 201

1320 5246e-05 200 5246e-05 200 3842e-05 200 3842e-05 200

1640 1316e-05 1316e-05 9649e-06 9649e-06

CPU(s) 3880 107191 4022 85058




4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)


4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)




h ) between the


h )






∆t

α=02 α=05



110 1089e-03 104 1089e-03 104 2961e-03 106 2961e-03 106

120 5304e-04 102 5304-04 102 1418e-03 104 1418e-03 104

140 2614e-04 101 2614e-04 101 6896e-04 103 6896e-04 103

180 1294e-04 1294e-04 3379e-04 3379e-04


h

α=02 α=05



180 9051e-04 196 9051e-04 196 6882e-04 198 6882e-04 198

1160 2319e-04 201 2319e-04 201 1746e-04 201 1746e-04 201

1320 5762e-05 200 5762e-05 200 4344e-05 200 4344e-05 200

1640 1447e-05 1447e-05 1091e-05 1091e-05

CPU(s) 3869 110658 4014 85587


h ) between the


h )




5 Conclusions





Acknowledgments




FC0 D

αt un=



+1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

=∆tminusα

n

Γ(1minusα)

(

un

1minusαminus(

α

1minusα+a0)u

nminus1minusnminus2

suml=1


1

nα)u0

)

(A1)

where

an =α∆tαNexp

sumj=1


j bn =α∆tαNexp

sumj=1


j



λ1j =

eminuss j∆t

s2j ∆t


eminuss j∆t

s2j ∆t



(FC0 D

αt gk)gk =

1

∆tαΓ(1minusα)

(

1

1minusα(gk)2minus

( α

1minusα+a0

)

gkminus1gk

minuskminus2

suml=1


bkminus2+1

kα

)

g0gk

)

ge 1

∆tαΓ(1minusα)

[

(

2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minus1

2

( α

1minusα+a0

)

(gkminus1)2minus 1

2

kminus2

suml=1


minus1

2

(

bkminus2+1

kα

)

(g0)2

)]

(A2)


∆tn

sumk=1

(FC0 D


Γ(1minusα)

[

n

sumk=2

(

1

1minusαminus α

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minusn

sumk=2

(

α

2(1minusα)+

a0

2

)

(gkminus1)2minus 1

2

n

sumk=2

kminus2

suml=1

(


)

(gl)2

+1



2

n

sumk=2

(

bkminus2+1

kα

)

(g0)2

]

=∆t1minusα

Γ(1minusα)

n

sumk=1

(

Ck(gk)2minusC0(g0)2

)

(A3)


Cnk =

1

2(1minusα)+

1

2

n

sumk=2

(

bkminus2+1

kα

)

k=0

1

2minus 1

2

nminus2

suml=0

(al+bl)+1

2bnminus2 k=1

1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1


2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα k=n

(A4)




1

t1+αminusεle

Nexp

sumj=1

ωjeminuss jtle 1

t1+α+ε (A5)


kminus2

suml=0


∆t

Nexp

sumj=1



(

1minus 1

kα

)


suml=0

(al+bl)le(

1minus 1

kα

)



Cn0 le

n1minusα

2(1minusα)+

α∆tαtnminus1ε

2

Cn1 ge

1

2minus 1

2

nminus2

suml=0

(al+bl)ge1


Cnk =1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1

2bnminuskminus1


Cnn ge

2minusα

2(1minusα)minus

nminus2

suml=0

(al+bl)minus1

2nαge 1


(A8)




FC0 D

αt ek

i =δ2xek


FC0 D

αt ek

0=2

h

[

δxek12minusFC

0 Dα2t ek

0

]

+Tk0 (B2)

FC0 D

αt ek

NS=

2

h

[

minusδxenNSminus 1

2minusFC

0 Dα2t ek

NS

]

+TkNS

(B3)





Tki =minus

[

C0 Dα

t u(xitk)minusFC0 D

αt Uk

i

]

+[


i

]


Tk0 =

uxx(x0tk)minus2

h

[


]

minus 2

h

[

C0 D

α2t u(x0tk)minusFC

0 D

α2t Uk

0

]

minus[

C0 Dα

t u(x0tk)minusFC0 D

αt Uk

0

]

1le kleNT

TkNS

=

uxx(xNStk)minus

2

h

[

ux(xNStk)minusδxUk

NSminus 12

]

minus 2

h

[

C0 D

α2t u(xNS

tk)minusFC0 D

α2t Uk

NS

]

minus[

C0 Dα

t u(xNStk)minusFC

0 Dαt Uk

NS

]

1le kleNT



∣

∣

∣Tk

i

∣

∣


∣

∣

∣Tk

0

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B6)

∣

∣

∣Tk

NS

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B7)


1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

le c21

2ν

(


)2+

2c21L

micro

(


le2c21

ν

(


+2c2

1L

micro

(


le4c21

(1

ν+

L

micro

)


(1

ν+

L

micro

)

ε2 (B8)


∆tn

sumk=1

ek2infin le ∆t(1+

radic

1+L2micro)

Lmicro

n

sumk=1

(

1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

)

(B9)




References


































































where

ρ=T1minusα


Tminusαminus2αεT

Γ(1minusα)

=T1minus α

2

2Γ(2minus α2 )

ν=Tminus α

2 minusαεT

Γ(1minus α2 )

(319)


have

hNSminus1

sumi=1

(FC0 D

αt uk

i )uki minush

NSminus1

sumi=1

(δ2xuk

i )uki =h

NSminus1

sumi=1

f ki uk

i

Multiplying h2 uk

0 and h2 uk



(FC0 D

αt ukuk)+

[

minus(δxvk12)uk

0minushNSminus1

sumi=1

(δ2xuk

i )uki +(δxuk

NSminus 12)uk

NS

]

+(FC0 D

α2t uk

0)uk0+(FC

0 D

α2t uk

NS)uk

NS=

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS (320)


minus(δxvk12)uk

0minushNSminus1

sumi=1

(δ2xuk

i )uki +(δxuk

NSminus 12)uk

NS=∥

∥

∥δxuk

∥

∥

∥

2 (321)



2Γ(1minusα)

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

tminus α


2Γ(1minus α2 )

n

sumk=1

[

(uk0)

2+(ukNS)2]

+∆tn

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2


2Γ(2minusα)

∥

∥u0∥

∥

2+

t1minus α

2n + α


2Γ(2minus α2 )

[

(u00)

2+(u0NS)2]

+∆tn

sumk=1

[

1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS

]

(322)


1

2(h f k

0 )uk0+h

NSminus1

sumi=1

f ki uk

i +1

2(h f k

NS)uk

NS



le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]

+hNSminus1

sumi=1

[


4Γ(1minusα)(uk

i )2+

Γ(1minusα)


( f ki )

2

]

le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]


4Γ(1minusα)

∥

∥

∥uk∥

∥

∥

2+h

NSminus1

sumi=1

Γ(1minusα)


( f ki )

2 (323)


micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2leρn

∥

∥u0∥

∥

2+n

[

(u00)

2+(u0NS)2]

+∆t

8νn

n

sumk=1

[

(h f k0 )

2+(h f kNS)2]

+∆t

micron

n

sumk=1

hNSminus1

sumi=1

( f ki )

2 (324)

where




Γ(1minusα)

n =t1minus α

2n + α


2Γ(2minus α2 )

νn =tminus α


Γ(1minus α2 )

(325)


θn = micron

4

(

ie θn = 2(

1+radic

1+L2micron)

(Lmicron))


∆tn

sumk=1

∥

∥

∥uk∥

∥

∥

2

infinle 2(

1+radic

1+L2micron)

Lmicron

(

micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2)

(326)












radic

∆tn

sumk=1

ek2infin le c2(h


where c22 =

4c21T(

1+radic

1+L2micro)

Lmicro

(

1ν +

Lmicro

)


defined in (319)







exp(minusx)t3+α+1]



f (xt)=






E(h∆t)=

radic

radic

radic

radic∆tNT

sumk=1

ek2infin rt = log2

E(h∆t)

E(h∆t2) rs = log2

E(h∆t)

E(h2∆t)




∆t

α=02 α=05



110 1570e-02 170 1570e-02 170 8151e-02 148 8151e-02 148

120 4846e-03 170 4844e-03 170 2922e-02 147 2923e-02 147

140 1489e-03 172 1488e-03 171 1052e-02 148 1052e-02 148

180 4524e-04 170 4541e-04 172 3781e-03 148 3784e-03 148

1160 1395e-04 1375e-04 1357e-03 1357e-03


h

α=02 α=05



π10 8862e-01 206 8632e-01 203 8652e-01 205 8035e-01 199

π20 2122e-01 201 2107e-01 201 2087e-01 201 2016e-01 199

π40 5258e-02 200 5244e-02 200 5177e-02 200 5064e-02 199

π80 1312e-02 200 1311e-02 200 1292e-02 200 1272e-02 199

π160 3280e-03 3277e-03 3229e-03 3195e-03

CPU(s) 4337 330466 4365 222606




30000











42 44 46 48 5 5205

1

15

2

25

3

35

4

Log10

(NT)

Log 10

(cpu

time)





C0 Dα

t u(xt)=uxx+ f (u)

u(x0)=u0(x)(41)





C0 Dα



(partx+3sα20 )

C0 Dα

t u+(3sα0partx+s

3α2

0 )u=(partx+3sα20 ) f (u) x= xr (44)

(partxminus3sα20 )

C0 Dα


3α2




C0 D

αt Un

i =δ2xUn


(δx+3sα20 )

C0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

C0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (48)


0 Dαt by our



FC0 D

αt Un

i =δ2xUn


(δx+3sα20 )

FC0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

FC0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (411)













∆t

α=02 α=05



110 8601e-04 102 8601e-04 102 2194e-03 109 2194e-03 109

120 4243e-04 101 4243e-04 101 1030e-03 106 1030e-03 106

140 2104e-04 101 2104e-04 101 4934e-04 104 4934e-04 104

180 1046e-04 1046e-04 2392e-04 2392e-04


h

α=02 α=05



180 8196e-04 196 8196e-04 196 6045e-04 197 6045e-04 197

1160 2112e-04 201 2112e-04 201 1547e-04 201 1547e-04 201

1320 5246e-05 200 5246e-05 200 3842e-05 200 3842e-05 200

1640 1316e-05 1316e-05 9649e-06 9649e-06

CPU(s) 3880 107191 4022 85058




4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)


4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)




h ) between the


h )






∆t

α=02 α=05



110 1089e-03 104 1089e-03 104 2961e-03 106 2961e-03 106

120 5304e-04 102 5304-04 102 1418e-03 104 1418e-03 104

140 2614e-04 101 2614e-04 101 6896e-04 103 6896e-04 103

180 1294e-04 1294e-04 3379e-04 3379e-04


h

α=02 α=05



180 9051e-04 196 9051e-04 196 6882e-04 198 6882e-04 198

1160 2319e-04 201 2319e-04 201 1746e-04 201 1746e-04 201

1320 5762e-05 200 5762e-05 200 4344e-05 200 4344e-05 200

1640 1447e-05 1447e-05 1091e-05 1091e-05

CPU(s) 3869 110658 4014 85587


h ) between the


h )




5 Conclusions





Acknowledgments




FC0 D

αt un=



+1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

=∆tminusα

n

Γ(1minusα)

(

un

1minusαminus(

α

1minusα+a0)u

nminus1minusnminus2

suml=1


1

nα)u0

)

(A1)

where

an =α∆tαNexp

sumj=1


j bn =α∆tαNexp

sumj=1


j



λ1j =

eminuss j∆t

s2j ∆t


eminuss j∆t

s2j ∆t



(FC0 D

αt gk)gk =

1

∆tαΓ(1minusα)

(

1

1minusα(gk)2minus

( α

1minusα+a0

)

gkminus1gk

minuskminus2

suml=1


bkminus2+1

kα

)

g0gk

)

ge 1

∆tαΓ(1minusα)

[

(

2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minus1

2

( α

1minusα+a0

)

(gkminus1)2minus 1

2

kminus2

suml=1


minus1

2

(

bkminus2+1

kα

)

(g0)2

)]

(A2)


∆tn

sumk=1

(FC0 D


Γ(1minusα)

[

n

sumk=2

(

1

1minusαminus α

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minusn

sumk=2

(

α

2(1minusα)+

a0

2

)

(gkminus1)2minus 1

2

n

sumk=2

kminus2

suml=1

(


)

(gl)2

+1



2

n

sumk=2

(

bkminus2+1

kα

)

(g0)2

]

=∆t1minusα

Γ(1minusα)

n

sumk=1

(

Ck(gk)2minusC0(g0)2

)

(A3)


Cnk =

1

2(1minusα)+

1

2

n

sumk=2

(

bkminus2+1

kα

)

k=0

1

2minus 1

2

nminus2

suml=0

(al+bl)+1

2bnminus2 k=1

1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1


2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα k=n

(A4)




1

t1+αminusεle

Nexp

sumj=1

ωjeminuss jtle 1

t1+α+ε (A5)


kminus2

suml=0


∆t

Nexp

sumj=1



(

1minus 1

kα

)


suml=0

(al+bl)le(

1minus 1

kα

)



Cn0 le

n1minusα

2(1minusα)+

α∆tαtnminus1ε

2

Cn1 ge

1

2minus 1

2

nminus2

suml=0

(al+bl)ge1


Cnk =1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1

2bnminuskminus1


Cnn ge

2minusα

2(1minusα)minus

nminus2

suml=0

(al+bl)minus1

2nαge 1


(A8)




FC0 D

αt ek

i =δ2xek


FC0 D

αt ek

0=2

h

[

δxek12minusFC

0 Dα2t ek

0

]

+Tk0 (B2)

FC0 D

αt ek

NS=

2

h

[

minusδxenNSminus 1

2minusFC

0 Dα2t ek

NS

]

+TkNS

(B3)





Tki =minus

[

C0 Dα

t u(xitk)minusFC0 D

αt Uk

i

]

+[


i

]


Tk0 =

uxx(x0tk)minus2

h

[


]

minus 2

h

[

C0 D

α2t u(x0tk)minusFC

0 D

α2t Uk

0

]

minus[

C0 Dα

t u(x0tk)minusFC0 D

αt Uk

0

]

1le kleNT

TkNS

=

uxx(xNStk)minus

2

h

[

ux(xNStk)minusδxUk

NSminus 12

]

minus 2

h

[

C0 D

α2t u(xNS

tk)minusFC0 D

α2t Uk

NS

]

minus[

C0 Dα

t u(xNStk)minusFC

0 Dαt Uk

NS

]

1le kleNT



∣

∣

∣Tk

i

∣

∣


∣

∣

∣Tk

0

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B6)

∣

∣

∣Tk

NS

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B7)


1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

le c21

2ν

(


)2+

2c21L

micro

(


le2c21

ν

(


+2c2

1L

micro

(


le4c21

(1

ν+

L

micro

)


(1

ν+

L

micro

)

ε2 (B8)


∆tn

sumk=1

ek2infin le ∆t(1+

radic

1+L2micro)

Lmicro

n

sumk=1

(

1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

)

(B9)




References


































































le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]

+hNSminus1

sumi=1

[


4Γ(1minusα)(uk

i )2+

Γ(1minusα)


( f ki )

2

]

le tminus α


2Γ(1minus α2 )

[

(uk0)

2+(ukNS)2]

+Γ(1minus α

2 )

8(tminus α


[

(h f k0 )

2+(h f kNS)2]


4Γ(1minusα)

∥

∥

∥uk∥

∥

∥

2+h

NSminus1

sumi=1

Γ(1minusα)


( f ki )

2 (323)


micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2leρn

∥

∥u0∥

∥

2+n

[

(u00)

2+(u0NS)2]

+∆t

8νn

n

sumk=1

[

(h f k0 )

2+(h f kNS)2]

+∆t

micron

n

sumk=1

hNSminus1

sumi=1

( f ki )

2 (324)

where




Γ(1minusα)

n =t1minus α

2n + α


2Γ(2minus α2 )

νn =tminus α


Γ(1minus α2 )

(325)


θn = micron

4

(

ie θn = 2(

1+radic

1+L2micron)

(Lmicron))


∆tn

sumk=1

∥

∥

∥uk∥

∥

∥

2

infinle 2(

1+radic

1+L2micron)

Lmicron

(

micron

4∆t

n

sumk=1

∥

∥

∥uk∥

∥

∥

2+∆t

n

sumk=1

∥

∥

∥δxuk

∥

∥

∥

2)

(326)












radic

∆tn

sumk=1

ek2infin le c2(h


where c22 =

4c21T(

1+radic

1+L2micro)

Lmicro

(

1ν +

Lmicro

)


defined in (319)







exp(minusx)t3+α+1]



f (xt)=






E(h∆t)=

radic

radic

radic

radic∆tNT

sumk=1

ek2infin rt = log2

E(h∆t)

E(h∆t2) rs = log2

E(h∆t)

E(h2∆t)




∆t

α=02 α=05



110 1570e-02 170 1570e-02 170 8151e-02 148 8151e-02 148

120 4846e-03 170 4844e-03 170 2922e-02 147 2923e-02 147

140 1489e-03 172 1488e-03 171 1052e-02 148 1052e-02 148

180 4524e-04 170 4541e-04 172 3781e-03 148 3784e-03 148

1160 1395e-04 1375e-04 1357e-03 1357e-03


h

α=02 α=05



π10 8862e-01 206 8632e-01 203 8652e-01 205 8035e-01 199

π20 2122e-01 201 2107e-01 201 2087e-01 201 2016e-01 199

π40 5258e-02 200 5244e-02 200 5177e-02 200 5064e-02 199

π80 1312e-02 200 1311e-02 200 1292e-02 200 1272e-02 199

π160 3280e-03 3277e-03 3229e-03 3195e-03

CPU(s) 4337 330466 4365 222606




30000











42 44 46 48 5 5205

1

15

2

25

3

35

4

Log10

(NT)

Log 10

(cpu

time)





C0 Dα

t u(xt)=uxx+ f (u)

u(x0)=u0(x)(41)





C0 Dα



(partx+3sα20 )

C0 Dα

t u+(3sα0partx+s

3α2

0 )u=(partx+3sα20 ) f (u) x= xr (44)

(partxminus3sα20 )

C0 Dα


3α2




C0 D

αt Un

i =δ2xUn


(δx+3sα20 )

C0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

C0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (48)


0 Dαt by our



FC0 D

αt Un

i =δ2xUn


(δx+3sα20 )

FC0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

FC0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (411)













∆t

α=02 α=05



110 8601e-04 102 8601e-04 102 2194e-03 109 2194e-03 109

120 4243e-04 101 4243e-04 101 1030e-03 106 1030e-03 106

140 2104e-04 101 2104e-04 101 4934e-04 104 4934e-04 104

180 1046e-04 1046e-04 2392e-04 2392e-04


h

α=02 α=05



180 8196e-04 196 8196e-04 196 6045e-04 197 6045e-04 197

1160 2112e-04 201 2112e-04 201 1547e-04 201 1547e-04 201

1320 5246e-05 200 5246e-05 200 3842e-05 200 3842e-05 200

1640 1316e-05 1316e-05 9649e-06 9649e-06

CPU(s) 3880 107191 4022 85058




4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)


4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)




h ) between the


h )






∆t

α=02 α=05



110 1089e-03 104 1089e-03 104 2961e-03 106 2961e-03 106

120 5304e-04 102 5304-04 102 1418e-03 104 1418e-03 104

140 2614e-04 101 2614e-04 101 6896e-04 103 6896e-04 103

180 1294e-04 1294e-04 3379e-04 3379e-04


h

α=02 α=05



180 9051e-04 196 9051e-04 196 6882e-04 198 6882e-04 198

1160 2319e-04 201 2319e-04 201 1746e-04 201 1746e-04 201

1320 5762e-05 200 5762e-05 200 4344e-05 200 4344e-05 200

1640 1447e-05 1447e-05 1091e-05 1091e-05

CPU(s) 3869 110658 4014 85587


h ) between the


h )




5 Conclusions





Acknowledgments




FC0 D

αt un=



+1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

=∆tminusα

n

Γ(1minusα)

(

un

1minusαminus(

α

1minusα+a0)u

nminus1minusnminus2

suml=1


1

nα)u0

)

(A1)

where

an =α∆tαNexp

sumj=1


j bn =α∆tαNexp

sumj=1


j



λ1j =

eminuss j∆t

s2j ∆t


eminuss j∆t

s2j ∆t



(FC0 D

αt gk)gk =

1

∆tαΓ(1minusα)

(

1

1minusα(gk)2minus

( α

1minusα+a0

)

gkminus1gk

minuskminus2

suml=1


bkminus2+1

kα

)

g0gk

)

ge 1

∆tαΓ(1minusα)

[

(

2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minus1

2

( α

1minusα+a0

)

(gkminus1)2minus 1

2

kminus2

suml=1


minus1

2

(

bkminus2+1

kα

)

(g0)2

)]

(A2)


∆tn

sumk=1

(FC0 D


Γ(1minusα)

[

n

sumk=2

(

1

1minusαminus α

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minusn

sumk=2

(

α

2(1minusα)+

a0

2

)

(gkminus1)2minus 1

2

n

sumk=2

kminus2

suml=1

(


)

(gl)2

+1



2

n

sumk=2

(

bkminus2+1

kα

)

(g0)2

]

=∆t1minusα

Γ(1minusα)

n

sumk=1

(

Ck(gk)2minusC0(g0)2

)

(A3)


Cnk =

1

2(1minusα)+

1

2

n

sumk=2

(

bkminus2+1

kα

)

k=0

1

2minus 1

2

nminus2

suml=0

(al+bl)+1

2bnminus2 k=1

1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1


2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα k=n

(A4)




1

t1+αminusεle

Nexp

sumj=1

ωjeminuss jtle 1

t1+α+ε (A5)


kminus2

suml=0


∆t

Nexp

sumj=1



(

1minus 1

kα

)


suml=0

(al+bl)le(

1minus 1

kα

)



Cn0 le

n1minusα

2(1minusα)+

α∆tαtnminus1ε

2

Cn1 ge

1

2minus 1

2

nminus2

suml=0

(al+bl)ge1


Cnk =1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1

2bnminuskminus1


Cnn ge

2minusα

2(1minusα)minus

nminus2

suml=0

(al+bl)minus1

2nαge 1


(A8)




FC0 D

αt ek

i =δ2xek


FC0 D

αt ek

0=2

h

[

δxek12minusFC

0 Dα2t ek

0

]

+Tk0 (B2)

FC0 D

αt ek

NS=

2

h

[

minusδxenNSminus 1

2minusFC

0 Dα2t ek

NS

]

+TkNS

(B3)





Tki =minus

[

C0 Dα

t u(xitk)minusFC0 D

αt Uk

i

]

+[


i

]


Tk0 =

uxx(x0tk)minus2

h

[


]

minus 2

h

[

C0 D

α2t u(x0tk)minusFC

0 D

α2t Uk

0

]

minus[

C0 Dα

t u(x0tk)minusFC0 D

αt Uk

0

]

1le kleNT

TkNS

=

uxx(xNStk)minus

2

h

[

ux(xNStk)minusδxUk

NSminus 12

]

minus 2

h

[

C0 D

α2t u(xNS

tk)minusFC0 D

α2t Uk

NS

]

minus[

C0 Dα

t u(xNStk)minusFC

0 Dαt Uk

NS

]

1le kleNT



∣

∣

∣Tk

i

∣

∣


∣

∣

∣Tk

0

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B6)

∣

∣

∣Tk

NS

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B7)


1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

le c21

2ν

(


)2+

2c21L

micro

(


le2c21

ν

(


+2c2

1L

micro

(


le4c21

(1

ν+

L

micro

)


(1

ν+

L

micro

)

ε2 (B8)


∆tn

sumk=1

ek2infin le ∆t(1+

radic

1+L2micro)

Lmicro

n

sumk=1

(

1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

)

(B9)




References





































































radic

∆tn

sumk=1

ek2infin le c2(h


where c22 =

4c21T(

1+radic

1+L2micro)

Lmicro

(

1ν +

Lmicro

)


defined in (319)







exp(minusx)t3+α+1]



f (xt)=






E(h∆t)=

radic

radic

radic

radic∆tNT

sumk=1

ek2infin rt = log2

E(h∆t)

E(h∆t2) rs = log2

E(h∆t)

E(h2∆t)




∆t

α=02 α=05



110 1570e-02 170 1570e-02 170 8151e-02 148 8151e-02 148

120 4846e-03 170 4844e-03 170 2922e-02 147 2923e-02 147

140 1489e-03 172 1488e-03 171 1052e-02 148 1052e-02 148

180 4524e-04 170 4541e-04 172 3781e-03 148 3784e-03 148

1160 1395e-04 1375e-04 1357e-03 1357e-03


h

α=02 α=05



π10 8862e-01 206 8632e-01 203 8652e-01 205 8035e-01 199

π20 2122e-01 201 2107e-01 201 2087e-01 201 2016e-01 199

π40 5258e-02 200 5244e-02 200 5177e-02 200 5064e-02 199

π80 1312e-02 200 1311e-02 200 1292e-02 200 1272e-02 199

π160 3280e-03 3277e-03 3229e-03 3195e-03

CPU(s) 4337 330466 4365 222606




30000











42 44 46 48 5 5205

1

15

2

25

3

35

4

Log10

(NT)

Log 10

(cpu

time)





C0 Dα

t u(xt)=uxx+ f (u)

u(x0)=u0(x)(41)





C0 Dα



(partx+3sα20 )

C0 Dα

t u+(3sα0partx+s

3α2

0 )u=(partx+3sα20 ) f (u) x= xr (44)

(partxminus3sα20 )

C0 Dα


3α2




C0 D

αt Un

i =δ2xUn


(δx+3sα20 )

C0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

C0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (48)


0 Dαt by our



FC0 D

αt Un

i =δ2xUn


(δx+3sα20 )

FC0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

FC0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (411)













∆t

α=02 α=05



110 8601e-04 102 8601e-04 102 2194e-03 109 2194e-03 109

120 4243e-04 101 4243e-04 101 1030e-03 106 1030e-03 106

140 2104e-04 101 2104e-04 101 4934e-04 104 4934e-04 104

180 1046e-04 1046e-04 2392e-04 2392e-04


h

α=02 α=05



180 8196e-04 196 8196e-04 196 6045e-04 197 6045e-04 197

1160 2112e-04 201 2112e-04 201 1547e-04 201 1547e-04 201

1320 5246e-05 200 5246e-05 200 3842e-05 200 3842e-05 200

1640 1316e-05 1316e-05 9649e-06 9649e-06

CPU(s) 3880 107191 4022 85058




4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)


4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)




h ) between the


h )






∆t

α=02 α=05



110 1089e-03 104 1089e-03 104 2961e-03 106 2961e-03 106

120 5304e-04 102 5304-04 102 1418e-03 104 1418e-03 104

140 2614e-04 101 2614e-04 101 6896e-04 103 6896e-04 103

180 1294e-04 1294e-04 3379e-04 3379e-04


h

α=02 α=05



180 9051e-04 196 9051e-04 196 6882e-04 198 6882e-04 198

1160 2319e-04 201 2319e-04 201 1746e-04 201 1746e-04 201

1320 5762e-05 200 5762e-05 200 4344e-05 200 4344e-05 200

1640 1447e-05 1447e-05 1091e-05 1091e-05

CPU(s) 3869 110658 4014 85587


h ) between the


h )




5 Conclusions





Acknowledgments




FC0 D

αt un=



+1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

=∆tminusα

n

Γ(1minusα)

(

un

1minusαminus(

α

1minusα+a0)u

nminus1minusnminus2

suml=1


1

nα)u0

)

(A1)

where

an =α∆tαNexp

sumj=1


j bn =α∆tαNexp

sumj=1


j



λ1j =

eminuss j∆t

s2j ∆t


eminuss j∆t

s2j ∆t



(FC0 D

αt gk)gk =

1

∆tαΓ(1minusα)

(

1

1minusα(gk)2minus

( α

1minusα+a0

)

gkminus1gk

minuskminus2

suml=1


bkminus2+1

kα

)

g0gk

)

ge 1

∆tαΓ(1minusα)

[

(

2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minus1

2

( α

1minusα+a0

)

(gkminus1)2minus 1

2

kminus2

suml=1


minus1

2

(

bkminus2+1

kα

)

(g0)2

)]

(A2)


∆tn

sumk=1

(FC0 D


Γ(1minusα)

[

n

sumk=2

(

1

1minusαminus α

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minusn

sumk=2

(

α

2(1minusα)+

a0

2

)

(gkminus1)2minus 1

2

n

sumk=2

kminus2

suml=1

(


)

(gl)2

+1



2

n

sumk=2

(

bkminus2+1

kα

)

(g0)2

]

=∆t1minusα

Γ(1minusα)

n

sumk=1

(

Ck(gk)2minusC0(g0)2

)

(A3)


Cnk =

1

2(1minusα)+

1

2

n

sumk=2

(

bkminus2+1

kα

)

k=0

1

2minus 1

2

nminus2

suml=0

(al+bl)+1

2bnminus2 k=1

1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1


2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα k=n

(A4)




1

t1+αminusεle

Nexp

sumj=1

ωjeminuss jtle 1

t1+α+ε (A5)


kminus2

suml=0


∆t

Nexp

sumj=1



(

1minus 1

kα

)


suml=0

(al+bl)le(

1minus 1

kα

)



Cn0 le

n1minusα

2(1minusα)+

α∆tαtnminus1ε

2

Cn1 ge

1

2minus 1

2

nminus2

suml=0

(al+bl)ge1


Cnk =1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1

2bnminuskminus1


Cnn ge

2minusα

2(1minusα)minus

nminus2

suml=0

(al+bl)minus1

2nαge 1


(A8)




FC0 D

αt ek

i =δ2xek


FC0 D

αt ek

0=2

h

[

δxek12minusFC

0 Dα2t ek

0

]

+Tk0 (B2)

FC0 D

αt ek

NS=

2

h

[

minusδxenNSminus 1

2minusFC

0 Dα2t ek

NS

]

+TkNS

(B3)





Tki =minus

[

C0 Dα

t u(xitk)minusFC0 D

αt Uk

i

]

+[


i

]


Tk0 =

uxx(x0tk)minus2

h

[


]

minus 2

h

[

C0 D

α2t u(x0tk)minusFC

0 D

α2t Uk

0

]

minus[

C0 Dα

t u(x0tk)minusFC0 D

αt Uk

0

]

1le kleNT

TkNS

=

uxx(xNStk)minus

2

h

[

ux(xNStk)minusδxUk

NSminus 12

]

minus 2

h

[

C0 D

α2t u(xNS

tk)minusFC0 D

α2t Uk

NS

]

minus[

C0 Dα

t u(xNStk)minusFC

0 Dαt Uk

NS

]

1le kleNT



∣

∣

∣Tk

i

∣

∣


∣

∣

∣Tk

0

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B6)

∣

∣

∣Tk

NS

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B7)


1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

le c21

2ν

(


)2+

2c21L

micro

(


le2c21

ν

(


+2c2

1L

micro

(


le4c21

(1

ν+

L

micro

)


(1

ν+

L

micro

)

ε2 (B8)


∆tn

sumk=1

ek2infin le ∆t(1+

radic

1+L2micro)

Lmicro

n

sumk=1

(

1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

)

(B9)




References



































































∆t

α=02 α=05



110 1570e-02 170 1570e-02 170 8151e-02 148 8151e-02 148

120 4846e-03 170 4844e-03 170 2922e-02 147 2923e-02 147

140 1489e-03 172 1488e-03 171 1052e-02 148 1052e-02 148

180 4524e-04 170 4541e-04 172 3781e-03 148 3784e-03 148

1160 1395e-04 1375e-04 1357e-03 1357e-03


h

α=02 α=05



π10 8862e-01 206 8632e-01 203 8652e-01 205 8035e-01 199

π20 2122e-01 201 2107e-01 201 2087e-01 201 2016e-01 199

π40 5258e-02 200 5244e-02 200 5177e-02 200 5064e-02 199

π80 1312e-02 200 1311e-02 200 1292e-02 200 1272e-02 199

π160 3280e-03 3277e-03 3229e-03 3195e-03

CPU(s) 4337 330466 4365 222606




30000











42 44 46 48 5 5205

1

15

2

25

3

35

4

Log10

(NT)

Log 10

(cpu

time)





C0 Dα

t u(xt)=uxx+ f (u)

u(x0)=u0(x)(41)





C0 Dα



(partx+3sα20 )

C0 Dα

t u+(3sα0partx+s

3α2

0 )u=(partx+3sα20 ) f (u) x= xr (44)

(partxminus3sα20 )

C0 Dα


3α2




C0 D

αt Un

i =δ2xUn


(δx+3sα20 )

C0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

C0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (48)


0 Dαt by our



FC0 D

αt Un

i =δ2xUn


(δx+3sα20 )

FC0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

FC0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (411)













∆t

α=02 α=05



110 8601e-04 102 8601e-04 102 2194e-03 109 2194e-03 109

120 4243e-04 101 4243e-04 101 1030e-03 106 1030e-03 106

140 2104e-04 101 2104e-04 101 4934e-04 104 4934e-04 104

180 1046e-04 1046e-04 2392e-04 2392e-04


h

α=02 α=05



180 8196e-04 196 8196e-04 196 6045e-04 197 6045e-04 197

1160 2112e-04 201 2112e-04 201 1547e-04 201 1547e-04 201

1320 5246e-05 200 5246e-05 200 3842e-05 200 3842e-05 200

1640 1316e-05 1316e-05 9649e-06 9649e-06

CPU(s) 3880 107191 4022 85058




4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)


4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)




h ) between the


h )






∆t

α=02 α=05



110 1089e-03 104 1089e-03 104 2961e-03 106 2961e-03 106

120 5304e-04 102 5304-04 102 1418e-03 104 1418e-03 104

140 2614e-04 101 2614e-04 101 6896e-04 103 6896e-04 103

180 1294e-04 1294e-04 3379e-04 3379e-04


h

α=02 α=05



180 9051e-04 196 9051e-04 196 6882e-04 198 6882e-04 198

1160 2319e-04 201 2319e-04 201 1746e-04 201 1746e-04 201

1320 5762e-05 200 5762e-05 200 4344e-05 200 4344e-05 200

1640 1447e-05 1447e-05 1091e-05 1091e-05

CPU(s) 3869 110658 4014 85587


h ) between the


h )




5 Conclusions





Acknowledgments




FC0 D

αt un=



+1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

=∆tminusα

n

Γ(1minusα)

(

un

1minusαminus(

α

1minusα+a0)u

nminus1minusnminus2

suml=1


1

nα)u0

)

(A1)

where

an =α∆tαNexp

sumj=1


j bn =α∆tαNexp

sumj=1


j



λ1j =

eminuss j∆t

s2j ∆t


eminuss j∆t

s2j ∆t



(FC0 D

αt gk)gk =

1

∆tαΓ(1minusα)

(

1

1minusα(gk)2minus

( α

1minusα+a0

)

gkminus1gk

minuskminus2

suml=1


bkminus2+1

kα

)

g0gk

)

ge 1

∆tαΓ(1minusα)

[

(

2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minus1

2

( α

1minusα+a0

)

(gkminus1)2minus 1

2

kminus2

suml=1


minus1

2

(

bkminus2+1

kα

)

(g0)2

)]

(A2)


∆tn

sumk=1

(FC0 D


Γ(1minusα)

[

n

sumk=2

(

1

1minusαminus α

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minusn

sumk=2

(

α

2(1minusα)+

a0

2

)

(gkminus1)2minus 1

2

n

sumk=2

kminus2

suml=1

(


)

(gl)2

+1



2

n

sumk=2

(

bkminus2+1

kα

)

(g0)2

]

=∆t1minusα

Γ(1minusα)

n

sumk=1

(

Ck(gk)2minusC0(g0)2

)

(A3)


Cnk =

1

2(1minusα)+

1

2

n

sumk=2

(

bkminus2+1

kα

)

k=0

1

2minus 1

2

nminus2

suml=0

(al+bl)+1

2bnminus2 k=1

1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1


2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα k=n

(A4)




1

t1+αminusεle

Nexp

sumj=1

ωjeminuss jtle 1

t1+α+ε (A5)


kminus2

suml=0


∆t

Nexp

sumj=1



(

1minus 1

kα

)


suml=0

(al+bl)le(

1minus 1

kα

)



Cn0 le

n1minusα

2(1minusα)+

α∆tαtnminus1ε

2

Cn1 ge

1

2minus 1

2

nminus2

suml=0

(al+bl)ge1


Cnk =1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1

2bnminuskminus1


Cnn ge

2minusα

2(1minusα)minus

nminus2

suml=0

(al+bl)minus1

2nαge 1


(A8)




FC0 D

αt ek

i =δ2xek


FC0 D

αt ek

0=2

h

[

δxek12minusFC

0 Dα2t ek

0

]

+Tk0 (B2)

FC0 D

αt ek

NS=

2

h

[

minusδxenNSminus 1

2minusFC

0 Dα2t ek

NS

]

+TkNS

(B3)





Tki =minus

[

C0 Dα

t u(xitk)minusFC0 D

αt Uk

i

]

+[


i

]


Tk0 =

uxx(x0tk)minus2

h

[


]

minus 2

h

[

C0 D

α2t u(x0tk)minusFC

0 D

α2t Uk

0

]

minus[

C0 Dα

t u(x0tk)minusFC0 D

αt Uk

0

]

1le kleNT

TkNS

=

uxx(xNStk)minus

2

h

[

ux(xNStk)minusδxUk

NSminus 12

]

minus 2

h

[

C0 D

α2t u(xNS

tk)minusFC0 D

α2t Uk

NS

]

minus[

C0 Dα

t u(xNStk)minusFC

0 Dαt Uk

NS

]

1le kleNT



∣

∣

∣Tk

i

∣

∣


∣

∣

∣Tk

0

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B6)

∣

∣

∣Tk

NS

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B7)


1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

le c21

2ν

(


)2+

2c21L

micro

(


le2c21

ν

(


+2c2

1L

micro

(


le4c21

(1

ν+

L

micro

)


(1

ν+

L

micro

)

ε2 (B8)


∆tn

sumk=1

ek2infin le ∆t(1+

radic

1+L2micro)

Lmicro

n

sumk=1

(

1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

)

(B9)




References







































































42 44 46 48 5 5205

1

15

2

25

3

35

4

Log10

(NT)

Log 10

(cpu

time)





C0 Dα

t u(xt)=uxx+ f (u)

u(x0)=u0(x)(41)





C0 Dα



(partx+3sα20 )

C0 Dα

t u+(3sα0partx+s

3α2

0 )u=(partx+3sα20 ) f (u) x= xr (44)

(partxminus3sα20 )

C0 Dα


3α2




C0 D

αt Un

i =δ2xUn


(δx+3sα20 )

C0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

C0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (48)


0 Dαt by our



FC0 D

αt Un

i =δ2xUn


(δx+3sα20 )

FC0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

FC0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (411)













∆t

α=02 α=05



110 8601e-04 102 8601e-04 102 2194e-03 109 2194e-03 109

120 4243e-04 101 4243e-04 101 1030e-03 106 1030e-03 106

140 2104e-04 101 2104e-04 101 4934e-04 104 4934e-04 104

180 1046e-04 1046e-04 2392e-04 2392e-04


h

α=02 α=05



180 8196e-04 196 8196e-04 196 6045e-04 197 6045e-04 197

1160 2112e-04 201 2112e-04 201 1547e-04 201 1547e-04 201

1320 5246e-05 200 5246e-05 200 3842e-05 200 3842e-05 200

1640 1316e-05 1316e-05 9649e-06 9649e-06

CPU(s) 3880 107191 4022 85058




4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)


4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)




h ) between the


h )






∆t

α=02 α=05



110 1089e-03 104 1089e-03 104 2961e-03 106 2961e-03 106

120 5304e-04 102 5304-04 102 1418e-03 104 1418e-03 104

140 2614e-04 101 2614e-04 101 6896e-04 103 6896e-04 103

180 1294e-04 1294e-04 3379e-04 3379e-04


h

α=02 α=05



180 9051e-04 196 9051e-04 196 6882e-04 198 6882e-04 198

1160 2319e-04 201 2319e-04 201 1746e-04 201 1746e-04 201

1320 5762e-05 200 5762e-05 200 4344e-05 200 4344e-05 200

1640 1447e-05 1447e-05 1091e-05 1091e-05

CPU(s) 3869 110658 4014 85587


h ) between the


h )




5 Conclusions





Acknowledgments




FC0 D

αt un=



+1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

=∆tminusα

n

Γ(1minusα)

(

un

1minusαminus(

α

1minusα+a0)u

nminus1minusnminus2

suml=1


1

nα)u0

)

(A1)

where

an =α∆tαNexp

sumj=1


j bn =α∆tαNexp

sumj=1


j



λ1j =

eminuss j∆t

s2j ∆t


eminuss j∆t

s2j ∆t



(FC0 D

αt gk)gk =

1

∆tαΓ(1minusα)

(

1

1minusα(gk)2minus

( α

1minusα+a0

)

gkminus1gk

minuskminus2

suml=1


bkminus2+1

kα

)

g0gk

)

ge 1

∆tαΓ(1minusα)

[

(

2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minus1

2

( α

1minusα+a0

)

(gkminus1)2minus 1

2

kminus2

suml=1


minus1

2

(

bkminus2+1

kα

)

(g0)2

)]

(A2)


∆tn

sumk=1

(FC0 D


Γ(1minusα)

[

n

sumk=2

(

1

1minusαminus α

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minusn

sumk=2

(

α

2(1minusα)+

a0

2

)

(gkminus1)2minus 1

2

n

sumk=2

kminus2

suml=1

(


)

(gl)2

+1



2

n

sumk=2

(

bkminus2+1

kα

)

(g0)2

]

=∆t1minusα

Γ(1minusα)

n

sumk=1

(

Ck(gk)2minusC0(g0)2

)

(A3)


Cnk =

1

2(1minusα)+

1

2

n

sumk=2

(

bkminus2+1

kα

)

k=0

1

2minus 1

2

nminus2

suml=0

(al+bl)+1

2bnminus2 k=1

1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1


2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα k=n

(A4)




1

t1+αminusεle

Nexp

sumj=1

ωjeminuss jtle 1

t1+α+ε (A5)


kminus2

suml=0


∆t

Nexp

sumj=1



(

1minus 1

kα

)


suml=0

(al+bl)le(

1minus 1

kα

)



Cn0 le

n1minusα

2(1minusα)+

α∆tαtnminus1ε

2

Cn1 ge

1

2minus 1

2

nminus2

suml=0

(al+bl)ge1


Cnk =1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1

2bnminuskminus1


Cnn ge

2minusα

2(1minusα)minus

nminus2

suml=0

(al+bl)minus1

2nαge 1


(A8)




FC0 D

αt ek

i =δ2xek


FC0 D

αt ek

0=2

h

[

δxek12minusFC

0 Dα2t ek

0

]

+Tk0 (B2)

FC0 D

αt ek

NS=

2

h

[

minusδxenNSminus 1

2minusFC

0 Dα2t ek

NS

]

+TkNS

(B3)





Tki =minus

[

C0 Dα

t u(xitk)minusFC0 D

αt Uk

i

]

+[


i

]


Tk0 =

uxx(x0tk)minus2

h

[


]

minus 2

h

[

C0 D

α2t u(x0tk)minusFC

0 D

α2t Uk

0

]

minus[

C0 Dα

t u(x0tk)minusFC0 D

αt Uk

0

]

1le kleNT

TkNS

=

uxx(xNStk)minus

2

h

[

ux(xNStk)minusδxUk

NSminus 12

]

minus 2

h

[

C0 D

α2t u(xNS

tk)minusFC0 D

α2t Uk

NS

]

minus[

C0 Dα

t u(xNStk)minusFC

0 Dαt Uk

NS

]

1le kleNT



∣

∣

∣Tk

i

∣

∣


∣

∣

∣Tk

0

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B6)

∣

∣

∣Tk

NS

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B7)


1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

le c21

2ν

(


)2+

2c21L

micro

(


le2c21

ν

(


+2c2

1L

micro

(


le4c21

(1

ν+

L

micro

)


(1

ν+

L

micro

)

ε2 (B8)


∆tn

sumk=1

ek2infin le ∆t(1+

radic

1+L2micro)

Lmicro

n

sumk=1

(

1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

)

(B9)




References




































































C0 Dα



(partx+3sα20 )

C0 Dα

t u+(3sα0partx+s

3α2

0 )u=(partx+3sα20 ) f (u) x= xr (44)

(partxminus3sα20 )

C0 Dα


3α2




C0 D

αt Un

i =δ2xUn


(δx+3sα20 )

C0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

C0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (48)


0 Dαt by our



FC0 D

αt Un

i =δ2xUn


(δx+3sα20 )

FC0 D

αt Un


3α2


α20 ) f (Unminus1


(δxminus3sα20 )

FC0 D

αt Unminus1

1 +(3sα0 δxminuss

3α2


α20 ) f (Unminus1

1 ) 1lenleNT (411)













∆t

α=02 α=05



110 8601e-04 102 8601e-04 102 2194e-03 109 2194e-03 109

120 4243e-04 101 4243e-04 101 1030e-03 106 1030e-03 106

140 2104e-04 101 2104e-04 101 4934e-04 104 4934e-04 104

180 1046e-04 1046e-04 2392e-04 2392e-04


h

α=02 α=05



180 8196e-04 196 8196e-04 196 6045e-04 197 6045e-04 197

1160 2112e-04 201 2112e-04 201 1547e-04 201 1547e-04 201

1320 5246e-05 200 5246e-05 200 3842e-05 200 3842e-05 200

1640 1316e-05 1316e-05 9649e-06 9649e-06

CPU(s) 3880 107191 4022 85058




4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)


4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)




h ) between the


h )






∆t

α=02 α=05



110 1089e-03 104 1089e-03 104 2961e-03 106 2961e-03 106

120 5304e-04 102 5304-04 102 1418e-03 104 1418e-03 104

140 2614e-04 101 2614e-04 101 6896e-04 103 6896e-04 103

180 1294e-04 1294e-04 3379e-04 3379e-04


h

α=02 α=05



180 9051e-04 196 9051e-04 196 6882e-04 198 6882e-04 198

1160 2319e-04 201 2319e-04 201 1746e-04 201 1746e-04 201

1320 5762e-05 200 5762e-05 200 4344e-05 200 4344e-05 200

1640 1447e-05 1447e-05 1091e-05 1091e-05

CPU(s) 3869 110658 4014 85587


h ) between the


h )




5 Conclusions





Acknowledgments




FC0 D

αt un=



+1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

=∆tminusα

n

Γ(1minusα)

(

un

1minusαminus(

α

1minusα+a0)u

nminus1minusnminus2

suml=1


1

nα)u0

)

(A1)

where

an =α∆tαNexp

sumj=1


j bn =α∆tαNexp

sumj=1


j



λ1j =

eminuss j∆t

s2j ∆t


eminuss j∆t

s2j ∆t



(FC0 D

αt gk)gk =

1

∆tαΓ(1minusα)

(

1

1minusα(gk)2minus

( α

1minusα+a0

)

gkminus1gk

minuskminus2

suml=1


bkminus2+1

kα

)

g0gk

)

ge 1

∆tαΓ(1minusα)

[

(

2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minus1

2

( α

1minusα+a0

)

(gkminus1)2minus 1

2

kminus2

suml=1


minus1

2

(

bkminus2+1

kα

)

(g0)2

)]

(A2)


∆tn

sumk=1

(FC0 D


Γ(1minusα)

[

n

sumk=2

(

1

1minusαminus α

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minusn

sumk=2

(

α

2(1minusα)+

a0

2

)

(gkminus1)2minus 1

2

n

sumk=2

kminus2

suml=1

(


)

(gl)2

+1



2

n

sumk=2

(

bkminus2+1

kα

)

(g0)2

]

=∆t1minusα

Γ(1minusα)

n

sumk=1

(

Ck(gk)2minusC0(g0)2

)

(A3)


Cnk =

1

2(1minusα)+

1

2

n

sumk=2

(

bkminus2+1

kα

)

k=0

1

2minus 1

2

nminus2

suml=0

(al+bl)+1

2bnminus2 k=1

1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1


2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα k=n

(A4)




1

t1+αminusεle

Nexp

sumj=1

ωjeminuss jtle 1

t1+α+ε (A5)


kminus2

suml=0


∆t

Nexp

sumj=1



(

1minus 1

kα

)


suml=0

(al+bl)le(

1minus 1

kα

)



Cn0 le

n1minusα

2(1minusα)+

α∆tαtnminus1ε

2

Cn1 ge

1

2minus 1

2

nminus2

suml=0

(al+bl)ge1


Cnk =1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1

2bnminuskminus1


Cnn ge

2minusα

2(1minusα)minus

nminus2

suml=0

(al+bl)minus1

2nαge 1


(A8)




FC0 D

αt ek

i =δ2xek


FC0 D

αt ek

0=2

h

[

δxek12minusFC

0 Dα2t ek

0

]

+Tk0 (B2)

FC0 D

αt ek

NS=

2

h

[

minusδxenNSminus 1

2minusFC

0 Dα2t ek

NS

]

+TkNS

(B3)





Tki =minus

[

C0 Dα

t u(xitk)minusFC0 D

αt Uk

i

]

+[


i

]


Tk0 =

uxx(x0tk)minus2

h

[


]

minus 2

h

[

C0 D

α2t u(x0tk)minusFC

0 D

α2t Uk

0

]

minus[

C0 Dα

t u(x0tk)minusFC0 D

αt Uk

0

]

1le kleNT

TkNS

=

uxx(xNStk)minus

2

h

[

ux(xNStk)minusδxUk

NSminus 12

]

minus 2

h

[

C0 D

α2t u(xNS

tk)minusFC0 D

α2t Uk

NS

]

minus[

C0 Dα

t u(xNStk)minusFC

0 Dαt Uk

NS

]

1le kleNT



∣

∣

∣Tk

i

∣

∣


∣

∣

∣Tk

0

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B6)

∣

∣

∣Tk

NS

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B7)


1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

le c21

2ν

(


)2+

2c21L

micro

(


le2c21

ν

(


+2c2

1L

micro

(


le4c21

(1

ν+

L

micro

)


(1

ν+

L

micro

)

ε2 (B8)


∆tn

sumk=1

ek2infin le ∆t(1+

radic

1+L2micro)

Lmicro

n

sumk=1

(

1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

)

(B9)




References








































































∆t

α=02 α=05



110 8601e-04 102 8601e-04 102 2194e-03 109 2194e-03 109

120 4243e-04 101 4243e-04 101 1030e-03 106 1030e-03 106

140 2104e-04 101 2104e-04 101 4934e-04 104 4934e-04 104

180 1046e-04 1046e-04 2392e-04 2392e-04


h

α=02 α=05



180 8196e-04 196 8196e-04 196 6045e-04 197 6045e-04 197

1160 2112e-04 201 2112e-04 201 1547e-04 201 1547e-04 201

1320 5246e-05 200 5246e-05 200 3842e-05 200 3842e-05 200

1640 1316e-05 1316e-05 9649e-06 9649e-06

CPU(s) 3880 107191 4022 85058




4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)


4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)




h ) between the


h )






∆t

α=02 α=05



110 1089e-03 104 1089e-03 104 2961e-03 106 2961e-03 106

120 5304e-04 102 5304-04 102 1418e-03 104 1418e-03 104

140 2614e-04 101 2614e-04 101 6896e-04 103 6896e-04 103

180 1294e-04 1294e-04 3379e-04 3379e-04


h

α=02 α=05



180 9051e-04 196 9051e-04 196 6882e-04 198 6882e-04 198

1160 2319e-04 201 2319e-04 201 1746e-04 201 1746e-04 201

1320 5762e-05 200 5762e-05 200 4344e-05 200 4344e-05 200

1640 1447e-05 1447e-05 1091e-05 1091e-05

CPU(s) 3869 110658 4014 85587


h ) between the


h )




5 Conclusions





Acknowledgments




FC0 D

αt un=



+1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

=∆tminusα

n

Γ(1minusα)

(

un

1minusαminus(

α

1minusα+a0)u

nminus1minusnminus2

suml=1


1

nα)u0

)

(A1)

where

an =α∆tαNexp

sumj=1


j bn =α∆tαNexp

sumj=1


j



λ1j =

eminuss j∆t

s2j ∆t


eminuss j∆t

s2j ∆t



(FC0 D

αt gk)gk =

1

∆tαΓ(1minusα)

(

1

1minusα(gk)2minus

( α

1minusα+a0

)

gkminus1gk

minuskminus2

suml=1


bkminus2+1

kα

)

g0gk

)

ge 1

∆tαΓ(1minusα)

[

(

2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minus1

2

( α

1minusα+a0

)

(gkminus1)2minus 1

2

kminus2

suml=1


minus1

2

(

bkminus2+1

kα

)

(g0)2

)]

(A2)


∆tn

sumk=1

(FC0 D


Γ(1minusα)

[

n

sumk=2

(

1

1minusαminus α

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minusn

sumk=2

(

α

2(1minusα)+

a0

2

)

(gkminus1)2minus 1

2

n

sumk=2

kminus2

suml=1

(


)

(gl)2

+1



2

n

sumk=2

(

bkminus2+1

kα

)

(g0)2

]

=∆t1minusα

Γ(1minusα)

n

sumk=1

(

Ck(gk)2minusC0(g0)2

)

(A3)


Cnk =

1

2(1minusα)+

1

2

n

sumk=2

(

bkminus2+1

kα

)

k=0

1

2minus 1

2

nminus2

suml=0

(al+bl)+1

2bnminus2 k=1

1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1


2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα k=n

(A4)




1

t1+αminusεle

Nexp

sumj=1

ωjeminuss jtle 1

t1+α+ε (A5)


kminus2

suml=0


∆t

Nexp

sumj=1



(

1minus 1

kα

)


suml=0

(al+bl)le(

1minus 1

kα

)



Cn0 le

n1minusα

2(1minusα)+

α∆tαtnminus1ε

2

Cn1 ge

1

2minus 1

2

nminus2

suml=0

(al+bl)ge1


Cnk =1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1

2bnminuskminus1


Cnn ge

2minusα

2(1minusα)minus

nminus2

suml=0

(al+bl)minus1

2nαge 1


(A8)




FC0 D

αt ek

i =δ2xek


FC0 D

αt ek

0=2

h

[

δxek12minusFC

0 Dα2t ek

0

]

+Tk0 (B2)

FC0 D

αt ek

NS=

2

h

[

minusδxenNSminus 1

2minusFC

0 Dα2t ek

NS

]

+TkNS

(B3)





Tki =minus

[

C0 Dα

t u(xitk)minusFC0 D

αt Uk

i

]

+[


i

]


Tk0 =

uxx(x0tk)minus2

h

[


]

minus 2

h

[

C0 D

α2t u(x0tk)minusFC

0 D

α2t Uk

0

]

minus[

C0 Dα

t u(x0tk)minusFC0 D

αt Uk

0

]

1le kleNT

TkNS

=

uxx(xNStk)minus

2

h

[

ux(xNStk)minusδxUk

NSminus 12

]

minus 2

h

[

C0 D

α2t u(xNS

tk)minusFC0 D

α2t Uk

NS

]

minus[

C0 Dα

t u(xNStk)minusFC

0 Dαt Uk

NS

]

1le kleNT



∣

∣

∣Tk

i

∣

∣


∣

∣

∣Tk

0

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B6)

∣

∣

∣Tk

NS

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B7)


1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

le c21

2ν

(


)2+

2c21L

micro

(


le2c21

ν

(


+2c2

1L

micro

(


le4c21

(1

ν+

L

micro

)


(1

ν+

L

micro

)

ε2 (B8)


∆tn

sumk=1

ek2infin le ∆t(1+

radic

1+L2micro)

Lmicro

n

sumk=1

(

1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

)

(B9)




References


































































4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)


4 42 44 46 480

05

1

15

2

25

3

Log10

(NT)

Log 10

(cpu

time)




h ) between the


h )






∆t

α=02 α=05



110 1089e-03 104 1089e-03 104 2961e-03 106 2961e-03 106

120 5304e-04 102 5304-04 102 1418e-03 104 1418e-03 104

140 2614e-04 101 2614e-04 101 6896e-04 103 6896e-04 103

180 1294e-04 1294e-04 3379e-04 3379e-04


h

α=02 α=05



180 9051e-04 196 9051e-04 196 6882e-04 198 6882e-04 198

1160 2319e-04 201 2319e-04 201 1746e-04 201 1746e-04 201

1320 5762e-05 200 5762e-05 200 4344e-05 200 4344e-05 200

1640 1447e-05 1447e-05 1091e-05 1091e-05

CPU(s) 3869 110658 4014 85587


h ) between the


h )




5 Conclusions





Acknowledgments




FC0 D

αt un=



+1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

=∆tminusα

n

Γ(1minusα)

(

un

1minusαminus(

α

1minusα+a0)u

nminus1minusnminus2

suml=1


1

nα)u0

)

(A1)

where

an =α∆tαNexp

sumj=1


j bn =α∆tαNexp

sumj=1


j



λ1j =

eminuss j∆t

s2j ∆t


eminuss j∆t

s2j ∆t



(FC0 D

αt gk)gk =

1

∆tαΓ(1minusα)

(

1

1minusα(gk)2minus

( α

1minusα+a0

)

gkminus1gk

minuskminus2

suml=1


bkminus2+1

kα

)

g0gk

)

ge 1

∆tαΓ(1minusα)

[

(

2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minus1

2

( α

1minusα+a0

)

(gkminus1)2minus 1

2

kminus2

suml=1


minus1

2

(

bkminus2+1

kα

)

(g0)2

)]

(A2)


∆tn

sumk=1

(FC0 D


Γ(1minusα)

[

n

sumk=2

(

1

1minusαminus α

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minusn

sumk=2

(

α

2(1minusα)+

a0

2

)

(gkminus1)2minus 1

2

n

sumk=2

kminus2

suml=1

(


)

(gl)2

+1



2

n

sumk=2

(

bkminus2+1

kα

)

(g0)2

]

=∆t1minusα

Γ(1minusα)

n

sumk=1

(

Ck(gk)2minusC0(g0)2

)

(A3)


Cnk =

1

2(1minusα)+

1

2

n

sumk=2

(

bkminus2+1

kα

)

k=0

1

2minus 1

2

nminus2

suml=0

(al+bl)+1

2bnminus2 k=1

1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1


2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα k=n

(A4)




1

t1+αminusεle

Nexp

sumj=1

ωjeminuss jtle 1

t1+α+ε (A5)


kminus2

suml=0


∆t

Nexp

sumj=1



(

1minus 1

kα

)


suml=0

(al+bl)le(

1minus 1

kα

)



Cn0 le

n1minusα

2(1minusα)+

α∆tαtnminus1ε

2

Cn1 ge

1

2minus 1

2

nminus2

suml=0

(al+bl)ge1


Cnk =1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1

2bnminuskminus1


Cnn ge

2minusα

2(1minusα)minus

nminus2

suml=0

(al+bl)minus1

2nαge 1


(A8)




FC0 D

αt ek

i =δ2xek


FC0 D

αt ek

0=2

h

[

δxek12minusFC

0 Dα2t ek

0

]

+Tk0 (B2)

FC0 D

αt ek

NS=

2

h

[

minusδxenNSminus 1

2minusFC

0 Dα2t ek

NS

]

+TkNS

(B3)





Tki =minus

[

C0 Dα

t u(xitk)minusFC0 D

αt Uk

i

]

+[


i

]


Tk0 =

uxx(x0tk)minus2

h

[


]

minus 2

h

[

C0 D

α2t u(x0tk)minusFC

0 D

α2t Uk

0

]

minus[

C0 Dα

t u(x0tk)minusFC0 D

αt Uk

0

]

1le kleNT

TkNS

=

uxx(xNStk)minus

2

h

[

ux(xNStk)minusδxUk

NSminus 12

]

minus 2

h

[

C0 D

α2t u(xNS

tk)minusFC0 D

α2t Uk

NS

]

minus[

C0 Dα

t u(xNStk)minusFC

0 Dαt Uk

NS

]

1le kleNT



∣

∣

∣Tk

i

∣

∣


∣

∣

∣Tk

0

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B6)

∣

∣

∣Tk

NS

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B7)


1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

le c21

2ν

(


)2+

2c21L

micro

(


le2c21

ν

(


+2c2

1L

micro

(


le4c21

(1

ν+

L

micro

)


(1

ν+

L

micro

)

ε2 (B8)


∆tn

sumk=1

ek2infin le ∆t(1+

radic

1+L2micro)

Lmicro

n

sumk=1

(

1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

)

(B9)




References



































































∆t

α=02 α=05



110 1089e-03 104 1089e-03 104 2961e-03 106 2961e-03 106

120 5304e-04 102 5304-04 102 1418e-03 104 1418e-03 104

140 2614e-04 101 2614e-04 101 6896e-04 103 6896e-04 103

180 1294e-04 1294e-04 3379e-04 3379e-04


h

α=02 α=05



180 9051e-04 196 9051e-04 196 6882e-04 198 6882e-04 198

1160 2319e-04 201 2319e-04 201 1746e-04 201 1746e-04 201

1320 5762e-05 200 5762e-05 200 4344e-05 200 4344e-05 200

1640 1447e-05 1447e-05 1091e-05 1091e-05

CPU(s) 3869 110658 4014 85587


h ) between the


h )




5 Conclusions





Acknowledgments




FC0 D

αt un=



+1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

=∆tminusα

n

Γ(1minusα)

(

un

1minusαminus(

α

1minusα+a0)u

nminus1minusnminus2

suml=1


1

nα)u0

)

(A1)

where

an =α∆tαNexp

sumj=1


j bn =α∆tαNexp

sumj=1


j



λ1j =

eminuss j∆t

s2j ∆t


eminuss j∆t

s2j ∆t



(FC0 D

αt gk)gk =

1

∆tαΓ(1minusα)

(

1

1minusα(gk)2minus

( α

1minusα+a0

)

gkminus1gk

minuskminus2

suml=1


bkminus2+1

kα

)

g0gk

)

ge 1

∆tαΓ(1minusα)

[

(

2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minus1

2

( α

1minusα+a0

)

(gkminus1)2minus 1

2

kminus2

suml=1


minus1

2

(

bkminus2+1

kα

)

(g0)2

)]

(A2)


∆tn

sumk=1

(FC0 D


Γ(1minusα)

[

n

sumk=2

(

1

1minusαminus α

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minusn

sumk=2

(

α

2(1minusα)+

a0

2

)

(gkminus1)2minus 1

2

n

sumk=2

kminus2

suml=1

(


)

(gl)2

+1



2

n

sumk=2

(

bkminus2+1

kα

)

(g0)2

]

=∆t1minusα

Γ(1minusα)

n

sumk=1

(

Ck(gk)2minusC0(g0)2

)

(A3)


Cnk =

1

2(1minusα)+

1

2

n

sumk=2

(

bkminus2+1

kα

)

k=0

1

2minus 1

2

nminus2

suml=0

(al+bl)+1

2bnminus2 k=1

1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1


2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα k=n

(A4)




1

t1+αminusεle

Nexp

sumj=1

ωjeminuss jtle 1

t1+α+ε (A5)


kminus2

suml=0


∆t

Nexp

sumj=1



(

1minus 1

kα

)


suml=0

(al+bl)le(

1minus 1

kα

)



Cn0 le

n1minusα

2(1minusα)+

α∆tαtnminus1ε

2

Cn1 ge

1

2minus 1

2

nminus2

suml=0

(al+bl)ge1


Cnk =1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1

2bnminuskminus1


Cnn ge

2minusα

2(1minusα)minus

nminus2

suml=0

(al+bl)minus1

2nαge 1


(A8)




FC0 D

αt ek

i =δ2xek


FC0 D

αt ek

0=2

h

[

δxek12minusFC

0 Dα2t ek

0

]

+Tk0 (B2)

FC0 D

αt ek

NS=

2

h

[

minusδxenNSminus 1

2minusFC

0 Dα2t ek

NS

]

+TkNS

(B3)





Tki =minus

[

C0 Dα

t u(xitk)minusFC0 D

αt Uk

i

]

+[


i

]


Tk0 =

uxx(x0tk)minus2

h

[


]

minus 2

h

[

C0 D

α2t u(x0tk)minusFC

0 D

α2t Uk

0

]

minus[

C0 Dα

t u(x0tk)minusFC0 D

αt Uk

0

]

1le kleNT

TkNS

=

uxx(xNStk)minus

2

h

[

ux(xNStk)minusδxUk

NSminus 12

]

minus 2

h

[

C0 D

α2t u(xNS

tk)minusFC0 D

α2t Uk

NS

]

minus[

C0 Dα

t u(xNStk)minusFC

0 Dαt Uk

NS

]

1le kleNT



∣

∣

∣Tk

i

∣

∣


∣

∣

∣Tk

0

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B6)

∣

∣

∣Tk

NS

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B7)


1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

le c21

2ν

(


)2+

2c21L

micro

(


le2c21

ν

(


+2c2

1L

micro

(


le4c21

(1

ν+

L

micro

)


(1

ν+

L

micro

)

ε2 (B8)


∆tn

sumk=1

ek2infin le ∆t(1+

radic

1+L2micro)

Lmicro

n

sumk=1

(

1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

)

(B9)




References


































































5 Conclusions





Acknowledgments




FC0 D

αt un=



+1

Γ(1minusα)

[

u(tnminus1)

∆tαn

minus u(t0)

tαn

minusαNexp

sumi=1

ωiUhisti(tn)

]

=∆tminusα

n

Γ(1minusα)

(

un

1minusαminus(

α

1minusα+a0)u

nminus1minusnminus2

suml=1


1

nα)u0

)

(A1)

where

an =α∆tαNexp

sumj=1


j bn =α∆tαNexp

sumj=1


j



λ1j =

eminuss j∆t

s2j ∆t


eminuss j∆t

s2j ∆t



(FC0 D

αt gk)gk =

1

∆tαΓ(1minusα)

(

1

1minusα(gk)2minus

( α

1minusα+a0

)

gkminus1gk

minuskminus2

suml=1


bkminus2+1

kα

)

g0gk

)

ge 1

∆tαΓ(1minusα)

[

(

2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minus1

2

( α

1minusα+a0

)

(gkminus1)2minus 1

2

kminus2

suml=1


minus1

2

(

bkminus2+1

kα

)

(g0)2

)]

(A2)


∆tn

sumk=1

(FC0 D


Γ(1minusα)

[

n

sumk=2

(

1

1minusαminus α

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minusn

sumk=2

(

α

2(1minusα)+

a0

2

)

(gkminus1)2minus 1

2

n

sumk=2

kminus2

suml=1

(


)

(gl)2

+1



2

n

sumk=2

(

bkminus2+1

kα

)

(g0)2

]

=∆t1minusα

Γ(1minusα)

n

sumk=1

(

Ck(gk)2minusC0(g0)2

)

(A3)


Cnk =

1

2(1minusα)+

1

2

n

sumk=2

(

bkminus2+1

kα

)

k=0

1

2minus 1

2

nminus2

suml=0

(al+bl)+1

2bnminus2 k=1

1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1


2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα k=n

(A4)




1

t1+αminusεle

Nexp

sumj=1

ωjeminuss jtle 1

t1+α+ε (A5)


kminus2

suml=0


∆t

Nexp

sumj=1



(

1minus 1

kα

)


suml=0

(al+bl)le(

1minus 1

kα

)



Cn0 le

n1minusα

2(1minusα)+

α∆tαtnminus1ε

2

Cn1 ge

1

2minus 1

2

nminus2

suml=0

(al+bl)ge1


Cnk =1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1

2bnminuskminus1


Cnn ge

2minusα

2(1minusα)minus

nminus2

suml=0

(al+bl)minus1

2nαge 1


(A8)




FC0 D

αt ek

i =δ2xek


FC0 D

αt ek

0=2

h

[

δxek12minusFC

0 Dα2t ek

0

]

+Tk0 (B2)

FC0 D

αt ek

NS=

2

h

[

minusδxenNSminus 1

2minusFC

0 Dα2t ek

NS

]

+TkNS

(B3)





Tki =minus

[

C0 Dα

t u(xitk)minusFC0 D

αt Uk

i

]

+[


i

]


Tk0 =

uxx(x0tk)minus2

h

[


]

minus 2

h

[

C0 D

α2t u(x0tk)minusFC

0 D

α2t Uk

0

]

minus[

C0 Dα

t u(x0tk)minusFC0 D

αt Uk

0

]

1le kleNT

TkNS

=

uxx(xNStk)minus

2

h

[

ux(xNStk)minusδxUk

NSminus 12

]

minus 2

h

[

C0 D

α2t u(xNS

tk)minusFC0 D

α2t Uk

NS

]

minus[

C0 Dα

t u(xNStk)minusFC

0 Dαt Uk

NS

]

1le kleNT



∣

∣

∣Tk

i

∣

∣


∣

∣

∣Tk

0

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B6)

∣

∣

∣Tk

NS

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B7)


1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

le c21

2ν

(


)2+

2c21L

micro

(


le2c21

ν

(


+2c2

1L

micro

(


le4c21

(1

ν+

L

micro

)


(1

ν+

L

micro

)

ε2 (B8)


∆tn

sumk=1

ek2infin le ∆t(1+

radic

1+L2micro)

Lmicro

n

sumk=1

(

1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

)

(B9)




References


































































λ1j =

eminuss j∆t

s2j ∆t


eminuss j∆t

s2j ∆t



(FC0 D

αt gk)gk =

1

∆tαΓ(1minusα)

(

1

1minusα(gk)2minus

( α

1minusα+a0

)

gkminus1gk

minuskminus2

suml=1


bkminus2+1

kα

)

g0gk

)

ge 1

∆tαΓ(1minusα)

[

(

2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minus1

2

( α

1minusα+a0

)

(gkminus1)2minus 1

2

kminus2

suml=1


minus1

2

(

bkminus2+1

kα

)

(g0)2

)]

(A2)


∆tn

sumk=1

(FC0 D


Γ(1minusα)

[

n

sumk=2

(

1

1minusαminus α

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα

)

(gk)2

minusn

sumk=2

(

α

2(1minusα)+

a0

2

)

(gkminus1)2minus 1

2

n

sumk=2

kminus2

suml=1

(


)

(gl)2

+1



2

n

sumk=2

(

bkminus2+1

kα

)

(g0)2

]

=∆t1minusα

Γ(1minusα)

n

sumk=1

(

Ck(gk)2minusC0(g0)2

)

(A3)


Cnk =

1

2(1minusα)+

1

2

n

sumk=2

(

bkminus2+1

kα

)

k=0

1

2minus 1

2

nminus2

suml=0

(al+bl)+1

2bnminus2 k=1

1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1


2minusα

2(1minusα)minus 1

2

kminus2

suml=0

(al+bl)minus1

2kα k=n

(A4)




1

t1+αminusεle

Nexp

sumj=1

ωjeminuss jtle 1

t1+α+ε (A5)


kminus2

suml=0


∆t

Nexp

sumj=1



(

1minus 1

kα

)


suml=0

(al+bl)le(

1minus 1

kα

)



Cn0 le

n1minusα

2(1minusα)+

α∆tαtnminus1ε

2

Cn1 ge

1

2minus 1

2

nminus2

suml=0

(al+bl)ge1


Cnk =1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1

2bnminuskminus1


Cnn ge

2minusα

2(1minusα)minus

nminus2

suml=0

(al+bl)minus1

2nαge 1


(A8)




FC0 D

αt ek

i =δ2xek


FC0 D

αt ek

0=2

h

[

δxek12minusFC

0 Dα2t ek

0

]

+Tk0 (B2)

FC0 D

αt ek

NS=

2

h

[

minusδxenNSminus 1

2minusFC

0 Dα2t ek

NS

]

+TkNS

(B3)





Tki =minus

[

C0 Dα

t u(xitk)minusFC0 D

αt Uk

i

]

+[


i

]


Tk0 =

uxx(x0tk)minus2

h

[


]

minus 2

h

[

C0 D

α2t u(x0tk)minusFC

0 D

α2t Uk

0

]

minus[

C0 Dα

t u(x0tk)minusFC0 D

αt Uk

0

]

1le kleNT

TkNS

=

uxx(xNStk)minus

2

h

[

ux(xNStk)minusδxUk

NSminus 12

]

minus 2

h

[

C0 D

α2t u(xNS

tk)minusFC0 D

α2t Uk

NS

]

minus[

C0 Dα

t u(xNStk)minusFC

0 Dαt Uk

NS

]

1le kleNT



∣

∣

∣Tk

i

∣

∣


∣

∣

∣Tk

0

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B6)

∣

∣

∣Tk

NS

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B7)


1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

le c21

2ν

(


)2+

2c21L

micro

(


le2c21

ν

(


+2c2

1L

micro

(


le4c21

(1

ν+

L

micro

)


(1

ν+

L

micro

)

ε2 (B8)


∆tn

sumk=1

ek2infin le ∆t(1+

radic

1+L2micro)

Lmicro

n

sumk=1

(

1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

)

(B9)




References



































































1

t1+αminusεle

Nexp

sumj=1

ωjeminuss jtle 1

t1+α+ε (A5)


kminus2

suml=0


∆t

Nexp

sumj=1



(

1minus 1

kα

)


suml=0

(al+bl)le(

1minus 1

kα

)



Cn0 le

n1minusα

2(1minusα)+

α∆tαtnminus1ε

2

Cn1 ge

1

2minus 1

2

nminus2

suml=0

(al+bl)ge1


Cnk =1minus 1

2

kminus2

suml=0

(al+bl)minus1

2kαminus 1

2

nminuskminus1

suml=0

(al+bl)+1

2bnminuskminus1


Cnn ge

2minusα

2(1minusα)minus

nminus2

suml=0

(al+bl)minus1

2nαge 1


(A8)




FC0 D

αt ek

i =δ2xek


FC0 D

αt ek

0=2

h

[

δxek12minusFC

0 Dα2t ek

0

]

+Tk0 (B2)

FC0 D

αt ek

NS=

2

h

[

minusδxenNSminus 1

2minusFC

0 Dα2t ek

NS

]

+TkNS

(B3)





Tki =minus

[

C0 Dα

t u(xitk)minusFC0 D

αt Uk

i

]

+[


i

]


Tk0 =

uxx(x0tk)minus2

h

[


]

minus 2

h

[

C0 D

α2t u(x0tk)minusFC

0 D

α2t Uk

0

]

minus[

C0 Dα

t u(x0tk)minusFC0 D

αt Uk

0

]

1le kleNT

TkNS

=

uxx(xNStk)minus

2

h

[

ux(xNStk)minusδxUk

NSminus 12

]

minus 2

h

[

C0 D

α2t u(xNS

tk)minusFC0 D

α2t Uk

NS

]

minus[

C0 Dα

t u(xNStk)minusFC

0 Dαt Uk

NS

]

1le kleNT



∣

∣

∣Tk

i

∣

∣


∣

∣

∣Tk

0

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B6)

∣

∣

∣Tk

NS

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B7)


1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

le c21

2ν

(


)2+

2c21L

micro

(


le2c21

ν

(


+2c2

1L

micro

(


le4c21

(1

ν+

L

micro

)


(1

ν+

L

micro

)

ε2 (B8)


∆tn

sumk=1

ek2infin le ∆t(1+

radic

1+L2micro)

Lmicro

n

sumk=1

(

1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

)

(B9)




References



































































Tki =minus

[

C0 Dα

t u(xitk)minusFC0 D

αt Uk

i

]

+[


i

]


Tk0 =

uxx(x0tk)minus2

h

[


]

minus 2

h

[

C0 D

α2t u(x0tk)minusFC

0 D

α2t Uk

0

]

minus[

C0 Dα

t u(x0tk)minusFC0 D

αt Uk

0

]

1le kleNT

TkNS

=

uxx(xNStk)minus

2

h

[

ux(xNStk)minusδxUk

NSminus 12

]

minus 2

h

[

C0 D

α2t u(xNS

tk)minusFC0 D

α2t Uk

NS

]

minus[

C0 Dα

t u(xNStk)minusFC

0 Dαt Uk

NS

]

1le kleNT



∣

∣

∣Tk

i

∣

∣


∣

∣

∣Tk

0

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B6)

∣

∣

∣Tk

NS

∣

∣


∆t2minusα2

h+

ε

h) 1le kleNT (B7)


1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

le c21

2ν

(


)2+

2c21L

micro

(


le2c21

ν

(


+2c2

1L

micro

(


le4c21

(1

ν+

L

micro

)


(1

ν+

L

micro

)

ε2 (B8)


∆tn

sumk=1

ek2infin le ∆t(1+

radic

1+L2micro)

Lmicro

n

sumk=1

(

1

4ν

[

(hTk0 )

2+(hTkNS)2]

+2

microh

NSminus1

sumi=1

(Tki )

2

)

(B9)




References


































































References

































































Fast Evaluation of the Caputo Fractional Derivative ...jiang/Papers/2017CiCP.pdf... approximation...

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Transcript of Fast Evaluation of the Caputo Fractional Derivative ...jiang/Papers/2017CiCP.pdf... approximation...