Research Article Legendre Wavelet Operational...
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Research Article Legendre Wavelet Operational Matrix Method for Solution of Riccati Differential Equation S. Balaji Department of Mathematics, SASTRA University, anjavur 613 401, India Correspondence should be addressed to S. Balaji; balaji [email protected] Received 12 February 2014; Revised 20 May 2014; Accepted 5 June 2014; Published 24 June 2014 Academic Editor: Petru Jebelean Copyright © 2014 S. Balaji. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A Legendre wavelet operational matrix method (LWM) is presented for the solution of nonlinear fractional-order Riccati differential equations, having variety of applications in quantum chemistry and quantum mechanics. e fractional-order Riccati differential equations converted into a system of algebraic equations using Legendre wavelet operational matrix. Solutions given by the proposed scheme are more accurate and reliable and they are compared with recently developed numerical, analytical, and stochastic approaches. Comparison shows that the proposed LWM approach has a greater performance and less computational effort for getting accurate solutions. Further existence and uniqueness of the proposed problem are given and moreover the condition of convergence is verified. 1. Introduction In recent years, use of fractional-order derivative goes very strongly in engineering and life sciences and also in other areas of science. One of the best advantages of use of fractional differential equation is modeling and control of many dynamic systems. Fractional-order derivatives are used in fruitful way to model many remarkable developments in those areas of science such as quantum chemistry, quantum mechanics, damping laws, rheology, and diffusion processes [1–5] described through the models of fractional differential equations (FDEs). Modeling of a physical phenomenon depends on two parameters such as the time instant and the prior time history; because of this reason, reasonable mod- eling through fractional calculus was successfully achieved. e abovementioned advantages and applications of FDEs attracted researchers to develop efficient methods to solve FDEs in order to get accurate solutions to such problems and more active research is still going on in those areas. Most of the FDEs are complicated in their structure; hence finding exact solutions for them cannot be simple. erefore, one can approach the best accurate solution of FDEs through analyti- cal and numerical methods. Designing accurate or best solu- tion to FDEs, many methods are developed in recent years; each method has its own advantages and limitations. is paper aims to solve a FDE called fractional-order Riccati differential equation, one of the important equations in the family of FDEs. e Riccati equations play an important role in engineering and applied science [6], especially in quantum mechanics [7] and quantum chemistry [8, 9]. erefore, solutions to the Riccati differential equations are important to scientists and engineers. Solving fractional-order Riccati dif- ferential equation, the most significant methods are Adomian decomposition method [10], homotopy perturbation method [11–14], homotopy analysis method [15, 16], Taylor matrix method [17], Haar wavelet method [18], and combination of Laplace, Adomian decomposition, and Pad´ e approximation [19] methods. Several numerical methods for approximating the solu- tion of nonlinear fractional-order Riccati differential equa- tions are known. Raja et al. [20] developed a stochastic tech- nique based on particle swarm optimization and simulated annealing. ey were used as a tool for rapid global search method and simulated annealing for efficient local search method. A fractional variational iteration method described in the Riemann-Liouville derivative has been applied in [21], to give an analytical approximate solution to nonlinear frac- tional Riccati differential equation. A combination of finite Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2014, Article ID 304745, 10 pages http://dx.doi.org/10.1155/2014/304745
Transcript of Research Article Legendre Wavelet Operational...
Research ArticleLegendre Wavelet Operational Matrix Method for Solution ofRiccati Differential Equation
S Balaji
Department of Mathematics SASTRA University Thanjavur 613 401 India
Correspondence should be addressed to S Balaji balaji mathsyahoocom
Received 12 February 2014 Revised 20 May 2014 Accepted 5 June 2014 Published 24 June 2014
Academic Editor Petru Jebelean
Copyright copy 2014 S Balaji This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
ALegendrewavelet operationalmatrixmethod (LWM) is presented for the solution of nonlinear fractional-orderRiccati differentialequations having variety of applications in quantum chemistry and quantum mechanics The fractional-order Riccati differentialequations converted into a system of algebraic equations using Legendre wavelet operational matrix Solutions given by theproposed scheme are more accurate and reliable and they are compared with recently developed numerical analytical andstochastic approaches Comparison shows that the proposed LWM approach has a greater performance and less computationaleffort for getting accurate solutions Further existence and uniqueness of the proposed problem are given and moreover thecondition of convergence is verified
1 Introduction
In recent years use of fractional-order derivative goes verystrongly in engineering and life sciences and also in otherareas of science One of the best advantages of use offractional differential equation is modeling and control ofmany dynamic systems Fractional-order derivatives are usedin fruitful way to model many remarkable developments inthose areas of science such as quantum chemistry quantummechanics damping laws rheology and diffusion processes[1ndash5] described through the models of fractional differentialequations (FDEs) Modeling of a physical phenomenondepends on two parameters such as the time instant and theprior time history because of this reason reasonable mod-eling through fractional calculus was successfully achievedThe abovementioned advantages and applications of FDEsattracted researchers to develop efficient methods to solveFDEs in order to get accurate solutions to such problems andmore active research is still going on in those areas Most ofthe FDEs are complicated in their structure hence findingexact solutions for them cannot be simpleTherefore one canapproach the best accurate solution of FDEs through analyti-cal and numerical methods Designing accurate or best solu-tion to FDEs many methods are developed in recent years
each method has its own advantages and limitations Thispaper aims to solve a FDE called fractional-order Riccatidifferential equation one of the important equations in thefamily of FDEs The Riccati equations play an important rolein engineering and applied science [6] especially in quantummechanics [7] and quantum chemistry [8 9] Thereforesolutions to the Riccati differential equations are important toscientists and engineers Solving fractional-order Riccati dif-ferential equation themost significantmethods are Adomiandecompositionmethod [10] homotopy perturbationmethod[11ndash14] homotopy analysis method [15 16] Taylor matrixmethod [17] Haar wavelet method [18] and combination ofLaplace Adomian decomposition and Pade approximation[19] methods
Several numerical methods for approximating the solu-tion of nonlinear fractional-order Riccati differential equa-tions are known Raja et al [20] developed a stochastic tech-nique based on particle swarm optimization and simulatedannealing They were used as a tool for rapid global searchmethod and simulated annealing for efficient local searchmethod A fractional variational iteration method describedin the Riemann-Liouville derivative has been applied in [21]to give an analytical approximate solution to nonlinear frac-tional Riccati differential equation A combination of finite
Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2014 Article ID 304745 10 pageshttpdxdoiorg1011552014304745
2 International Journal of Mathematics and Mathematical Sciences
difference method and Pade-variational iteration numericalscheme was proposed by Sweilam et al [22] Moreoveran analytical scheme comprising the Laplace transformthe Adomian decomposition method (ADM) and the Padeapproximation is given in [19]
However the abovementioned methods have somerestrictions and disadvantages in their performance Forexample very complicated and toughest Adomian polynomi-als are constructed in the Adomian decomposition methodIn the variational iteration method identification Lagrangemultiplier yields an underlying accuracy The homotopyperturbation method needs a linear functional equation ineach iteration to solve nonlinear equations forming thesefunctional equations is very difficult The performance of thehomotopy analysis method verymuch depends on the choiceof the auxiliary parameter ℎ of the zero-order deformationequation Moreover the convergence region and implemen-tation of these results are very small
In recent years wavelets theory is one of the growingand predominantly newmethods in the area of mathematicaland engineering research It has been applied in vast rangeof engineering sciences particularly they are used verysuccessfully for waveform representation and segmentationsin signal analysis and time-frequency analysis and in themathematical sciences it is used in thriving manner forsolving variety of linear and nonlinear differential and partialdifferential equations and fast algorithms for easy implemen-tation [23] Moreover wavelets build a connection with fastnumerical algorithms [24] this is due to the fact that waveletsadmit the exact representation of a variety of functions andoperatorsThe application of Legendre wavelet and its opera-tional matrix for solving differential integral and fractional-order differential equations is thoroughly considered in[25 26]
In this work the nonlinear Riccati differential equationsof fractional-order are approached analytically by using Leg-endre wavelets method The operational matrix of Legendrewavelet is generalized for fractional calculus in order tosolve fractional and classical Riccati differential equationsThe Legendre wavelet method (LWM) is illustrated by appli-cation and obtained results are compared with recentlyproposed method for the fractional-order Riccati differen-tial equation We have adopted Legendre wavelet methodto solve Riccati differential equations not only due to itsemerging application but also due to its greater convergenceregion
The rest of the paper is as follows In Section 2 definitionsandmathematical preliminaries of fractional calculus are pre-sented In Section 3 Legendre wavelet its properties func-tion approximations and generalized Legendre wavelet oper-ational matrix fractional calculus are discussed Section 4establishes application of proposed method in the solutionof Riccati differential equations existence and uniquenesssolution of the proposed problem and convergence analysesof the proposed approach Section 5 deals with the illustrativeexamples and their solutions by the proposed approachSection 6 ends with our conclusion
2 Preliminaries and Notations
The notations definitions and preliminary facts present inthis section will be used in forthcoming sections of this workSeveral definitions of fractional integrals and derivatives havebeen proposed after the logical definition given by LiouvilleImportant and few of these definitions include the Riemann-Liouville the Caputo the Weyl the Hadamard the Mar-chaud the Riesz the Grunwald-Letnikov and the Erdelyi-Kober As stated in [25] the Caputo fractional derivative usesinitial and boundary conditions of integer order derivativeshaving some physical interpretations Because of this specificreason in this work we will use the Caputo fractionalderivative 119863
120572 proposed by Caputo [27] in the theory ofviscoelasticity
The Caputo fractional derivative of order 120572 gt 0(120572 isin
119877 119899minus1 lt 120572 le 119899 119899 isin 119873) and ℎ (0infin) rarr 119877 is continuousand is defined by
119863120572
119891 (119905) = 119868119899minus120572
(119889119899
119889119905119899119891 (119905)) (1)
where
119868120572
119891 (119905) =1
Γ (120572)int
119905
0
(119905 minus 119904)120572minus1
119891 (119904) 119889119904 (2)
is the Riemann-Liouville fractional integral operator of order120572 gt 0 and Γ is the gamma function
The fractional integral of 119905120573 120573 gt minus1 is given as
119868120572
(119905 minus 120572)120573
=Γ (120573 + 1)
Γ (120573 + 120572 + 1)(119905 minus 119886)
120573+120572
119886 ge 0 (3)
Properties of fractional integrals and derivatives are as follows[28] for 120572 120573 gt 0
The fractional-order integral satisfies the semigroupproperty
119868120572
(119868120573
119891 (119905)) = 119868120573
(119868120572
119891 (119905)) = 119868120572+120573
119891 (119905) (4)
The integer order derivative 119863119899 and fractional-order deriva-tive119863120572 commute with each other
119863119899
(119863120572
119891 (119905)) = 119863120572
(119863119899
119891 (119905)) = 119863119899+120572
119891 (119905) (5)
The fractional integral operator and fractional derivativeoperator do not satisfy the commutative property In general
119868120572
(119863120572
119891 (119905)) = 119891 (119905) minus
119899minus1
sum
119896=0
119891(119896)
(0)119905119896
119896 (6)
But in the reverse way we have
119863120572
(119868120573
119891 (119905)) = 119863120572minus120573
119891 (119905) (7)
International Journal of Mathematics and Mathematical Sciences 3
3 Generalized Legendre Wavelet OperationalMatrix to Fractional Integration
31 Legendre Wavelets A family of functions were consti-tuted by wavelets and constructed from dilation and trans-lation of a single function called mother wavelet When theparameters 119886 of dilation and 119887 of translation vary continu-ously following are the family of continuous wavelets [29]
120595119886119887
(119905) = |119886|minus12
120595(119905 minus 119887
119886) 119886 119887 isin R 119886 = 0 (8)
If the parameters 119886 and 119887 are restricted to discrete values as119886 = 119886
minus119896
0 119887 = 119899119887
0119886minus119896
0 1198860gt 1 119887
0gt 0 and 119899 and 119896 are positive
integers following are the family of discrete wavelets
120595119896119899
(119905) =10038161003816100381610038161198860
10038161003816100381610038161198962
120595 (119886119896
0119905 minus 119899119887
0) (9)
where 120595119896119899(119905) form a wavelet basis for 1198712(119877) In particular
when 1198860= 2 and 119887
0= 1 120595
119896119899(119905) form an orthonormal basis
[29]Legendre wavelets 120595
119899119898(119905) = 120595(119896 119899119898 119905) have four argu-
ments 119899 = 2119899 minus 1 119899 = 1 2 3 2119896minus1
119896 can assume anypositive integer119898 is the order for Legendre polynomials and119905 is the normalized timeThey are defined on the interval [01)as [30 31]
120595119899119898
(119905)
=
radic119898 +1
221198962
119875119898(2119896
119905 minus 119899) for 119899 minus 1
2119896le 119905 lt
119899 + 1
2119896
0 otherwise(10)
where 119898 = 0 1 119872 minus 1 and 119899 = 1 2 3 2119896minus1 The
coefficient radic119898 + (12) is for orthonormality the dilationparameter is 119886 = 2
minus119896 and translation parameter is 119887 =
119899 2minus119896 119875119898(119905) are the well-known Legendre polynomials of
ordermdefined on the interval [minus1 1] and can be determinedwith the aid of the following recurrence formulae
1198750(119905) = 1 119875
1(119905) = 119905
119875119898+1
(119905) = (2119898 + 1
119898 + 1) 119905119875119898(119905) minus (
119898
119898 + 1)119875119898minus1
(119905)
119898 = 1 2 3
(11)
The Legendre wavelet series representation of the function119891(119905) defined over [0 1) is given by
119891 (119905) =
infin
sum
119899=1
infin
sum
119898=0
119888119899119898120595119899119898
(119905) (12)
where 119888119899119898
= ⟨119891(119905) 120595119899119898(119905)⟩ in which ⟨sdot sdot⟩ denotes the inner
product If the infinite series in (12) is truncated then (12) canbe written as
119891 (119905) cong
2119896minus1
sum
119899=1
119872minus1
sum
119898=0
119888119899119898120595119899119898
(119905) = 119862119879
Ψ (119905) = 119891 (119905) (13)
where 119862 andΨ(119905) are 2119896minus1119872times 1matrices given by
119862 = [11988810 11988811 119888
1119872minus1 11988820 11988821 119888
2119872minus1
1198882119896minus10 1198882119896minus11 119888
2119896minus1119872minus1
]119879
Ψ (119905) = [12059510(119905) 12059511(119905) 120595
1119872minus1(119905) 12059520(119905)
12059521(119905) 120595
2119872minus1(119905)
1205952119896minus10(119905) 1205952119896minus11(119905) 120595
2119896minus1119872minus1
(119905) ]119879
(14)
Taking suitable collocation points as
119905119894= cos((2119894 + 1) 120587
2119896119872) 119894 = 1 2 2
119896minus1
119872 (15)
we defined the -square Legendre matrix
120601times
= [Ψ(cos( 3120587
2119896119872)) Ψ(cos( 5120587
2119896119872))
sdot sdot sdot Ψ(cos((2119896
119872+ 1)120587
2119896119872))]
(16)
where = 2119896minus1
119872 correspondingly we have
119891 = [119891(cos( 3120587
2119896119872))
times 119891(cos( 5120587
2119896119872)) sdot sdot sdot 119891(cos(
(2119896
119872+ 1)120587
2119896119872))]
= 119862119879
120601times
(17)
The Legendre matrix 120601times
is an invertible matrix and thecoefficient vector 119862119879 is obtained by 119862119879 = 119891 120601
minus1
times
32 Operational Matrix of the Fractional Integration Theintegration of the Ψ(119905) defined in (14) can be approximatedby Legendre wavelet series with Legendre wavelet coefficientmatrix 119875
int
119905
0
Ψ (119905) 119889119905 = 119875times
Ψ (119905) (18)
where the -square matrix 119875 is called Legendre waveletoperational matrix of integration
The 119898-set of block-pulse functions is defined on [0 l) asfollows
119887119894(119905) =
1119894
119898le 119905 lt
(119894 + 1)
119898
0 otherwise(19)
where 119894 = 0 1 2 3 119898
4 International Journal of Mathematics and Mathematical Sciences
The functions 119887119894are disjoint and orthogonal That is
119887119894(119905) 119887119895(119905) =
119887119894(119905) 119894 = 119895
0 119894 = 119895
int
1
0
119887119894(119905) 119887119895(119905) 119889119905 =
1
119898 119894 = 119895
0 119894 = 119895
(20)
The orthogonality property of block-pulse function isobtained from the disjointness property
An arbitrary function 119891 isin 1198712[0 1) can be expanded into
block-pulse functions as
119891 (119905) asymp
119898minus1
sum
119894=0
119891119894119887119894(119905) = 119891
119879
119861 (119905) (21)
where119891119894are the coefficients of the block-pulse function given
by
119891119894=119898
119897int
119897
0
119891 (119905) 119887119894(119905) (22)
The Legendre wavelets can be expanded into119898-set of block-pulse functions as
Ψ (119905) = 120601times
119861 (119905) (23)
where 119861(119905) = [1198870(119905) 1198871(119905) sdot sdot sdot 119887
119894(119905) sdot sdot sdot 119887
119898minus1(119905)]119879
The fractional integral of block-pulse function vector canbe written as
(119868120572
119861) (119905) = 119865120572
119898times119898119861 (119905) (24)
where 119865120572119898times119898
is given in [32]Now we introduce the derivation process of the Legendre
wavelet operational matrix of the fractional integration
(119868120572
Ψ) (119905) asymp 119875120572
timesΨ (119905) (25)
where the -square matrix 119875120572times
is called Legendre waveletoperational matrix of the fractional integration
Using (23) and (25) we have
(119868120572
Ψ) (119905) asymp (119868120572
120601times
119861) (119905) = 120601times
(119868120572
119861) (119905)
asymp 120601times
(119865120572
119861) (119905)
(26)
From (25) and (26) we get
119875120572
timesΨ (119905) = 120601
times119865120572
119861 (119905) (27)
and by (23) (27) becomes
119875120572
times120601times
119861 (119905) = 120601times
119865120572
119861 (119905) (28)
Then the Legendre wavelet operational matrix119875120572times
of frac-tional integration is given by
119875120572
times= 120601times
119865120572
120601minus1
times (29)
Following is the Legendre wavelet operationalmatrix119875120572times
offractional-order integration for the particular values of 119896 = 2119872 = 3 and 120572 = 05
11987505
6times6= (
(
05415 04324 01819 minus00871 minus00179 00154
0 05415 0 01819 0 minus00179
minus02046 0071 02243 minus00449 00798 00119
0 minus02046 0 02243 0 00798
01781 02506 minus00252 minus00652 01555 00143
0 01781 0 minus00252 0 01555
)
)
(30)
4 Application to Fractional RiccatiDifferential Equation
In this section we use the generalized Legendre waveletoperational matrix to solve nonlinear Riccati differentialequation and we discuss the existence and uniqueness ofsolutions with initial conditions and convergence criteria ofthe proposed LWM approach
Consider the fractional-order Riccati differential equa-tion of the form
119863120572
119910 (119905) = 119875 (119905) 1199102
+ 119876 (119905) 119910 + 119877 (119905)
119905 gt 0 0 lt 120572 le 1
(31)
subject to the initial condition
119910 (0) = 119896 (32)
Let us suppose that the functions119863120572119910(119905) 119875(119905)119876(119905) and 119877(119905)are approximated using Legendre wavelet as follows
119863120572
119910 (119905) = 119880119879
Ψ (119905) 119875 (119905) = 119881119879
Ψ (119905)
119876 (119905) = 119882119879
Ψ (119905) 119877 (119905) = 119883119879
Ψ (119905)
(33)
where 119880 119881119882119883 and Ψ(119905) are given in (14)Using (6) we can write
119910 (119905) = 119868120572
(119863120572
119910 (119905)) minus 119910 (0) (34)
International Journal of Mathematics and Mathematical Sciences 5
By (25) and (32) (34) leads to
119910 (119905) asymp 119880119879
119875120572
Ψ (119905) + 119884119879
0Ψ (119905) = 119862
119879
Ψ (119905) (35)
where 119910(0) = 119896 asymp 119884119879
0Ψ(119905) 119862 = (119880
119879
119875120572
times+ 119884119879
0)119879
Substituting (33) and (35) into (31) we have
119880119879
Ψ (119905) = 119881119879
Ψ (119905) [119862119879
Ψ (119905)]2
+119882119879
Ψ (119905) 119862119879
Ψ (119905) + 119883119879
Ψ (119905)
(36)
Substituting (23) into (36) we have
119880119879
120601times
= 119881119879
[119862119879
120601times
]2
+119882119879
119862119879
120601times
+ 119883119879
(37)
where 119862 119881 119882 and 120601times
are known Equation (37) repre-sents a system of nonlinear equations with unknown vector119880 This system of nonlinear equations can be solved byNewton method for the unknown vector 119880 and we can getthe approximation solution by including 119880 into (35)
41 Existence and Uniqueness of Solutions Consider thefractional-order Riccati differential equation of the forms(31) and (32) The nonlinear term in (31) is 119910
2 and119875(119905) 119876(119905) and 119877(119905) are known functions For 120572 = 1 thefractional-order Riccati converts into the classical Riccatidifferential equation
Definition 1 Let 119868 = [0 119897] 119897 lt infin and 119862(119868) be the class of allcontinuous functions defined on 119868 with the norm
10038171003817100381710038171199101003817100381710038171003817 = sup119905isin119868
10038161003816100381610038161003816119890minusℎ119905
119910 (119905)10038161003816100381610038161003816 ℎ gt 0 (38)
which is equivalent to the sup norm of 119910 That is 119910 =
sup119905isin119868
|119890minusℎ119905
119910(119905)|
RemarkAssume that solution 119910(119905) of fractional-order Riccatidifferential equations (31) and (32) belongs to the space 119878 =
119910 isin 119877 |119910| le 119888 119888 is any constant in order to study theexistence and uniqueness of the initial value problem
Definition 2 The space of integrable functions 1198711[0 119897] in the
interval [0 119897] is defined as
1198711[0 119897] = 119906 (119905) int
119897
0
|119906 (119905)| 119889119905 lt infin (39)
Theorem 3 The initial value problem given by (31) and (32)has a unique solution
119910 isin 119862 (119868) 1199101015840
isin 119883 = 119910 isin 1198711[0 119897]
10038171003817100381710038171199101003817100381710038171003817 =
10038171003817100381710038171003817119890minusℎ119905
119910(119905)100381710038171003817100381710038171198711
(40)
Proof By (1) the fractional differential equation (31) can bewritten as
1198681minus120572
119889119910 (119905)
119889119905= 119875 (119905) 119910
2
+ 119876 (119905) 119910 + 119877 (119905) (41)
and becomes
119910 (119905) = 119868120572
(119875 (119905) 1199102
+ 119876 (119905) 119910 + 119877 (119905)) (42)
Now we define the operator Θ 119862(119868) rarr 119862(119868) by
Θ119910 (119905) = 119868120572
(119875 (119905) 1199102
+ 119876 (119905) 119910 + 119877 (119905)) (43)
and then
119890minusℎ119905
(Θ119910 minus Θ119908)
= 119890minusℎ119905
119868120572
[(1198751199102
(119905) + 119876119910 (119905) + 119877)
minus (1198751199082
(119905) + 119876119908 (119905) + 119877)]
le1
Γ (120572)int
119905
0
(119905 minus 119904)120572minus1
119890minusℎ(119905minus119904)
times [(119910 (119904) minus 119908 (119904)) (119910 (119904) + 119908 (119904))
minus119896 (119910 (119904) minus 119908 (119904))] 119890minusℎ119904
119889119904
le1003817100381710038171003817119910 minus 119908
1003817100381710038171003817
1
Γ (120572)int
119905
0
119904120572minus1
119890minusℎ119904
119889119904
(44)
hence we have1003817100381710038171003817Θ119910 minus Θ119908
1003817100381710038171003817 lt1003817100381710038171003817119910 minus 119908
1003817100381710038171003817 (45)
which implies that the operator given by (43) has a uniquefixed point and consequently the given integral equation hasa unique solution 119910(119905) isin 119862(119868) Also we can see that
119868120572
(119875(119905)1199102
+ 119876(119905)119910 + 119877(119905))10038161003816100381610038161003816119905=0
= 119896 (46)
Now from (42) we have
119910 (119905) = [119905120572
Γ (120572 + 1)(1198751199102
0+ 1198761199100+ 119877)
+ 119868120572+1
(1198751015840
1199102
+ 21199101015840
119875 + 1198761015840
119910 + 1198761199101015840
+ 1198771015840
) ]
119889119910
119889119905= [
119905120572minus1
Γ (120572)(1198751199102
0+ 1198761199100+ 119877)
+ 119868120572
(1198751015840
1199102
+ 21199101015840
119875 + 1198761015840
119910 + 1198761199101015840
+ 1198771015840
) ]
119890minusℎ119905
1199101015840
(119905) = 119890minusℎ119905
[119905120572minus1
Γ (120572)(1198751199102
0+ 1198761199100+ 119877)
+ 119868120572
(1198751015840
1199102
+ 21199101015840
119875 + 1198761015840
119910 + 1198761199101015840
+ 1198771015840
) ]
(47)
from which we can deduce that 1199101015840(119905) isin 119862(119868) and 1199101015840
(119905) isin 119878
6 International Journal of Mathematics and Mathematical Sciences
Now again from (42) (43) and (46) we get
119889119910
119889119905=
119889
119889119905119868120572
[1198751199102
(119905) + 119876119910 (119905) + 119877]
1198681minus120572
119889119910
119889119905= 1198681minus120572
119889
119889119905119868120572
[1198751199102
(119905) + 119876119910 (119905) + 119877]
=119889
1198891199051198681minus120572
119868120572
[1198751199102
(119905) + 119876119910 (119905) + 119877]
119863120572
119910 (119905) =119889
119889119905119868 [1198751199102
(119905) + 119876119910 (119905) + 119877]
= 1198751199102
(119905) + 119876119910 (119905) + 119877
119910 (0) = 119868120572
(1198751199102
(119905) + 119876119910 (119905) + 119877)10038161003816100381610038161003816119905=0
= 119896
(48)
which implies that the integral equation (46) is equivalent tothe initial value problem (32) and the theorem is proved
42 Convergence Analyses Let
120595119896119899
(119905) =10038161003816100381610038161198860
10038161003816100381610038161198962
120595 (119886119896
0119905 minus 119899119887
0) (49)
where 120595119896119899(119905) form a wavelet basis for 1198712(119877) In particular
when 1198860= 2 and 119887
0= 1 120595
119896119899(119905) form an orthonormal basis
[29]By (14) let 119910(119905) = sum
119872minus1
119894=111988811198941205951119894(119905) be the solution of (31)
where 1198881119894= ⟨119910(119905) 120595
1119894(119905)⟩ for 119896 = 1 in which ⟨sdot sdot⟩ denotes
the inner product
119910 (119905) =
119899
sum
119894=1
⟨119910 (119905) 1205951119894(119905)⟩ 120595
1119894(119905) (50)
Let 120573119895= ⟨119910(119905) 120595(119905)⟩ where 120595(119905) = 120595
1119894(119905)
Let119909119899= sum119899
119895=1120573119895120595(119905119895) be a sequence of partial sumsThen
⟨119910 (119905) 119909119899⟩ = ⟨119910 (119905)
119899
sum
119895=1
120573119895120595 (119905119895)⟩
=
119899
sum
119895=1
120573119895⟨119910 (119905) 120595 (119905
119895)⟩
=
119899
sum
119895=1
120573119895120573119895
=
119899
sum
119895=1
10038161003816100381610038161003816120573119895
10038161003816100381610038161003816
2
(51)
Further
1003817100381710038171003817119909119899 minus 119909119898
10038171003817100381710038172
=
10038171003817100381710038171003817100381710038171003817100381710038171003817
119899
sum
119895=119898+1
120573119895120595 (119905119895)
10038171003817100381710038171003817100381710038171003817100381710038171003817
2
= ⟨
119899
sum
119894=119898+1
120573119894120595 (119905119894)
119899
sum
119895=119898+1
120573119895120595 (119905119895)⟩
=
119899
sum
119894=119898+1
119899
sum
119895=119898+1
120573119894120573119895⟨120595 (119905119894) 120595 (119905
119895)⟩
=
119899
sum
119895=119898+1
10038161003816100381610038161003816120573119895
10038161003816100381610038161003816
2
(52)
As 119899 rarr infin from Besselrsquos inequality we have suminfin119895=1
|120573119895|2 is
convergentIt implies that 119909
119899 is a Cauchy sequence and it converges
to 119909 (say)Also
⟨119909 minus 119910 (119905) 120595 (119905119895)⟩ = ⟨119909 120595 (119905
119895)⟩ minus ⟨119910 (119905) 120595 (119905
119895)⟩
= ⟨ lim119899rarrinfin
119909119899 120595 (119905119895)⟩ minus 120573
119895
= lim119899rarrinfin
⟨119909119899 120595 (119905119895)⟩ minus 120573
119895
= lim119899rarrinfin
⟨
119899
sum
119895=1
120573119895120595 (119905119895) 120595 (119905
119895)⟩ minus 120573
119895
= 120573119895minus 120573119895= 0
(53)
which is possible only if 119910(119905) = 119909 That is both 119910(119905) and 119909119899
converge to the same value which indeed give the guaranteeof convergence of LWM
5 Numerical Examples
In order to show the effectiveness of the Legendre waveletsmethod (LWM) we implement LWM to the nonlinearfractional Riccati differential equations All the numericalexperiments were carried out on a personal computer withsomeMATLAB codes The specifications of PC are Intel corei5 processor and with Turbo boost up to 31 GHz and 4GB ofDDR3 memory The following problems of nonlinear Riccatidifferential equations are solved with real coefficients
Example 1 Consider the following nonlinear fractional Ric-cati differential equation
119863120572
119910 (119905) = 1 + 2119910 (119905) minus 1199102
(119905) 0 lt 120572 le 1 (54)
with initial condition
119910 (0) = 0 (55)
Exact solution for 120572 = 1 was found to be
119910 (119905) = 1 + radic2 tanh(radic2119905 +1
2log(
radic2 minus 1
radic2 + 1)) (56)
International Journal of Mathematics and Mathematical Sciences 7
01 02 03 04 05 06 07 08 09 10
02
04
06
08
1
12
14
16
18
Exact
LWM
t
y(t)
Figure 1 Numerical results of Example 1 by LWM for 120572 = 1
The integral representation of (54) and (55) is given by
119868120572
(119863120572
119910 (119905)) = 119868120572
(1 + 2119910 (119905) minus 1199102
(119905)) (57)
119910 (119905) = 119910 (0) +119905120572
Γ (120572 + 1)+ 2119868120572
119910 (119905) minus 119868120572
1199102
(119905) (58)
Let
119910 (119905) = 119862119879
Ψ (119905) (59)
and then
119868120572
119910 (119905) = 119862119879
119868120572
Ψ (119905)
= 119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(60)
By substituting (59) and (60) into (58) we get the followingsystem of algebraic equations
119862119879
Ψ (119905) =119905120572
Γ (120572 + 1)+ 2119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
minus 119862119879
1198752120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(61)
By solving the above system of linear equations we can findthe vector 119862 Numerical results are obtained for differentvalues of 119896 119872 and 120572 Solution obtained by the proposedLWM approach for 120572 = 1 119896 = 1 and 119872 = 3 is given inFigure 1 and for different values of 120572 = 06 07 08 and 09and for 119896 = 2 and 119872 = 5 is graphically given in Figure 2It can be seen from Figure 1 that the solution obtained bythe proposed LWM approach is more close to the exactsolution Table 1 describes the efficiency of the proposedmethod by comparing with the methods in [20 22] throughtheir absolute errorThe following is used for the errors of theapproximation119910(119905) of 119910(119905) that is 119910minus119910 = max |119910(119905)minus119910(119905)|Table 1 shows that very high accuracies are obtained for 119896= 3and119872= 5 by the present method
0204
0608
10
09 08 07 06 05
t
y(t)
120572
minus05
00
05
15
20
10
Figure 2 Numerical results of Example 1 by LWM for differentvalues of 120572
Example 2 Consider another fractional-order Riccati differ-ential equation
119863120572
119910 (119905) = 1 minus 1199102
(119905) 0 lt 120572 le 1 (62)
with initial condition
119910 (0) = 0 (63)
Exact solution for the above equation was found to be
119910 (119905) =1198902119905
minus 1
1198902119905 + 1 (64)
The integral representation of (62) and (63) is given by
119910 (119905) = 119910 (0) +119905120572
Γ (120572 + 1)minus 119868120572
1199102
(119905) (65)
Let
119910 (119905) = 119862119879
Ψ (119905) (66)
and then
119868120572
119910 (119905) = 119862119879
119868120572
Ψ (119905)
= 119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(67)
By substituting (66) and (67) in (62) we get the followingsystem of algebraic equations
119862119879
Ψ (119905) =119905120572
Γ (120572 + 1)minus 119862119879
1198752120572
2119896minus1119872times2119896minus1119872Ψ (119905) (68)
By solving the above system of linear equations we can findthe vector 119862 Numerical results are obtained for differentvalues of 119896119872 and 120572 Results obtained by LWM for 120572 = 1 119896 =2 and119872= 3 are shown in Figure 3 and it can be seen from thefigure that solution given by the LWMmerely coincides withthe exact solution Figure 4 shows the obtained results of (62)and (63) by LWM for different values of 120572 and for 119896 = 2 and
8 International Journal of Mathematics and Mathematical Sciences
Table 1 Numerical results of Example 1 for 120572 = 1
119905 Exact solution Absolute error in [22] Absolute error in [20] LWM LWM LWM119872 = 2 119896 = 2 119872 = 3 119896 = 2 119872 = 5 119896 = 3
01 0099667 751119864 minus 09 930119864 minus 05 0 0 002 0197375 152119864 minus 06 293119864 minus 02 191119864 minus 08 0 003 0291312 393119864 minus 05 378119864 minus 03 272119864 minus 08 145119864 minus 13 004 0379948 432119864 minus 04 281119864 minus 03 165119864 minus 08 117119864 minus 13 194119864 minus 16
05 0462117 841119864 minus 04 980119864 minus 04 131119864 minus 08 328119864 minus 13 320119864 minus 16
06 0537049 294119864 minus 05 793119864 minus 03 198119864 minus 08 497119864 minus 13 124119864 minus 16
07 0604367 335119864 minus 04 944119864 minus 03 252119864 minus 08 632119864 minus 13 158119864 minus 16
08 0664036 544119864 minus 04 117119864 minus 02 294119864 minus 08 736119864 minus 13 184119864 minus 16
09 0716297 656119864 minus 09 396119864 minus 02 323119864 minus 08 812119864 minus 13 194119864 minus 16
10 0761594 253119864 minus 06 295119864 minus 02 263119864 minus 08 462119864 minus 13 199119864 minus 16
Table 2 Numerical results of Example 2 for 120572 = 1
119905 Exact solution Absolute error in [22] Absolute error in [20] LWM LWM LWM119872 = 2 119896 = 2 119872 = 3 119896 = 2 119872 = 5 119896 = 3
01 0110295 123119864 minus 15 281119864 minus 03 0 0 002 0241976 524119864 minus 15 383119864 minus 04 0 0 003 0395104 816119864 minus 15 123119864 minus 04 0 0 004 0567812 115119864 minus 12 286119864 minus 03 168119864 minus 12 166119864 minus 15 143119864 minus 1605 0756014 617119864 minus 12 438119864 minus 04 222119864 minus 12 212119864 minus 15 166119864 minus 1606 0953566 455119864 minus 11 519119864 minus 02 115119864 minus 12 110119864 minus 15 154119864 minus 1607 1152946 757119864 minus 10 214119864 minus 02 127119864 minus 12 111119864 minus 15 133119864 minus 1608 1346363 633119864 minus 09 142119864 minus 02 187119864 minus 12 201119864 minus 15 175119864 minus 1609 1526911 367119864 minus 08 698119864 minus 03 193119864 minus 12 266119864 minus 15 187119864 minus 1610 1689498 164119864 minus 07 496119864 minus 03 156119864 minus 12 166119864 minus 15 164119864 minus 16
119872 = 5 Table 2 describes the efficiency of the proposedmethod by comparing with the methods in [20 22] throughtheir absolute error Table 1 shows that very high accuraciesare obtained for 119896 = 3 and 119872 = 5 by the present methodand from these results we can identify that guarantee ofconvergence of the proposed LWM approach is very high
Example 3 Let us consider another problem of nonlinearRiccati differential equation
119863120572
119910 (119905) = 1199052
+ 1199102
(119905) 0 lt 120572 le 1 119905 ge 0 (69)
with initial condition
119910 (0) = 1 (70)
When 120572 = 1 its exact solution is given by
119910 (119905) =119905 (119869minus34
(1199052
2) Γ (14) + 211986934
(1199052
2) Γ (34))
11986914
(11990522) Γ (14) minus 2119869minus14
(11990522) Γ (34)
(71)
where 119869119899(119905) is the Bessel function of first kind
0
01
02
03
04
05
06
07
08
LWM
01 02 03 04 05 06 07 08 09 1
Exact
t
y(t)
Figure 3 Numerical results of Example 2 by LWM for 120572 = 1
The integral representation of (69) and (70) is given by
119910 (119905) = 1 +2
Γ (120572 + 1) + 2Γ (120572)119905120572+2
+ 119868120572
1199102
(119905) (72)
International Journal of Mathematics and Mathematical Sciences 9
0204
060810
09 08 07 06 05
t
y(t)
120572
00
01
02
03
04
0605
0807
081
Figure 4 Numerical results of Example 2 by LWM for differentvalues of 120572
Let
119910 (119905) = 119862119879
Ψ (119905) (73)
and then
119868120572
119910 (119905) = 119862119879
119868120572
Ψ (119905)
= 119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(74)
By substituting (73) and (74) into (69) we get the followingsystem of algebraic equations
119862119879
Ψ (119905) = 1 +2
Γ (120572 + 1) + 2Γ (120572)(119862119879
Ψ (119905))120572+2
+ 119862119879
1198752120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(75)
By solving the above system of linear equations we can findthe vector 119862 Numerical results are obtained for differentvalues of 119896 119872 and 120572 Obtained results for (69) and (70)are shown in Figures 5 and 6 Figure 5 shows the solutionsobtained by LWM for different values of 120572 and for 119896 = 2 and119872 = 4 Figure 6 compares the solution obtained by LWMwith the exact solution of (69) and (70) when 120572 = 1 119896 = 1and 119872 = 2 So far there are no published results of absoluteerror for this problem and hence we are unable to compareabsolute error of ourmethodwith the existingmethods Fromthese results we can see that the proposed LWM approachgives the solution which is very close to the exact solutionand outperformed recently developed approaches for thenonlinear fractional-order Riccati differential equations interms of solution quality and convergence criteria
6 Conclusions
Nonlinear fractional-order Riccati differential equations playan important role in the modeling of many biologicalphysical chemical and real life problemsTherefore it is nec-essary to develop a method which would give more accuratesolutions to such type of problems with greater convergence
0204
0608
10
09 0807 06
05
t
y(t)
120572
10
0
10
20
30
40
0 406
08
0 9 t
Figure 5 Numerical results of Example 3 by LWM for differentvalues of 120572
1
15
2
25
3
35
4
45
5
55
6
LWM
01 02 03 04 05 06 07 08 09 1
Exact
t
y(t)
Figure 6 Numerical results of Example 3 by LWM for 120572 = 1
criteria In this work a Legendrersquos wavelet operational matrixmethod called LWM was proposed for solving nonlinearfractional-order Riccati differential equations Comparisonwas made for the solutions obtained by the proposedmethodand with the other recent approaches developed for thesame problem through their error analysis obtained resultsshow that the proposed LWM yields more accurate andreliable solutions even for small values of 119872 and 119896 whichassures the best approximate solution in less computationaleffort Further we have discussed the convergence criteriaof proposed scheme which indeed provides the guaranteeof consistency and stability of the proposed LWM schemefor the solutions of nonlinear fractional Riccati differentialequations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
10 International Journal of Mathematics and Mathematical Sciences
References
[1] N A Khan M Jamil A Ara and S Das ldquoExplicit solution fortime-fractional batch reactor systemrdquo International Journal ofChemical Reactor Engineering vol 9 article A91 2011
[2] V Feliu-Batlle R R Perez and L S Rodrıguez ldquoFractionalrobust control of main irrigation canals with variable dynamicparametersrdquoControl Engineering Practice vol 15 no 6 pp 673ndash686 2007
[3] I Podlubny ldquoFractional-order systems and 119875119868120582
119863120583-controllersrdquo
IEEE Transactions on Automatic Control vol 44 no 1 pp 208ndash214 1999
[4] R Garrappa ldquoOn some explicit Adams multistep methods forfractional differential equationsrdquo Journal of Computational andApplied Mathematics vol 229 no 2 pp 392ndash399 2009
[5] M Jamil and N A Khan ldquoSlip effects on fractional viscoelasticfluidsrdquo International Journal of Differential Equations vol 2011Article ID 193813 19 pages 2011
[6] F Mohammadi and M M Hosseini ldquoA comparative study ofnumerical methods for solving quadratic Riccati differentialequationsrdquo Journal of the Franklin Institute vol 348 no 2 pp156ndash164 2011
[7] R Shankar Principles of Quantum Mechanics Plenum PressNew York NY USA 1980
[8] S FragaM J Garcıa de la Vega and E S FragaTheSchrodingerand Riccati Equations vol 70 of Lect Notes Chem 1999
[9] L B Burrows and M Cohen ldquoSchrodingerrsquos wave equation-A lie algebra treatmentrdquo in Fundamental World of QuantumChemistry A Tribute to the Memory of Per-Olov Lowdin EJ Brandas and E S Kryachko Eds Kluwer Dordrecht TheNetherlands 2004
[10] S Abbasbandy ldquoHomotopy perturbation method for quadraticRiccati differential equation and comparison with Adomianrsquosdecomposition methodrdquo Applied Mathematics and Computa-tion vol 172 no 1 pp 485ndash490 2006
[11] Z Odibat and S Momani ldquoModified homotopy perturbationmethod application to quadratic Riccati differential equationof fractional orderrdquo Chaos Solitons amp Fractals vol 36 no 1 pp167ndash174 2008
[12] N A Khan A Ara and M Jamil ldquoAn efficient approach forsolving the Riccati equation with fractional ordersrdquo Computersamp Mathematics with Applications vol 61 no 9 pp 2683ndash26892011
[13] H Aminikhah and M Hemmatnezhad ldquoAn efficient methodfor quadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 4 pp835ndash839 2010
[14] S Abbasbandy ldquoIterated Hersquos homotopy perturbation methodfor quadratic Riccati differential equationrdquo Applied Mathemat-ics and Computation vol 175 no 1 pp 581ndash589 2006
[15] J Cang Y Tan H Xu and S Liao ldquoSeries solutions of non-lin-ear Riccati differential equations with fractional orderrdquo ChaosSolitons and Fractals vol 40 no 1 pp 1ndash9 2009
[16] Y Tan and S Abbasbandy ldquoHomotopy analysis method forquadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 3 pp539ndash546 2008
[17] M Gulsu and M Sezer ldquoOn the solution of the Riccati equa-tion by the Taylor matrix methodrdquo Applied Mathematics andComputation vol 176 no 2 pp 414ndash421 2006
[18] Y Li and LHu ldquoSolving fractional Riccati differential equationsusing Haar waveletrdquo in Proceedings of the 3rd InternationalConference on Information and Computing (ICIC 10) pp 314ndash317 Wuxi China June 2010
[19] N A Khan and A Ara ldquoFractional-order Riccati differentialequation analytical approximation and numerical resultsrdquoAdvances in Difference Equations vol 2013 article 185 2013
[20] M A Z Raja J A Khan and I M Qureshi ldquoA new stochasticapproach for solution of Riccati differential equation of frac-tional orderrdquo Annals of Mathematics and Artificial Intelligencevol 60 no 3-4 pp 229ndash250 2010
[21] M Merdan ldquoOn the solutions fractional Riccati differentialequation with modified RIEmann-Liouville derivativerdquo Inter-national Journal of Differential Equations vol 2012 Article ID346089 17 pages 2012
[22] N H Sweilam M M Khader and A M S Mahdy ldquoNumericalstudies for solving fractional Riccati differential equationrdquoApplications and AppliedMathematics vol 7 no 2 pp 595ndash6082012
[23] C K ChuiWavelets A Mathematical Tool for Signal ProcessingSIAM Philadelphia Pa USA 1997
[24] G Beylkin R Coifman and V Rokhlin ldquoFast wavelet trans-forms and numerical algorithms Irdquo Communications on Pureand Applied Mathematics vol 44 no 2 pp 141ndash183 1991
[25] M ur Rehman and R Ali Khan ldquoThe Legendre wavelet methodfor solving fractional differential equationsrdquoCommunications inNonlinear Science and Numerical Simulation vol 16 no 11 pp4163ndash4173 2011
[26] S Balaji ldquoA new approach for solving Duffing equationsinvolving both integral and non-integral forcing termsrdquo AinShams Engineering Journal 2014
[27] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent IIrdquo Geophysical Journal of the RoyalAstronomical Society vol 13 pp 529ndash539 1967
[28] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-Holland Mathematics Studies Elsevier 2006
[29] J S Gu and W S Jiang ldquoThe Haar wavelets operational matrixof integrationrdquo International Journal of Systems Science vol 27no 7 pp 623ndash628 1996
[30] M Razzaghi and S Yousefi ldquoLegendre wavelets direct methodfor variational problemsrdquo Mathematics and Computers in Sim-ulation vol 53 no 3 pp 185ndash192 2000
[31] M Razzaghi and S Yousefi ldquoLegendre wavelets method forconstrained optimal control problemsrdquo Mathematical Methodsin the Applied Sciences vol 25 no 7 pp 529ndash539 2002
[32] A Kilicman and Z A A Al Zhour ldquoKronecker operationalmatrices for fractional calculus and some applicationsrdquo AppliedMathematics and Computation vol 187 no 1 pp 250ndash265 2007
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 International Journal of Mathematics and Mathematical Sciences
difference method and Pade-variational iteration numericalscheme was proposed by Sweilam et al [22] Moreoveran analytical scheme comprising the Laplace transformthe Adomian decomposition method (ADM) and the Padeapproximation is given in [19]
However the abovementioned methods have somerestrictions and disadvantages in their performance Forexample very complicated and toughest Adomian polynomi-als are constructed in the Adomian decomposition methodIn the variational iteration method identification Lagrangemultiplier yields an underlying accuracy The homotopyperturbation method needs a linear functional equation ineach iteration to solve nonlinear equations forming thesefunctional equations is very difficult The performance of thehomotopy analysis method verymuch depends on the choiceof the auxiliary parameter ℎ of the zero-order deformationequation Moreover the convergence region and implemen-tation of these results are very small
In recent years wavelets theory is one of the growingand predominantly newmethods in the area of mathematicaland engineering research It has been applied in vast rangeof engineering sciences particularly they are used verysuccessfully for waveform representation and segmentationsin signal analysis and time-frequency analysis and in themathematical sciences it is used in thriving manner forsolving variety of linear and nonlinear differential and partialdifferential equations and fast algorithms for easy implemen-tation [23] Moreover wavelets build a connection with fastnumerical algorithms [24] this is due to the fact that waveletsadmit the exact representation of a variety of functions andoperatorsThe application of Legendre wavelet and its opera-tional matrix for solving differential integral and fractional-order differential equations is thoroughly considered in[25 26]
In this work the nonlinear Riccati differential equationsof fractional-order are approached analytically by using Leg-endre wavelets method The operational matrix of Legendrewavelet is generalized for fractional calculus in order tosolve fractional and classical Riccati differential equationsThe Legendre wavelet method (LWM) is illustrated by appli-cation and obtained results are compared with recentlyproposed method for the fractional-order Riccati differen-tial equation We have adopted Legendre wavelet methodto solve Riccati differential equations not only due to itsemerging application but also due to its greater convergenceregion
The rest of the paper is as follows In Section 2 definitionsandmathematical preliminaries of fractional calculus are pre-sented In Section 3 Legendre wavelet its properties func-tion approximations and generalized Legendre wavelet oper-ational matrix fractional calculus are discussed Section 4establishes application of proposed method in the solutionof Riccati differential equations existence and uniquenesssolution of the proposed problem and convergence analysesof the proposed approach Section 5 deals with the illustrativeexamples and their solutions by the proposed approachSection 6 ends with our conclusion
2 Preliminaries and Notations
The notations definitions and preliminary facts present inthis section will be used in forthcoming sections of this workSeveral definitions of fractional integrals and derivatives havebeen proposed after the logical definition given by LiouvilleImportant and few of these definitions include the Riemann-Liouville the Caputo the Weyl the Hadamard the Mar-chaud the Riesz the Grunwald-Letnikov and the Erdelyi-Kober As stated in [25] the Caputo fractional derivative usesinitial and boundary conditions of integer order derivativeshaving some physical interpretations Because of this specificreason in this work we will use the Caputo fractionalderivative 119863
120572 proposed by Caputo [27] in the theory ofviscoelasticity
The Caputo fractional derivative of order 120572 gt 0(120572 isin
119877 119899minus1 lt 120572 le 119899 119899 isin 119873) and ℎ (0infin) rarr 119877 is continuousand is defined by
119863120572
119891 (119905) = 119868119899minus120572
(119889119899
119889119905119899119891 (119905)) (1)
where
119868120572
119891 (119905) =1
Γ (120572)int
119905
0
(119905 minus 119904)120572minus1
119891 (119904) 119889119904 (2)
is the Riemann-Liouville fractional integral operator of order120572 gt 0 and Γ is the gamma function
The fractional integral of 119905120573 120573 gt minus1 is given as
119868120572
(119905 minus 120572)120573
=Γ (120573 + 1)
Γ (120573 + 120572 + 1)(119905 minus 119886)
120573+120572
119886 ge 0 (3)
Properties of fractional integrals and derivatives are as follows[28] for 120572 120573 gt 0
The fractional-order integral satisfies the semigroupproperty
119868120572
(119868120573
119891 (119905)) = 119868120573
(119868120572
119891 (119905)) = 119868120572+120573
119891 (119905) (4)
The integer order derivative 119863119899 and fractional-order deriva-tive119863120572 commute with each other
119863119899
(119863120572
119891 (119905)) = 119863120572
(119863119899
119891 (119905)) = 119863119899+120572
119891 (119905) (5)
The fractional integral operator and fractional derivativeoperator do not satisfy the commutative property In general
119868120572
(119863120572
119891 (119905)) = 119891 (119905) minus
119899minus1
sum
119896=0
119891(119896)
(0)119905119896
119896 (6)
But in the reverse way we have
119863120572
(119868120573
119891 (119905)) = 119863120572minus120573
119891 (119905) (7)
International Journal of Mathematics and Mathematical Sciences 3
3 Generalized Legendre Wavelet OperationalMatrix to Fractional Integration
31 Legendre Wavelets A family of functions were consti-tuted by wavelets and constructed from dilation and trans-lation of a single function called mother wavelet When theparameters 119886 of dilation and 119887 of translation vary continu-ously following are the family of continuous wavelets [29]
120595119886119887
(119905) = |119886|minus12
120595(119905 minus 119887
119886) 119886 119887 isin R 119886 = 0 (8)
If the parameters 119886 and 119887 are restricted to discrete values as119886 = 119886
minus119896
0 119887 = 119899119887
0119886minus119896
0 1198860gt 1 119887
0gt 0 and 119899 and 119896 are positive
integers following are the family of discrete wavelets
120595119896119899
(119905) =10038161003816100381610038161198860
10038161003816100381610038161198962
120595 (119886119896
0119905 minus 119899119887
0) (9)
where 120595119896119899(119905) form a wavelet basis for 1198712(119877) In particular
when 1198860= 2 and 119887
0= 1 120595
119896119899(119905) form an orthonormal basis
[29]Legendre wavelets 120595
119899119898(119905) = 120595(119896 119899119898 119905) have four argu-
ments 119899 = 2119899 minus 1 119899 = 1 2 3 2119896minus1
119896 can assume anypositive integer119898 is the order for Legendre polynomials and119905 is the normalized timeThey are defined on the interval [01)as [30 31]
120595119899119898
(119905)
=
radic119898 +1
221198962
119875119898(2119896
119905 minus 119899) for 119899 minus 1
2119896le 119905 lt
119899 + 1
2119896
0 otherwise(10)
where 119898 = 0 1 119872 minus 1 and 119899 = 1 2 3 2119896minus1 The
coefficient radic119898 + (12) is for orthonormality the dilationparameter is 119886 = 2
minus119896 and translation parameter is 119887 =
119899 2minus119896 119875119898(119905) are the well-known Legendre polynomials of
ordermdefined on the interval [minus1 1] and can be determinedwith the aid of the following recurrence formulae
1198750(119905) = 1 119875
1(119905) = 119905
119875119898+1
(119905) = (2119898 + 1
119898 + 1) 119905119875119898(119905) minus (
119898
119898 + 1)119875119898minus1
(119905)
119898 = 1 2 3
(11)
The Legendre wavelet series representation of the function119891(119905) defined over [0 1) is given by
119891 (119905) =
infin
sum
119899=1
infin
sum
119898=0
119888119899119898120595119899119898
(119905) (12)
where 119888119899119898
= ⟨119891(119905) 120595119899119898(119905)⟩ in which ⟨sdot sdot⟩ denotes the inner
product If the infinite series in (12) is truncated then (12) canbe written as
119891 (119905) cong
2119896minus1
sum
119899=1
119872minus1
sum
119898=0
119888119899119898120595119899119898
(119905) = 119862119879
Ψ (119905) = 119891 (119905) (13)
where 119862 andΨ(119905) are 2119896minus1119872times 1matrices given by
119862 = [11988810 11988811 119888
1119872minus1 11988820 11988821 119888
2119872minus1
1198882119896minus10 1198882119896minus11 119888
2119896minus1119872minus1
]119879
Ψ (119905) = [12059510(119905) 12059511(119905) 120595
1119872minus1(119905) 12059520(119905)
12059521(119905) 120595
2119872minus1(119905)
1205952119896minus10(119905) 1205952119896minus11(119905) 120595
2119896minus1119872minus1
(119905) ]119879
(14)
Taking suitable collocation points as
119905119894= cos((2119894 + 1) 120587
2119896119872) 119894 = 1 2 2
119896minus1
119872 (15)
we defined the -square Legendre matrix
120601times
= [Ψ(cos( 3120587
2119896119872)) Ψ(cos( 5120587
2119896119872))
sdot sdot sdot Ψ(cos((2119896
119872+ 1)120587
2119896119872))]
(16)
where = 2119896minus1
119872 correspondingly we have
119891 = [119891(cos( 3120587
2119896119872))
times 119891(cos( 5120587
2119896119872)) sdot sdot sdot 119891(cos(
(2119896
119872+ 1)120587
2119896119872))]
= 119862119879
120601times
(17)
The Legendre matrix 120601times
is an invertible matrix and thecoefficient vector 119862119879 is obtained by 119862119879 = 119891 120601
minus1
times
32 Operational Matrix of the Fractional Integration Theintegration of the Ψ(119905) defined in (14) can be approximatedby Legendre wavelet series with Legendre wavelet coefficientmatrix 119875
int
119905
0
Ψ (119905) 119889119905 = 119875times
Ψ (119905) (18)
where the -square matrix 119875 is called Legendre waveletoperational matrix of integration
The 119898-set of block-pulse functions is defined on [0 l) asfollows
119887119894(119905) =
1119894
119898le 119905 lt
(119894 + 1)
119898
0 otherwise(19)
where 119894 = 0 1 2 3 119898
4 International Journal of Mathematics and Mathematical Sciences
The functions 119887119894are disjoint and orthogonal That is
119887119894(119905) 119887119895(119905) =
119887119894(119905) 119894 = 119895
0 119894 = 119895
int
1
0
119887119894(119905) 119887119895(119905) 119889119905 =
1
119898 119894 = 119895
0 119894 = 119895
(20)
The orthogonality property of block-pulse function isobtained from the disjointness property
An arbitrary function 119891 isin 1198712[0 1) can be expanded into
block-pulse functions as
119891 (119905) asymp
119898minus1
sum
119894=0
119891119894119887119894(119905) = 119891
119879
119861 (119905) (21)
where119891119894are the coefficients of the block-pulse function given
by
119891119894=119898
119897int
119897
0
119891 (119905) 119887119894(119905) (22)
The Legendre wavelets can be expanded into119898-set of block-pulse functions as
Ψ (119905) = 120601times
119861 (119905) (23)
where 119861(119905) = [1198870(119905) 1198871(119905) sdot sdot sdot 119887
119894(119905) sdot sdot sdot 119887
119898minus1(119905)]119879
The fractional integral of block-pulse function vector canbe written as
(119868120572
119861) (119905) = 119865120572
119898times119898119861 (119905) (24)
where 119865120572119898times119898
is given in [32]Now we introduce the derivation process of the Legendre
wavelet operational matrix of the fractional integration
(119868120572
Ψ) (119905) asymp 119875120572
timesΨ (119905) (25)
where the -square matrix 119875120572times
is called Legendre waveletoperational matrix of the fractional integration
Using (23) and (25) we have
(119868120572
Ψ) (119905) asymp (119868120572
120601times
119861) (119905) = 120601times
(119868120572
119861) (119905)
asymp 120601times
(119865120572
119861) (119905)
(26)
From (25) and (26) we get
119875120572
timesΨ (119905) = 120601
times119865120572
119861 (119905) (27)
and by (23) (27) becomes
119875120572
times120601times
119861 (119905) = 120601times
119865120572
119861 (119905) (28)
Then the Legendre wavelet operational matrix119875120572times
of frac-tional integration is given by
119875120572
times= 120601times
119865120572
120601minus1
times (29)
Following is the Legendre wavelet operationalmatrix119875120572times
offractional-order integration for the particular values of 119896 = 2119872 = 3 and 120572 = 05
11987505
6times6= (
(
05415 04324 01819 minus00871 minus00179 00154
0 05415 0 01819 0 minus00179
minus02046 0071 02243 minus00449 00798 00119
0 minus02046 0 02243 0 00798
01781 02506 minus00252 minus00652 01555 00143
0 01781 0 minus00252 0 01555
)
)
(30)
4 Application to Fractional RiccatiDifferential Equation
In this section we use the generalized Legendre waveletoperational matrix to solve nonlinear Riccati differentialequation and we discuss the existence and uniqueness ofsolutions with initial conditions and convergence criteria ofthe proposed LWM approach
Consider the fractional-order Riccati differential equa-tion of the form
119863120572
119910 (119905) = 119875 (119905) 1199102
+ 119876 (119905) 119910 + 119877 (119905)
119905 gt 0 0 lt 120572 le 1
(31)
subject to the initial condition
119910 (0) = 119896 (32)
Let us suppose that the functions119863120572119910(119905) 119875(119905)119876(119905) and 119877(119905)are approximated using Legendre wavelet as follows
119863120572
119910 (119905) = 119880119879
Ψ (119905) 119875 (119905) = 119881119879
Ψ (119905)
119876 (119905) = 119882119879
Ψ (119905) 119877 (119905) = 119883119879
Ψ (119905)
(33)
where 119880 119881119882119883 and Ψ(119905) are given in (14)Using (6) we can write
119910 (119905) = 119868120572
(119863120572
119910 (119905)) minus 119910 (0) (34)
International Journal of Mathematics and Mathematical Sciences 5
By (25) and (32) (34) leads to
119910 (119905) asymp 119880119879
119875120572
Ψ (119905) + 119884119879
0Ψ (119905) = 119862
119879
Ψ (119905) (35)
where 119910(0) = 119896 asymp 119884119879
0Ψ(119905) 119862 = (119880
119879
119875120572
times+ 119884119879
0)119879
Substituting (33) and (35) into (31) we have
119880119879
Ψ (119905) = 119881119879
Ψ (119905) [119862119879
Ψ (119905)]2
+119882119879
Ψ (119905) 119862119879
Ψ (119905) + 119883119879
Ψ (119905)
(36)
Substituting (23) into (36) we have
119880119879
120601times
= 119881119879
[119862119879
120601times
]2
+119882119879
119862119879
120601times
+ 119883119879
(37)
where 119862 119881 119882 and 120601times
are known Equation (37) repre-sents a system of nonlinear equations with unknown vector119880 This system of nonlinear equations can be solved byNewton method for the unknown vector 119880 and we can getthe approximation solution by including 119880 into (35)
41 Existence and Uniqueness of Solutions Consider thefractional-order Riccati differential equation of the forms(31) and (32) The nonlinear term in (31) is 119910
2 and119875(119905) 119876(119905) and 119877(119905) are known functions For 120572 = 1 thefractional-order Riccati converts into the classical Riccatidifferential equation
Definition 1 Let 119868 = [0 119897] 119897 lt infin and 119862(119868) be the class of allcontinuous functions defined on 119868 with the norm
10038171003817100381710038171199101003817100381710038171003817 = sup119905isin119868
10038161003816100381610038161003816119890minusℎ119905
119910 (119905)10038161003816100381610038161003816 ℎ gt 0 (38)
which is equivalent to the sup norm of 119910 That is 119910 =
sup119905isin119868
|119890minusℎ119905
119910(119905)|
RemarkAssume that solution 119910(119905) of fractional-order Riccatidifferential equations (31) and (32) belongs to the space 119878 =
119910 isin 119877 |119910| le 119888 119888 is any constant in order to study theexistence and uniqueness of the initial value problem
Definition 2 The space of integrable functions 1198711[0 119897] in the
interval [0 119897] is defined as
1198711[0 119897] = 119906 (119905) int
119897
0
|119906 (119905)| 119889119905 lt infin (39)
Theorem 3 The initial value problem given by (31) and (32)has a unique solution
119910 isin 119862 (119868) 1199101015840
isin 119883 = 119910 isin 1198711[0 119897]
10038171003817100381710038171199101003817100381710038171003817 =
10038171003817100381710038171003817119890minusℎ119905
119910(119905)100381710038171003817100381710038171198711
(40)
Proof By (1) the fractional differential equation (31) can bewritten as
1198681minus120572
119889119910 (119905)
119889119905= 119875 (119905) 119910
2
+ 119876 (119905) 119910 + 119877 (119905) (41)
and becomes
119910 (119905) = 119868120572
(119875 (119905) 1199102
+ 119876 (119905) 119910 + 119877 (119905)) (42)
Now we define the operator Θ 119862(119868) rarr 119862(119868) by
Θ119910 (119905) = 119868120572
(119875 (119905) 1199102
+ 119876 (119905) 119910 + 119877 (119905)) (43)
and then
119890minusℎ119905
(Θ119910 minus Θ119908)
= 119890minusℎ119905
119868120572
[(1198751199102
(119905) + 119876119910 (119905) + 119877)
minus (1198751199082
(119905) + 119876119908 (119905) + 119877)]
le1
Γ (120572)int
119905
0
(119905 minus 119904)120572minus1
119890minusℎ(119905minus119904)
times [(119910 (119904) minus 119908 (119904)) (119910 (119904) + 119908 (119904))
minus119896 (119910 (119904) minus 119908 (119904))] 119890minusℎ119904
119889119904
le1003817100381710038171003817119910 minus 119908
1003817100381710038171003817
1
Γ (120572)int
119905
0
119904120572minus1
119890minusℎ119904
119889119904
(44)
hence we have1003817100381710038171003817Θ119910 minus Θ119908
1003817100381710038171003817 lt1003817100381710038171003817119910 minus 119908
1003817100381710038171003817 (45)
which implies that the operator given by (43) has a uniquefixed point and consequently the given integral equation hasa unique solution 119910(119905) isin 119862(119868) Also we can see that
119868120572
(119875(119905)1199102
+ 119876(119905)119910 + 119877(119905))10038161003816100381610038161003816119905=0
= 119896 (46)
Now from (42) we have
119910 (119905) = [119905120572
Γ (120572 + 1)(1198751199102
0+ 1198761199100+ 119877)
+ 119868120572+1
(1198751015840
1199102
+ 21199101015840
119875 + 1198761015840
119910 + 1198761199101015840
+ 1198771015840
) ]
119889119910
119889119905= [
119905120572minus1
Γ (120572)(1198751199102
0+ 1198761199100+ 119877)
+ 119868120572
(1198751015840
1199102
+ 21199101015840
119875 + 1198761015840
119910 + 1198761199101015840
+ 1198771015840
) ]
119890minusℎ119905
1199101015840
(119905) = 119890minusℎ119905
[119905120572minus1
Γ (120572)(1198751199102
0+ 1198761199100+ 119877)
+ 119868120572
(1198751015840
1199102
+ 21199101015840
119875 + 1198761015840
119910 + 1198761199101015840
+ 1198771015840
) ]
(47)
from which we can deduce that 1199101015840(119905) isin 119862(119868) and 1199101015840
(119905) isin 119878
6 International Journal of Mathematics and Mathematical Sciences
Now again from (42) (43) and (46) we get
119889119910
119889119905=
119889
119889119905119868120572
[1198751199102
(119905) + 119876119910 (119905) + 119877]
1198681minus120572
119889119910
119889119905= 1198681minus120572
119889
119889119905119868120572
[1198751199102
(119905) + 119876119910 (119905) + 119877]
=119889
1198891199051198681minus120572
119868120572
[1198751199102
(119905) + 119876119910 (119905) + 119877]
119863120572
119910 (119905) =119889
119889119905119868 [1198751199102
(119905) + 119876119910 (119905) + 119877]
= 1198751199102
(119905) + 119876119910 (119905) + 119877
119910 (0) = 119868120572
(1198751199102
(119905) + 119876119910 (119905) + 119877)10038161003816100381610038161003816119905=0
= 119896
(48)
which implies that the integral equation (46) is equivalent tothe initial value problem (32) and the theorem is proved
42 Convergence Analyses Let
120595119896119899
(119905) =10038161003816100381610038161198860
10038161003816100381610038161198962
120595 (119886119896
0119905 minus 119899119887
0) (49)
where 120595119896119899(119905) form a wavelet basis for 1198712(119877) In particular
when 1198860= 2 and 119887
0= 1 120595
119896119899(119905) form an orthonormal basis
[29]By (14) let 119910(119905) = sum
119872minus1
119894=111988811198941205951119894(119905) be the solution of (31)
where 1198881119894= ⟨119910(119905) 120595
1119894(119905)⟩ for 119896 = 1 in which ⟨sdot sdot⟩ denotes
the inner product
119910 (119905) =
119899
sum
119894=1
⟨119910 (119905) 1205951119894(119905)⟩ 120595
1119894(119905) (50)
Let 120573119895= ⟨119910(119905) 120595(119905)⟩ where 120595(119905) = 120595
1119894(119905)
Let119909119899= sum119899
119895=1120573119895120595(119905119895) be a sequence of partial sumsThen
⟨119910 (119905) 119909119899⟩ = ⟨119910 (119905)
119899
sum
119895=1
120573119895120595 (119905119895)⟩
=
119899
sum
119895=1
120573119895⟨119910 (119905) 120595 (119905
119895)⟩
=
119899
sum
119895=1
120573119895120573119895
=
119899
sum
119895=1
10038161003816100381610038161003816120573119895
10038161003816100381610038161003816
2
(51)
Further
1003817100381710038171003817119909119899 minus 119909119898
10038171003817100381710038172
=
10038171003817100381710038171003817100381710038171003817100381710038171003817
119899
sum
119895=119898+1
120573119895120595 (119905119895)
10038171003817100381710038171003817100381710038171003817100381710038171003817
2
= ⟨
119899
sum
119894=119898+1
120573119894120595 (119905119894)
119899
sum
119895=119898+1
120573119895120595 (119905119895)⟩
=
119899
sum
119894=119898+1
119899
sum
119895=119898+1
120573119894120573119895⟨120595 (119905119894) 120595 (119905
119895)⟩
=
119899
sum
119895=119898+1
10038161003816100381610038161003816120573119895
10038161003816100381610038161003816
2
(52)
As 119899 rarr infin from Besselrsquos inequality we have suminfin119895=1
|120573119895|2 is
convergentIt implies that 119909
119899 is a Cauchy sequence and it converges
to 119909 (say)Also
⟨119909 minus 119910 (119905) 120595 (119905119895)⟩ = ⟨119909 120595 (119905
119895)⟩ minus ⟨119910 (119905) 120595 (119905
119895)⟩
= ⟨ lim119899rarrinfin
119909119899 120595 (119905119895)⟩ minus 120573
119895
= lim119899rarrinfin
⟨119909119899 120595 (119905119895)⟩ minus 120573
119895
= lim119899rarrinfin
⟨
119899
sum
119895=1
120573119895120595 (119905119895) 120595 (119905
119895)⟩ minus 120573
119895
= 120573119895minus 120573119895= 0
(53)
which is possible only if 119910(119905) = 119909 That is both 119910(119905) and 119909119899
converge to the same value which indeed give the guaranteeof convergence of LWM
5 Numerical Examples
In order to show the effectiveness of the Legendre waveletsmethod (LWM) we implement LWM to the nonlinearfractional Riccati differential equations All the numericalexperiments were carried out on a personal computer withsomeMATLAB codes The specifications of PC are Intel corei5 processor and with Turbo boost up to 31 GHz and 4GB ofDDR3 memory The following problems of nonlinear Riccatidifferential equations are solved with real coefficients
Example 1 Consider the following nonlinear fractional Ric-cati differential equation
119863120572
119910 (119905) = 1 + 2119910 (119905) minus 1199102
(119905) 0 lt 120572 le 1 (54)
with initial condition
119910 (0) = 0 (55)
Exact solution for 120572 = 1 was found to be
119910 (119905) = 1 + radic2 tanh(radic2119905 +1
2log(
radic2 minus 1
radic2 + 1)) (56)
International Journal of Mathematics and Mathematical Sciences 7
01 02 03 04 05 06 07 08 09 10
02
04
06
08
1
12
14
16
18
Exact
LWM
t
y(t)
Figure 1 Numerical results of Example 1 by LWM for 120572 = 1
The integral representation of (54) and (55) is given by
119868120572
(119863120572
119910 (119905)) = 119868120572
(1 + 2119910 (119905) minus 1199102
(119905)) (57)
119910 (119905) = 119910 (0) +119905120572
Γ (120572 + 1)+ 2119868120572
119910 (119905) minus 119868120572
1199102
(119905) (58)
Let
119910 (119905) = 119862119879
Ψ (119905) (59)
and then
119868120572
119910 (119905) = 119862119879
119868120572
Ψ (119905)
= 119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(60)
By substituting (59) and (60) into (58) we get the followingsystem of algebraic equations
119862119879
Ψ (119905) =119905120572
Γ (120572 + 1)+ 2119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
minus 119862119879
1198752120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(61)
By solving the above system of linear equations we can findthe vector 119862 Numerical results are obtained for differentvalues of 119896 119872 and 120572 Solution obtained by the proposedLWM approach for 120572 = 1 119896 = 1 and 119872 = 3 is given inFigure 1 and for different values of 120572 = 06 07 08 and 09and for 119896 = 2 and 119872 = 5 is graphically given in Figure 2It can be seen from Figure 1 that the solution obtained bythe proposed LWM approach is more close to the exactsolution Table 1 describes the efficiency of the proposedmethod by comparing with the methods in [20 22] throughtheir absolute errorThe following is used for the errors of theapproximation119910(119905) of 119910(119905) that is 119910minus119910 = max |119910(119905)minus119910(119905)|Table 1 shows that very high accuracies are obtained for 119896= 3and119872= 5 by the present method
0204
0608
10
09 08 07 06 05
t
y(t)
120572
minus05
00
05
15
20
10
Figure 2 Numerical results of Example 1 by LWM for differentvalues of 120572
Example 2 Consider another fractional-order Riccati differ-ential equation
119863120572
119910 (119905) = 1 minus 1199102
(119905) 0 lt 120572 le 1 (62)
with initial condition
119910 (0) = 0 (63)
Exact solution for the above equation was found to be
119910 (119905) =1198902119905
minus 1
1198902119905 + 1 (64)
The integral representation of (62) and (63) is given by
119910 (119905) = 119910 (0) +119905120572
Γ (120572 + 1)minus 119868120572
1199102
(119905) (65)
Let
119910 (119905) = 119862119879
Ψ (119905) (66)
and then
119868120572
119910 (119905) = 119862119879
119868120572
Ψ (119905)
= 119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(67)
By substituting (66) and (67) in (62) we get the followingsystem of algebraic equations
119862119879
Ψ (119905) =119905120572
Γ (120572 + 1)minus 119862119879
1198752120572
2119896minus1119872times2119896minus1119872Ψ (119905) (68)
By solving the above system of linear equations we can findthe vector 119862 Numerical results are obtained for differentvalues of 119896119872 and 120572 Results obtained by LWM for 120572 = 1 119896 =2 and119872= 3 are shown in Figure 3 and it can be seen from thefigure that solution given by the LWMmerely coincides withthe exact solution Figure 4 shows the obtained results of (62)and (63) by LWM for different values of 120572 and for 119896 = 2 and
8 International Journal of Mathematics and Mathematical Sciences
Table 1 Numerical results of Example 1 for 120572 = 1
119905 Exact solution Absolute error in [22] Absolute error in [20] LWM LWM LWM119872 = 2 119896 = 2 119872 = 3 119896 = 2 119872 = 5 119896 = 3
01 0099667 751119864 minus 09 930119864 minus 05 0 0 002 0197375 152119864 minus 06 293119864 minus 02 191119864 minus 08 0 003 0291312 393119864 minus 05 378119864 minus 03 272119864 minus 08 145119864 minus 13 004 0379948 432119864 minus 04 281119864 minus 03 165119864 minus 08 117119864 minus 13 194119864 minus 16
05 0462117 841119864 minus 04 980119864 minus 04 131119864 minus 08 328119864 minus 13 320119864 minus 16
06 0537049 294119864 minus 05 793119864 minus 03 198119864 minus 08 497119864 minus 13 124119864 minus 16
07 0604367 335119864 minus 04 944119864 minus 03 252119864 minus 08 632119864 minus 13 158119864 minus 16
08 0664036 544119864 minus 04 117119864 minus 02 294119864 minus 08 736119864 minus 13 184119864 minus 16
09 0716297 656119864 minus 09 396119864 minus 02 323119864 minus 08 812119864 minus 13 194119864 minus 16
10 0761594 253119864 minus 06 295119864 minus 02 263119864 minus 08 462119864 minus 13 199119864 minus 16
Table 2 Numerical results of Example 2 for 120572 = 1
119905 Exact solution Absolute error in [22] Absolute error in [20] LWM LWM LWM119872 = 2 119896 = 2 119872 = 3 119896 = 2 119872 = 5 119896 = 3
01 0110295 123119864 minus 15 281119864 minus 03 0 0 002 0241976 524119864 minus 15 383119864 minus 04 0 0 003 0395104 816119864 minus 15 123119864 minus 04 0 0 004 0567812 115119864 minus 12 286119864 minus 03 168119864 minus 12 166119864 minus 15 143119864 minus 1605 0756014 617119864 minus 12 438119864 minus 04 222119864 minus 12 212119864 minus 15 166119864 minus 1606 0953566 455119864 minus 11 519119864 minus 02 115119864 minus 12 110119864 minus 15 154119864 minus 1607 1152946 757119864 minus 10 214119864 minus 02 127119864 minus 12 111119864 minus 15 133119864 minus 1608 1346363 633119864 minus 09 142119864 minus 02 187119864 minus 12 201119864 minus 15 175119864 minus 1609 1526911 367119864 minus 08 698119864 minus 03 193119864 minus 12 266119864 minus 15 187119864 minus 1610 1689498 164119864 minus 07 496119864 minus 03 156119864 minus 12 166119864 minus 15 164119864 minus 16
119872 = 5 Table 2 describes the efficiency of the proposedmethod by comparing with the methods in [20 22] throughtheir absolute error Table 1 shows that very high accuraciesare obtained for 119896 = 3 and 119872 = 5 by the present methodand from these results we can identify that guarantee ofconvergence of the proposed LWM approach is very high
Example 3 Let us consider another problem of nonlinearRiccati differential equation
119863120572
119910 (119905) = 1199052
+ 1199102
(119905) 0 lt 120572 le 1 119905 ge 0 (69)
with initial condition
119910 (0) = 1 (70)
When 120572 = 1 its exact solution is given by
119910 (119905) =119905 (119869minus34
(1199052
2) Γ (14) + 211986934
(1199052
2) Γ (34))
11986914
(11990522) Γ (14) minus 2119869minus14
(11990522) Γ (34)
(71)
where 119869119899(119905) is the Bessel function of first kind
0
01
02
03
04
05
06
07
08
LWM
01 02 03 04 05 06 07 08 09 1
Exact
t
y(t)
Figure 3 Numerical results of Example 2 by LWM for 120572 = 1
The integral representation of (69) and (70) is given by
119910 (119905) = 1 +2
Γ (120572 + 1) + 2Γ (120572)119905120572+2
+ 119868120572
1199102
(119905) (72)
International Journal of Mathematics and Mathematical Sciences 9
0204
060810
09 08 07 06 05
t
y(t)
120572
00
01
02
03
04
0605
0807
081
Figure 4 Numerical results of Example 2 by LWM for differentvalues of 120572
Let
119910 (119905) = 119862119879
Ψ (119905) (73)
and then
119868120572
119910 (119905) = 119862119879
119868120572
Ψ (119905)
= 119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(74)
By substituting (73) and (74) into (69) we get the followingsystem of algebraic equations
119862119879
Ψ (119905) = 1 +2
Γ (120572 + 1) + 2Γ (120572)(119862119879
Ψ (119905))120572+2
+ 119862119879
1198752120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(75)
By solving the above system of linear equations we can findthe vector 119862 Numerical results are obtained for differentvalues of 119896 119872 and 120572 Obtained results for (69) and (70)are shown in Figures 5 and 6 Figure 5 shows the solutionsobtained by LWM for different values of 120572 and for 119896 = 2 and119872 = 4 Figure 6 compares the solution obtained by LWMwith the exact solution of (69) and (70) when 120572 = 1 119896 = 1and 119872 = 2 So far there are no published results of absoluteerror for this problem and hence we are unable to compareabsolute error of ourmethodwith the existingmethods Fromthese results we can see that the proposed LWM approachgives the solution which is very close to the exact solutionand outperformed recently developed approaches for thenonlinear fractional-order Riccati differential equations interms of solution quality and convergence criteria
6 Conclusions
Nonlinear fractional-order Riccati differential equations playan important role in the modeling of many biologicalphysical chemical and real life problemsTherefore it is nec-essary to develop a method which would give more accuratesolutions to such type of problems with greater convergence
0204
0608
10
09 0807 06
05
t
y(t)
120572
10
0
10
20
30
40
0 406
08
0 9 t
Figure 5 Numerical results of Example 3 by LWM for differentvalues of 120572
1
15
2
25
3
35
4
45
5
55
6
LWM
01 02 03 04 05 06 07 08 09 1
Exact
t
y(t)
Figure 6 Numerical results of Example 3 by LWM for 120572 = 1
criteria In this work a Legendrersquos wavelet operational matrixmethod called LWM was proposed for solving nonlinearfractional-order Riccati differential equations Comparisonwas made for the solutions obtained by the proposedmethodand with the other recent approaches developed for thesame problem through their error analysis obtained resultsshow that the proposed LWM yields more accurate andreliable solutions even for small values of 119872 and 119896 whichassures the best approximate solution in less computationaleffort Further we have discussed the convergence criteriaof proposed scheme which indeed provides the guaranteeof consistency and stability of the proposed LWM schemefor the solutions of nonlinear fractional Riccati differentialequations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
10 International Journal of Mathematics and Mathematical Sciences
References
[1] N A Khan M Jamil A Ara and S Das ldquoExplicit solution fortime-fractional batch reactor systemrdquo International Journal ofChemical Reactor Engineering vol 9 article A91 2011
[2] V Feliu-Batlle R R Perez and L S Rodrıguez ldquoFractionalrobust control of main irrigation canals with variable dynamicparametersrdquoControl Engineering Practice vol 15 no 6 pp 673ndash686 2007
[3] I Podlubny ldquoFractional-order systems and 119875119868120582
119863120583-controllersrdquo
IEEE Transactions on Automatic Control vol 44 no 1 pp 208ndash214 1999
[4] R Garrappa ldquoOn some explicit Adams multistep methods forfractional differential equationsrdquo Journal of Computational andApplied Mathematics vol 229 no 2 pp 392ndash399 2009
[5] M Jamil and N A Khan ldquoSlip effects on fractional viscoelasticfluidsrdquo International Journal of Differential Equations vol 2011Article ID 193813 19 pages 2011
[6] F Mohammadi and M M Hosseini ldquoA comparative study ofnumerical methods for solving quadratic Riccati differentialequationsrdquo Journal of the Franklin Institute vol 348 no 2 pp156ndash164 2011
[7] R Shankar Principles of Quantum Mechanics Plenum PressNew York NY USA 1980
[8] S FragaM J Garcıa de la Vega and E S FragaTheSchrodingerand Riccati Equations vol 70 of Lect Notes Chem 1999
[9] L B Burrows and M Cohen ldquoSchrodingerrsquos wave equation-A lie algebra treatmentrdquo in Fundamental World of QuantumChemistry A Tribute to the Memory of Per-Olov Lowdin EJ Brandas and E S Kryachko Eds Kluwer Dordrecht TheNetherlands 2004
[10] S Abbasbandy ldquoHomotopy perturbation method for quadraticRiccati differential equation and comparison with Adomianrsquosdecomposition methodrdquo Applied Mathematics and Computa-tion vol 172 no 1 pp 485ndash490 2006
[11] Z Odibat and S Momani ldquoModified homotopy perturbationmethod application to quadratic Riccati differential equationof fractional orderrdquo Chaos Solitons amp Fractals vol 36 no 1 pp167ndash174 2008
[12] N A Khan A Ara and M Jamil ldquoAn efficient approach forsolving the Riccati equation with fractional ordersrdquo Computersamp Mathematics with Applications vol 61 no 9 pp 2683ndash26892011
[13] H Aminikhah and M Hemmatnezhad ldquoAn efficient methodfor quadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 4 pp835ndash839 2010
[14] S Abbasbandy ldquoIterated Hersquos homotopy perturbation methodfor quadratic Riccati differential equationrdquo Applied Mathemat-ics and Computation vol 175 no 1 pp 581ndash589 2006
[15] J Cang Y Tan H Xu and S Liao ldquoSeries solutions of non-lin-ear Riccati differential equations with fractional orderrdquo ChaosSolitons and Fractals vol 40 no 1 pp 1ndash9 2009
[16] Y Tan and S Abbasbandy ldquoHomotopy analysis method forquadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 3 pp539ndash546 2008
[17] M Gulsu and M Sezer ldquoOn the solution of the Riccati equa-tion by the Taylor matrix methodrdquo Applied Mathematics andComputation vol 176 no 2 pp 414ndash421 2006
[18] Y Li and LHu ldquoSolving fractional Riccati differential equationsusing Haar waveletrdquo in Proceedings of the 3rd InternationalConference on Information and Computing (ICIC 10) pp 314ndash317 Wuxi China June 2010
[19] N A Khan and A Ara ldquoFractional-order Riccati differentialequation analytical approximation and numerical resultsrdquoAdvances in Difference Equations vol 2013 article 185 2013
[20] M A Z Raja J A Khan and I M Qureshi ldquoA new stochasticapproach for solution of Riccati differential equation of frac-tional orderrdquo Annals of Mathematics and Artificial Intelligencevol 60 no 3-4 pp 229ndash250 2010
[21] M Merdan ldquoOn the solutions fractional Riccati differentialequation with modified RIEmann-Liouville derivativerdquo Inter-national Journal of Differential Equations vol 2012 Article ID346089 17 pages 2012
[22] N H Sweilam M M Khader and A M S Mahdy ldquoNumericalstudies for solving fractional Riccati differential equationrdquoApplications and AppliedMathematics vol 7 no 2 pp 595ndash6082012
[23] C K ChuiWavelets A Mathematical Tool for Signal ProcessingSIAM Philadelphia Pa USA 1997
[24] G Beylkin R Coifman and V Rokhlin ldquoFast wavelet trans-forms and numerical algorithms Irdquo Communications on Pureand Applied Mathematics vol 44 no 2 pp 141ndash183 1991
[25] M ur Rehman and R Ali Khan ldquoThe Legendre wavelet methodfor solving fractional differential equationsrdquoCommunications inNonlinear Science and Numerical Simulation vol 16 no 11 pp4163ndash4173 2011
[26] S Balaji ldquoA new approach for solving Duffing equationsinvolving both integral and non-integral forcing termsrdquo AinShams Engineering Journal 2014
[27] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent IIrdquo Geophysical Journal of the RoyalAstronomical Society vol 13 pp 529ndash539 1967
[28] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-Holland Mathematics Studies Elsevier 2006
[29] J S Gu and W S Jiang ldquoThe Haar wavelets operational matrixof integrationrdquo International Journal of Systems Science vol 27no 7 pp 623ndash628 1996
[30] M Razzaghi and S Yousefi ldquoLegendre wavelets direct methodfor variational problemsrdquo Mathematics and Computers in Sim-ulation vol 53 no 3 pp 185ndash192 2000
[31] M Razzaghi and S Yousefi ldquoLegendre wavelets method forconstrained optimal control problemsrdquo Mathematical Methodsin the Applied Sciences vol 25 no 7 pp 529ndash539 2002
[32] A Kilicman and Z A A Al Zhour ldquoKronecker operationalmatrices for fractional calculus and some applicationsrdquo AppliedMathematics and Computation vol 187 no 1 pp 250ndash265 2007
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
Volume 2014
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Mathematics and Mathematical Sciences 3
3 Generalized Legendre Wavelet OperationalMatrix to Fractional Integration
31 Legendre Wavelets A family of functions were consti-tuted by wavelets and constructed from dilation and trans-lation of a single function called mother wavelet When theparameters 119886 of dilation and 119887 of translation vary continu-ously following are the family of continuous wavelets [29]
120595119886119887
(119905) = |119886|minus12
120595(119905 minus 119887
119886) 119886 119887 isin R 119886 = 0 (8)
If the parameters 119886 and 119887 are restricted to discrete values as119886 = 119886
minus119896
0 119887 = 119899119887
0119886minus119896
0 1198860gt 1 119887
0gt 0 and 119899 and 119896 are positive
integers following are the family of discrete wavelets
120595119896119899
(119905) =10038161003816100381610038161198860
10038161003816100381610038161198962
120595 (119886119896
0119905 minus 119899119887
0) (9)
where 120595119896119899(119905) form a wavelet basis for 1198712(119877) In particular
when 1198860= 2 and 119887
0= 1 120595
119896119899(119905) form an orthonormal basis
[29]Legendre wavelets 120595
119899119898(119905) = 120595(119896 119899119898 119905) have four argu-
ments 119899 = 2119899 minus 1 119899 = 1 2 3 2119896minus1
119896 can assume anypositive integer119898 is the order for Legendre polynomials and119905 is the normalized timeThey are defined on the interval [01)as [30 31]
120595119899119898
(119905)
=
radic119898 +1
221198962
119875119898(2119896
119905 minus 119899) for 119899 minus 1
2119896le 119905 lt
119899 + 1
2119896
0 otherwise(10)
where 119898 = 0 1 119872 minus 1 and 119899 = 1 2 3 2119896minus1 The
coefficient radic119898 + (12) is for orthonormality the dilationparameter is 119886 = 2
minus119896 and translation parameter is 119887 =
119899 2minus119896 119875119898(119905) are the well-known Legendre polynomials of
ordermdefined on the interval [minus1 1] and can be determinedwith the aid of the following recurrence formulae
1198750(119905) = 1 119875
1(119905) = 119905
119875119898+1
(119905) = (2119898 + 1
119898 + 1) 119905119875119898(119905) minus (
119898
119898 + 1)119875119898minus1
(119905)
119898 = 1 2 3
(11)
The Legendre wavelet series representation of the function119891(119905) defined over [0 1) is given by
119891 (119905) =
infin
sum
119899=1
infin
sum
119898=0
119888119899119898120595119899119898
(119905) (12)
where 119888119899119898
= ⟨119891(119905) 120595119899119898(119905)⟩ in which ⟨sdot sdot⟩ denotes the inner
product If the infinite series in (12) is truncated then (12) canbe written as
119891 (119905) cong
2119896minus1
sum
119899=1
119872minus1
sum
119898=0
119888119899119898120595119899119898
(119905) = 119862119879
Ψ (119905) = 119891 (119905) (13)
where 119862 andΨ(119905) are 2119896minus1119872times 1matrices given by
119862 = [11988810 11988811 119888
1119872minus1 11988820 11988821 119888
2119872minus1
1198882119896minus10 1198882119896minus11 119888
2119896minus1119872minus1
]119879
Ψ (119905) = [12059510(119905) 12059511(119905) 120595
1119872minus1(119905) 12059520(119905)
12059521(119905) 120595
2119872minus1(119905)
1205952119896minus10(119905) 1205952119896minus11(119905) 120595
2119896minus1119872minus1
(119905) ]119879
(14)
Taking suitable collocation points as
119905119894= cos((2119894 + 1) 120587
2119896119872) 119894 = 1 2 2
119896minus1
119872 (15)
we defined the -square Legendre matrix
120601times
= [Ψ(cos( 3120587
2119896119872)) Ψ(cos( 5120587
2119896119872))
sdot sdot sdot Ψ(cos((2119896
119872+ 1)120587
2119896119872))]
(16)
where = 2119896minus1
119872 correspondingly we have
119891 = [119891(cos( 3120587
2119896119872))
times 119891(cos( 5120587
2119896119872)) sdot sdot sdot 119891(cos(
(2119896
119872+ 1)120587
2119896119872))]
= 119862119879
120601times
(17)
The Legendre matrix 120601times
is an invertible matrix and thecoefficient vector 119862119879 is obtained by 119862119879 = 119891 120601
minus1
times
32 Operational Matrix of the Fractional Integration Theintegration of the Ψ(119905) defined in (14) can be approximatedby Legendre wavelet series with Legendre wavelet coefficientmatrix 119875
int
119905
0
Ψ (119905) 119889119905 = 119875times
Ψ (119905) (18)
where the -square matrix 119875 is called Legendre waveletoperational matrix of integration
The 119898-set of block-pulse functions is defined on [0 l) asfollows
119887119894(119905) =
1119894
119898le 119905 lt
(119894 + 1)
119898
0 otherwise(19)
where 119894 = 0 1 2 3 119898
4 International Journal of Mathematics and Mathematical Sciences
The functions 119887119894are disjoint and orthogonal That is
119887119894(119905) 119887119895(119905) =
119887119894(119905) 119894 = 119895
0 119894 = 119895
int
1
0
119887119894(119905) 119887119895(119905) 119889119905 =
1
119898 119894 = 119895
0 119894 = 119895
(20)
The orthogonality property of block-pulse function isobtained from the disjointness property
An arbitrary function 119891 isin 1198712[0 1) can be expanded into
block-pulse functions as
119891 (119905) asymp
119898minus1
sum
119894=0
119891119894119887119894(119905) = 119891
119879
119861 (119905) (21)
where119891119894are the coefficients of the block-pulse function given
by
119891119894=119898
119897int
119897
0
119891 (119905) 119887119894(119905) (22)
The Legendre wavelets can be expanded into119898-set of block-pulse functions as
Ψ (119905) = 120601times
119861 (119905) (23)
where 119861(119905) = [1198870(119905) 1198871(119905) sdot sdot sdot 119887
119894(119905) sdot sdot sdot 119887
119898minus1(119905)]119879
The fractional integral of block-pulse function vector canbe written as
(119868120572
119861) (119905) = 119865120572
119898times119898119861 (119905) (24)
where 119865120572119898times119898
is given in [32]Now we introduce the derivation process of the Legendre
wavelet operational matrix of the fractional integration
(119868120572
Ψ) (119905) asymp 119875120572
timesΨ (119905) (25)
where the -square matrix 119875120572times
is called Legendre waveletoperational matrix of the fractional integration
Using (23) and (25) we have
(119868120572
Ψ) (119905) asymp (119868120572
120601times
119861) (119905) = 120601times
(119868120572
119861) (119905)
asymp 120601times
(119865120572
119861) (119905)
(26)
From (25) and (26) we get
119875120572
timesΨ (119905) = 120601
times119865120572
119861 (119905) (27)
and by (23) (27) becomes
119875120572
times120601times
119861 (119905) = 120601times
119865120572
119861 (119905) (28)
Then the Legendre wavelet operational matrix119875120572times
of frac-tional integration is given by
119875120572
times= 120601times
119865120572
120601minus1
times (29)
Following is the Legendre wavelet operationalmatrix119875120572times
offractional-order integration for the particular values of 119896 = 2119872 = 3 and 120572 = 05
11987505
6times6= (
(
05415 04324 01819 minus00871 minus00179 00154
0 05415 0 01819 0 minus00179
minus02046 0071 02243 minus00449 00798 00119
0 minus02046 0 02243 0 00798
01781 02506 minus00252 minus00652 01555 00143
0 01781 0 minus00252 0 01555
)
)
(30)
4 Application to Fractional RiccatiDifferential Equation
In this section we use the generalized Legendre waveletoperational matrix to solve nonlinear Riccati differentialequation and we discuss the existence and uniqueness ofsolutions with initial conditions and convergence criteria ofthe proposed LWM approach
Consider the fractional-order Riccati differential equa-tion of the form
119863120572
119910 (119905) = 119875 (119905) 1199102
+ 119876 (119905) 119910 + 119877 (119905)
119905 gt 0 0 lt 120572 le 1
(31)
subject to the initial condition
119910 (0) = 119896 (32)
Let us suppose that the functions119863120572119910(119905) 119875(119905)119876(119905) and 119877(119905)are approximated using Legendre wavelet as follows
119863120572
119910 (119905) = 119880119879
Ψ (119905) 119875 (119905) = 119881119879
Ψ (119905)
119876 (119905) = 119882119879
Ψ (119905) 119877 (119905) = 119883119879
Ψ (119905)
(33)
where 119880 119881119882119883 and Ψ(119905) are given in (14)Using (6) we can write
119910 (119905) = 119868120572
(119863120572
119910 (119905)) minus 119910 (0) (34)
International Journal of Mathematics and Mathematical Sciences 5
By (25) and (32) (34) leads to
119910 (119905) asymp 119880119879
119875120572
Ψ (119905) + 119884119879
0Ψ (119905) = 119862
119879
Ψ (119905) (35)
where 119910(0) = 119896 asymp 119884119879
0Ψ(119905) 119862 = (119880
119879
119875120572
times+ 119884119879
0)119879
Substituting (33) and (35) into (31) we have
119880119879
Ψ (119905) = 119881119879
Ψ (119905) [119862119879
Ψ (119905)]2
+119882119879
Ψ (119905) 119862119879
Ψ (119905) + 119883119879
Ψ (119905)
(36)
Substituting (23) into (36) we have
119880119879
120601times
= 119881119879
[119862119879
120601times
]2
+119882119879
119862119879
120601times
+ 119883119879
(37)
where 119862 119881 119882 and 120601times
are known Equation (37) repre-sents a system of nonlinear equations with unknown vector119880 This system of nonlinear equations can be solved byNewton method for the unknown vector 119880 and we can getthe approximation solution by including 119880 into (35)
41 Existence and Uniqueness of Solutions Consider thefractional-order Riccati differential equation of the forms(31) and (32) The nonlinear term in (31) is 119910
2 and119875(119905) 119876(119905) and 119877(119905) are known functions For 120572 = 1 thefractional-order Riccati converts into the classical Riccatidifferential equation
Definition 1 Let 119868 = [0 119897] 119897 lt infin and 119862(119868) be the class of allcontinuous functions defined on 119868 with the norm
10038171003817100381710038171199101003817100381710038171003817 = sup119905isin119868
10038161003816100381610038161003816119890minusℎ119905
119910 (119905)10038161003816100381610038161003816 ℎ gt 0 (38)
which is equivalent to the sup norm of 119910 That is 119910 =
sup119905isin119868
|119890minusℎ119905
119910(119905)|
RemarkAssume that solution 119910(119905) of fractional-order Riccatidifferential equations (31) and (32) belongs to the space 119878 =
119910 isin 119877 |119910| le 119888 119888 is any constant in order to study theexistence and uniqueness of the initial value problem
Definition 2 The space of integrable functions 1198711[0 119897] in the
interval [0 119897] is defined as
1198711[0 119897] = 119906 (119905) int
119897
0
|119906 (119905)| 119889119905 lt infin (39)
Theorem 3 The initial value problem given by (31) and (32)has a unique solution
119910 isin 119862 (119868) 1199101015840
isin 119883 = 119910 isin 1198711[0 119897]
10038171003817100381710038171199101003817100381710038171003817 =
10038171003817100381710038171003817119890minusℎ119905
119910(119905)100381710038171003817100381710038171198711
(40)
Proof By (1) the fractional differential equation (31) can bewritten as
1198681minus120572
119889119910 (119905)
119889119905= 119875 (119905) 119910
2
+ 119876 (119905) 119910 + 119877 (119905) (41)
and becomes
119910 (119905) = 119868120572
(119875 (119905) 1199102
+ 119876 (119905) 119910 + 119877 (119905)) (42)
Now we define the operator Θ 119862(119868) rarr 119862(119868) by
Θ119910 (119905) = 119868120572
(119875 (119905) 1199102
+ 119876 (119905) 119910 + 119877 (119905)) (43)
and then
119890minusℎ119905
(Θ119910 minus Θ119908)
= 119890minusℎ119905
119868120572
[(1198751199102
(119905) + 119876119910 (119905) + 119877)
minus (1198751199082
(119905) + 119876119908 (119905) + 119877)]
le1
Γ (120572)int
119905
0
(119905 minus 119904)120572minus1
119890minusℎ(119905minus119904)
times [(119910 (119904) minus 119908 (119904)) (119910 (119904) + 119908 (119904))
minus119896 (119910 (119904) minus 119908 (119904))] 119890minusℎ119904
119889119904
le1003817100381710038171003817119910 minus 119908
1003817100381710038171003817
1
Γ (120572)int
119905
0
119904120572minus1
119890minusℎ119904
119889119904
(44)
hence we have1003817100381710038171003817Θ119910 minus Θ119908
1003817100381710038171003817 lt1003817100381710038171003817119910 minus 119908
1003817100381710038171003817 (45)
which implies that the operator given by (43) has a uniquefixed point and consequently the given integral equation hasa unique solution 119910(119905) isin 119862(119868) Also we can see that
119868120572
(119875(119905)1199102
+ 119876(119905)119910 + 119877(119905))10038161003816100381610038161003816119905=0
= 119896 (46)
Now from (42) we have
119910 (119905) = [119905120572
Γ (120572 + 1)(1198751199102
0+ 1198761199100+ 119877)
+ 119868120572+1
(1198751015840
1199102
+ 21199101015840
119875 + 1198761015840
119910 + 1198761199101015840
+ 1198771015840
) ]
119889119910
119889119905= [
119905120572minus1
Γ (120572)(1198751199102
0+ 1198761199100+ 119877)
+ 119868120572
(1198751015840
1199102
+ 21199101015840
119875 + 1198761015840
119910 + 1198761199101015840
+ 1198771015840
) ]
119890minusℎ119905
1199101015840
(119905) = 119890minusℎ119905
[119905120572minus1
Γ (120572)(1198751199102
0+ 1198761199100+ 119877)
+ 119868120572
(1198751015840
1199102
+ 21199101015840
119875 + 1198761015840
119910 + 1198761199101015840
+ 1198771015840
) ]
(47)
from which we can deduce that 1199101015840(119905) isin 119862(119868) and 1199101015840
(119905) isin 119878
6 International Journal of Mathematics and Mathematical Sciences
Now again from (42) (43) and (46) we get
119889119910
119889119905=
119889
119889119905119868120572
[1198751199102
(119905) + 119876119910 (119905) + 119877]
1198681minus120572
119889119910
119889119905= 1198681minus120572
119889
119889119905119868120572
[1198751199102
(119905) + 119876119910 (119905) + 119877]
=119889
1198891199051198681minus120572
119868120572
[1198751199102
(119905) + 119876119910 (119905) + 119877]
119863120572
119910 (119905) =119889
119889119905119868 [1198751199102
(119905) + 119876119910 (119905) + 119877]
= 1198751199102
(119905) + 119876119910 (119905) + 119877
119910 (0) = 119868120572
(1198751199102
(119905) + 119876119910 (119905) + 119877)10038161003816100381610038161003816119905=0
= 119896
(48)
which implies that the integral equation (46) is equivalent tothe initial value problem (32) and the theorem is proved
42 Convergence Analyses Let
120595119896119899
(119905) =10038161003816100381610038161198860
10038161003816100381610038161198962
120595 (119886119896
0119905 minus 119899119887
0) (49)
where 120595119896119899(119905) form a wavelet basis for 1198712(119877) In particular
when 1198860= 2 and 119887
0= 1 120595
119896119899(119905) form an orthonormal basis
[29]By (14) let 119910(119905) = sum
119872minus1
119894=111988811198941205951119894(119905) be the solution of (31)
where 1198881119894= ⟨119910(119905) 120595
1119894(119905)⟩ for 119896 = 1 in which ⟨sdot sdot⟩ denotes
the inner product
119910 (119905) =
119899
sum
119894=1
⟨119910 (119905) 1205951119894(119905)⟩ 120595
1119894(119905) (50)
Let 120573119895= ⟨119910(119905) 120595(119905)⟩ where 120595(119905) = 120595
1119894(119905)
Let119909119899= sum119899
119895=1120573119895120595(119905119895) be a sequence of partial sumsThen
⟨119910 (119905) 119909119899⟩ = ⟨119910 (119905)
119899
sum
119895=1
120573119895120595 (119905119895)⟩
=
119899
sum
119895=1
120573119895⟨119910 (119905) 120595 (119905
119895)⟩
=
119899
sum
119895=1
120573119895120573119895
=
119899
sum
119895=1
10038161003816100381610038161003816120573119895
10038161003816100381610038161003816
2
(51)
Further
1003817100381710038171003817119909119899 minus 119909119898
10038171003817100381710038172
=
10038171003817100381710038171003817100381710038171003817100381710038171003817
119899
sum
119895=119898+1
120573119895120595 (119905119895)
10038171003817100381710038171003817100381710038171003817100381710038171003817
2
= ⟨
119899
sum
119894=119898+1
120573119894120595 (119905119894)
119899
sum
119895=119898+1
120573119895120595 (119905119895)⟩
=
119899
sum
119894=119898+1
119899
sum
119895=119898+1
120573119894120573119895⟨120595 (119905119894) 120595 (119905
119895)⟩
=
119899
sum
119895=119898+1
10038161003816100381610038161003816120573119895
10038161003816100381610038161003816
2
(52)
As 119899 rarr infin from Besselrsquos inequality we have suminfin119895=1
|120573119895|2 is
convergentIt implies that 119909
119899 is a Cauchy sequence and it converges
to 119909 (say)Also
⟨119909 minus 119910 (119905) 120595 (119905119895)⟩ = ⟨119909 120595 (119905
119895)⟩ minus ⟨119910 (119905) 120595 (119905
119895)⟩
= ⟨ lim119899rarrinfin
119909119899 120595 (119905119895)⟩ minus 120573
119895
= lim119899rarrinfin
⟨119909119899 120595 (119905119895)⟩ minus 120573
119895
= lim119899rarrinfin
⟨
119899
sum
119895=1
120573119895120595 (119905119895) 120595 (119905
119895)⟩ minus 120573
119895
= 120573119895minus 120573119895= 0
(53)
which is possible only if 119910(119905) = 119909 That is both 119910(119905) and 119909119899
converge to the same value which indeed give the guaranteeof convergence of LWM
5 Numerical Examples
In order to show the effectiveness of the Legendre waveletsmethod (LWM) we implement LWM to the nonlinearfractional Riccati differential equations All the numericalexperiments were carried out on a personal computer withsomeMATLAB codes The specifications of PC are Intel corei5 processor and with Turbo boost up to 31 GHz and 4GB ofDDR3 memory The following problems of nonlinear Riccatidifferential equations are solved with real coefficients
Example 1 Consider the following nonlinear fractional Ric-cati differential equation
119863120572
119910 (119905) = 1 + 2119910 (119905) minus 1199102
(119905) 0 lt 120572 le 1 (54)
with initial condition
119910 (0) = 0 (55)
Exact solution for 120572 = 1 was found to be
119910 (119905) = 1 + radic2 tanh(radic2119905 +1
2log(
radic2 minus 1
radic2 + 1)) (56)
International Journal of Mathematics and Mathematical Sciences 7
01 02 03 04 05 06 07 08 09 10
02
04
06
08
1
12
14
16
18
Exact
LWM
t
y(t)
Figure 1 Numerical results of Example 1 by LWM for 120572 = 1
The integral representation of (54) and (55) is given by
119868120572
(119863120572
119910 (119905)) = 119868120572
(1 + 2119910 (119905) minus 1199102
(119905)) (57)
119910 (119905) = 119910 (0) +119905120572
Γ (120572 + 1)+ 2119868120572
119910 (119905) minus 119868120572
1199102
(119905) (58)
Let
119910 (119905) = 119862119879
Ψ (119905) (59)
and then
119868120572
119910 (119905) = 119862119879
119868120572
Ψ (119905)
= 119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(60)
By substituting (59) and (60) into (58) we get the followingsystem of algebraic equations
119862119879
Ψ (119905) =119905120572
Γ (120572 + 1)+ 2119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
minus 119862119879
1198752120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(61)
By solving the above system of linear equations we can findthe vector 119862 Numerical results are obtained for differentvalues of 119896 119872 and 120572 Solution obtained by the proposedLWM approach for 120572 = 1 119896 = 1 and 119872 = 3 is given inFigure 1 and for different values of 120572 = 06 07 08 and 09and for 119896 = 2 and 119872 = 5 is graphically given in Figure 2It can be seen from Figure 1 that the solution obtained bythe proposed LWM approach is more close to the exactsolution Table 1 describes the efficiency of the proposedmethod by comparing with the methods in [20 22] throughtheir absolute errorThe following is used for the errors of theapproximation119910(119905) of 119910(119905) that is 119910minus119910 = max |119910(119905)minus119910(119905)|Table 1 shows that very high accuracies are obtained for 119896= 3and119872= 5 by the present method
0204
0608
10
09 08 07 06 05
t
y(t)
120572
minus05
00
05
15
20
10
Figure 2 Numerical results of Example 1 by LWM for differentvalues of 120572
Example 2 Consider another fractional-order Riccati differ-ential equation
119863120572
119910 (119905) = 1 minus 1199102
(119905) 0 lt 120572 le 1 (62)
with initial condition
119910 (0) = 0 (63)
Exact solution for the above equation was found to be
119910 (119905) =1198902119905
minus 1
1198902119905 + 1 (64)
The integral representation of (62) and (63) is given by
119910 (119905) = 119910 (0) +119905120572
Γ (120572 + 1)minus 119868120572
1199102
(119905) (65)
Let
119910 (119905) = 119862119879
Ψ (119905) (66)
and then
119868120572
119910 (119905) = 119862119879
119868120572
Ψ (119905)
= 119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(67)
By substituting (66) and (67) in (62) we get the followingsystem of algebraic equations
119862119879
Ψ (119905) =119905120572
Γ (120572 + 1)minus 119862119879
1198752120572
2119896minus1119872times2119896minus1119872Ψ (119905) (68)
By solving the above system of linear equations we can findthe vector 119862 Numerical results are obtained for differentvalues of 119896119872 and 120572 Results obtained by LWM for 120572 = 1 119896 =2 and119872= 3 are shown in Figure 3 and it can be seen from thefigure that solution given by the LWMmerely coincides withthe exact solution Figure 4 shows the obtained results of (62)and (63) by LWM for different values of 120572 and for 119896 = 2 and
8 International Journal of Mathematics and Mathematical Sciences
Table 1 Numerical results of Example 1 for 120572 = 1
119905 Exact solution Absolute error in [22] Absolute error in [20] LWM LWM LWM119872 = 2 119896 = 2 119872 = 3 119896 = 2 119872 = 5 119896 = 3
01 0099667 751119864 minus 09 930119864 minus 05 0 0 002 0197375 152119864 minus 06 293119864 minus 02 191119864 minus 08 0 003 0291312 393119864 minus 05 378119864 minus 03 272119864 minus 08 145119864 minus 13 004 0379948 432119864 minus 04 281119864 minus 03 165119864 minus 08 117119864 minus 13 194119864 minus 16
05 0462117 841119864 minus 04 980119864 minus 04 131119864 minus 08 328119864 minus 13 320119864 minus 16
06 0537049 294119864 minus 05 793119864 minus 03 198119864 minus 08 497119864 minus 13 124119864 minus 16
07 0604367 335119864 minus 04 944119864 minus 03 252119864 minus 08 632119864 minus 13 158119864 minus 16
08 0664036 544119864 minus 04 117119864 minus 02 294119864 minus 08 736119864 minus 13 184119864 minus 16
09 0716297 656119864 minus 09 396119864 minus 02 323119864 minus 08 812119864 minus 13 194119864 minus 16
10 0761594 253119864 minus 06 295119864 minus 02 263119864 minus 08 462119864 minus 13 199119864 minus 16
Table 2 Numerical results of Example 2 for 120572 = 1
119905 Exact solution Absolute error in [22] Absolute error in [20] LWM LWM LWM119872 = 2 119896 = 2 119872 = 3 119896 = 2 119872 = 5 119896 = 3
01 0110295 123119864 minus 15 281119864 minus 03 0 0 002 0241976 524119864 minus 15 383119864 minus 04 0 0 003 0395104 816119864 minus 15 123119864 minus 04 0 0 004 0567812 115119864 minus 12 286119864 minus 03 168119864 minus 12 166119864 minus 15 143119864 minus 1605 0756014 617119864 minus 12 438119864 minus 04 222119864 minus 12 212119864 minus 15 166119864 minus 1606 0953566 455119864 minus 11 519119864 minus 02 115119864 minus 12 110119864 minus 15 154119864 minus 1607 1152946 757119864 minus 10 214119864 minus 02 127119864 minus 12 111119864 minus 15 133119864 minus 1608 1346363 633119864 minus 09 142119864 minus 02 187119864 minus 12 201119864 minus 15 175119864 minus 1609 1526911 367119864 minus 08 698119864 minus 03 193119864 minus 12 266119864 minus 15 187119864 minus 1610 1689498 164119864 minus 07 496119864 minus 03 156119864 minus 12 166119864 minus 15 164119864 minus 16
119872 = 5 Table 2 describes the efficiency of the proposedmethod by comparing with the methods in [20 22] throughtheir absolute error Table 1 shows that very high accuraciesare obtained for 119896 = 3 and 119872 = 5 by the present methodand from these results we can identify that guarantee ofconvergence of the proposed LWM approach is very high
Example 3 Let us consider another problem of nonlinearRiccati differential equation
119863120572
119910 (119905) = 1199052
+ 1199102
(119905) 0 lt 120572 le 1 119905 ge 0 (69)
with initial condition
119910 (0) = 1 (70)
When 120572 = 1 its exact solution is given by
119910 (119905) =119905 (119869minus34
(1199052
2) Γ (14) + 211986934
(1199052
2) Γ (34))
11986914
(11990522) Γ (14) minus 2119869minus14
(11990522) Γ (34)
(71)
where 119869119899(119905) is the Bessel function of first kind
0
01
02
03
04
05
06
07
08
LWM
01 02 03 04 05 06 07 08 09 1
Exact
t
y(t)
Figure 3 Numerical results of Example 2 by LWM for 120572 = 1
The integral representation of (69) and (70) is given by
119910 (119905) = 1 +2
Γ (120572 + 1) + 2Γ (120572)119905120572+2
+ 119868120572
1199102
(119905) (72)
International Journal of Mathematics and Mathematical Sciences 9
0204
060810
09 08 07 06 05
t
y(t)
120572
00
01
02
03
04
0605
0807
081
Figure 4 Numerical results of Example 2 by LWM for differentvalues of 120572
Let
119910 (119905) = 119862119879
Ψ (119905) (73)
and then
119868120572
119910 (119905) = 119862119879
119868120572
Ψ (119905)
= 119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(74)
By substituting (73) and (74) into (69) we get the followingsystem of algebraic equations
119862119879
Ψ (119905) = 1 +2
Γ (120572 + 1) + 2Γ (120572)(119862119879
Ψ (119905))120572+2
+ 119862119879
1198752120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(75)
By solving the above system of linear equations we can findthe vector 119862 Numerical results are obtained for differentvalues of 119896 119872 and 120572 Obtained results for (69) and (70)are shown in Figures 5 and 6 Figure 5 shows the solutionsobtained by LWM for different values of 120572 and for 119896 = 2 and119872 = 4 Figure 6 compares the solution obtained by LWMwith the exact solution of (69) and (70) when 120572 = 1 119896 = 1and 119872 = 2 So far there are no published results of absoluteerror for this problem and hence we are unable to compareabsolute error of ourmethodwith the existingmethods Fromthese results we can see that the proposed LWM approachgives the solution which is very close to the exact solutionand outperformed recently developed approaches for thenonlinear fractional-order Riccati differential equations interms of solution quality and convergence criteria
6 Conclusions
Nonlinear fractional-order Riccati differential equations playan important role in the modeling of many biologicalphysical chemical and real life problemsTherefore it is nec-essary to develop a method which would give more accuratesolutions to such type of problems with greater convergence
0204
0608
10
09 0807 06
05
t
y(t)
120572
10
0
10
20
30
40
0 406
08
0 9 t
Figure 5 Numerical results of Example 3 by LWM for differentvalues of 120572
1
15
2
25
3
35
4
45
5
55
6
LWM
01 02 03 04 05 06 07 08 09 1
Exact
t
y(t)
Figure 6 Numerical results of Example 3 by LWM for 120572 = 1
criteria In this work a Legendrersquos wavelet operational matrixmethod called LWM was proposed for solving nonlinearfractional-order Riccati differential equations Comparisonwas made for the solutions obtained by the proposedmethodand with the other recent approaches developed for thesame problem through their error analysis obtained resultsshow that the proposed LWM yields more accurate andreliable solutions even for small values of 119872 and 119896 whichassures the best approximate solution in less computationaleffort Further we have discussed the convergence criteriaof proposed scheme which indeed provides the guaranteeof consistency and stability of the proposed LWM schemefor the solutions of nonlinear fractional Riccati differentialequations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
10 International Journal of Mathematics and Mathematical Sciences
References
[1] N A Khan M Jamil A Ara and S Das ldquoExplicit solution fortime-fractional batch reactor systemrdquo International Journal ofChemical Reactor Engineering vol 9 article A91 2011
[2] V Feliu-Batlle R R Perez and L S Rodrıguez ldquoFractionalrobust control of main irrigation canals with variable dynamicparametersrdquoControl Engineering Practice vol 15 no 6 pp 673ndash686 2007
[3] I Podlubny ldquoFractional-order systems and 119875119868120582
119863120583-controllersrdquo
IEEE Transactions on Automatic Control vol 44 no 1 pp 208ndash214 1999
[4] R Garrappa ldquoOn some explicit Adams multistep methods forfractional differential equationsrdquo Journal of Computational andApplied Mathematics vol 229 no 2 pp 392ndash399 2009
[5] M Jamil and N A Khan ldquoSlip effects on fractional viscoelasticfluidsrdquo International Journal of Differential Equations vol 2011Article ID 193813 19 pages 2011
[6] F Mohammadi and M M Hosseini ldquoA comparative study ofnumerical methods for solving quadratic Riccati differentialequationsrdquo Journal of the Franklin Institute vol 348 no 2 pp156ndash164 2011
[7] R Shankar Principles of Quantum Mechanics Plenum PressNew York NY USA 1980
[8] S FragaM J Garcıa de la Vega and E S FragaTheSchrodingerand Riccati Equations vol 70 of Lect Notes Chem 1999
[9] L B Burrows and M Cohen ldquoSchrodingerrsquos wave equation-A lie algebra treatmentrdquo in Fundamental World of QuantumChemistry A Tribute to the Memory of Per-Olov Lowdin EJ Brandas and E S Kryachko Eds Kluwer Dordrecht TheNetherlands 2004
[10] S Abbasbandy ldquoHomotopy perturbation method for quadraticRiccati differential equation and comparison with Adomianrsquosdecomposition methodrdquo Applied Mathematics and Computa-tion vol 172 no 1 pp 485ndash490 2006
[11] Z Odibat and S Momani ldquoModified homotopy perturbationmethod application to quadratic Riccati differential equationof fractional orderrdquo Chaos Solitons amp Fractals vol 36 no 1 pp167ndash174 2008
[12] N A Khan A Ara and M Jamil ldquoAn efficient approach forsolving the Riccati equation with fractional ordersrdquo Computersamp Mathematics with Applications vol 61 no 9 pp 2683ndash26892011
[13] H Aminikhah and M Hemmatnezhad ldquoAn efficient methodfor quadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 4 pp835ndash839 2010
[14] S Abbasbandy ldquoIterated Hersquos homotopy perturbation methodfor quadratic Riccati differential equationrdquo Applied Mathemat-ics and Computation vol 175 no 1 pp 581ndash589 2006
[15] J Cang Y Tan H Xu and S Liao ldquoSeries solutions of non-lin-ear Riccati differential equations with fractional orderrdquo ChaosSolitons and Fractals vol 40 no 1 pp 1ndash9 2009
[16] Y Tan and S Abbasbandy ldquoHomotopy analysis method forquadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 3 pp539ndash546 2008
[17] M Gulsu and M Sezer ldquoOn the solution of the Riccati equa-tion by the Taylor matrix methodrdquo Applied Mathematics andComputation vol 176 no 2 pp 414ndash421 2006
[18] Y Li and LHu ldquoSolving fractional Riccati differential equationsusing Haar waveletrdquo in Proceedings of the 3rd InternationalConference on Information and Computing (ICIC 10) pp 314ndash317 Wuxi China June 2010
[19] N A Khan and A Ara ldquoFractional-order Riccati differentialequation analytical approximation and numerical resultsrdquoAdvances in Difference Equations vol 2013 article 185 2013
[20] M A Z Raja J A Khan and I M Qureshi ldquoA new stochasticapproach for solution of Riccati differential equation of frac-tional orderrdquo Annals of Mathematics and Artificial Intelligencevol 60 no 3-4 pp 229ndash250 2010
[21] M Merdan ldquoOn the solutions fractional Riccati differentialequation with modified RIEmann-Liouville derivativerdquo Inter-national Journal of Differential Equations vol 2012 Article ID346089 17 pages 2012
[22] N H Sweilam M M Khader and A M S Mahdy ldquoNumericalstudies for solving fractional Riccati differential equationrdquoApplications and AppliedMathematics vol 7 no 2 pp 595ndash6082012
[23] C K ChuiWavelets A Mathematical Tool for Signal ProcessingSIAM Philadelphia Pa USA 1997
[24] G Beylkin R Coifman and V Rokhlin ldquoFast wavelet trans-forms and numerical algorithms Irdquo Communications on Pureand Applied Mathematics vol 44 no 2 pp 141ndash183 1991
[25] M ur Rehman and R Ali Khan ldquoThe Legendre wavelet methodfor solving fractional differential equationsrdquoCommunications inNonlinear Science and Numerical Simulation vol 16 no 11 pp4163ndash4173 2011
[26] S Balaji ldquoA new approach for solving Duffing equationsinvolving both integral and non-integral forcing termsrdquo AinShams Engineering Journal 2014
[27] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent IIrdquo Geophysical Journal of the RoyalAstronomical Society vol 13 pp 529ndash539 1967
[28] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-Holland Mathematics Studies Elsevier 2006
[29] J S Gu and W S Jiang ldquoThe Haar wavelets operational matrixof integrationrdquo International Journal of Systems Science vol 27no 7 pp 623ndash628 1996
[30] M Razzaghi and S Yousefi ldquoLegendre wavelets direct methodfor variational problemsrdquo Mathematics and Computers in Sim-ulation vol 53 no 3 pp 185ndash192 2000
[31] M Razzaghi and S Yousefi ldquoLegendre wavelets method forconstrained optimal control problemsrdquo Mathematical Methodsin the Applied Sciences vol 25 no 7 pp 529ndash539 2002
[32] A Kilicman and Z A A Al Zhour ldquoKronecker operationalmatrices for fractional calculus and some applicationsrdquo AppliedMathematics and Computation vol 187 no 1 pp 250ndash265 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 International Journal of Mathematics and Mathematical Sciences
The functions 119887119894are disjoint and orthogonal That is
119887119894(119905) 119887119895(119905) =
119887119894(119905) 119894 = 119895
0 119894 = 119895
int
1
0
119887119894(119905) 119887119895(119905) 119889119905 =
1
119898 119894 = 119895
0 119894 = 119895
(20)
The orthogonality property of block-pulse function isobtained from the disjointness property
An arbitrary function 119891 isin 1198712[0 1) can be expanded into
block-pulse functions as
119891 (119905) asymp
119898minus1
sum
119894=0
119891119894119887119894(119905) = 119891
119879
119861 (119905) (21)
where119891119894are the coefficients of the block-pulse function given
by
119891119894=119898
119897int
119897
0
119891 (119905) 119887119894(119905) (22)
The Legendre wavelets can be expanded into119898-set of block-pulse functions as
Ψ (119905) = 120601times
119861 (119905) (23)
where 119861(119905) = [1198870(119905) 1198871(119905) sdot sdot sdot 119887
119894(119905) sdot sdot sdot 119887
119898minus1(119905)]119879
The fractional integral of block-pulse function vector canbe written as
(119868120572
119861) (119905) = 119865120572
119898times119898119861 (119905) (24)
where 119865120572119898times119898
is given in [32]Now we introduce the derivation process of the Legendre
wavelet operational matrix of the fractional integration
(119868120572
Ψ) (119905) asymp 119875120572
timesΨ (119905) (25)
where the -square matrix 119875120572times
is called Legendre waveletoperational matrix of the fractional integration
Using (23) and (25) we have
(119868120572
Ψ) (119905) asymp (119868120572
120601times
119861) (119905) = 120601times
(119868120572
119861) (119905)
asymp 120601times
(119865120572
119861) (119905)
(26)
From (25) and (26) we get
119875120572
timesΨ (119905) = 120601
times119865120572
119861 (119905) (27)
and by (23) (27) becomes
119875120572
times120601times
119861 (119905) = 120601times
119865120572
119861 (119905) (28)
Then the Legendre wavelet operational matrix119875120572times
of frac-tional integration is given by
119875120572
times= 120601times
119865120572
120601minus1
times (29)
Following is the Legendre wavelet operationalmatrix119875120572times
offractional-order integration for the particular values of 119896 = 2119872 = 3 and 120572 = 05
11987505
6times6= (
(
05415 04324 01819 minus00871 minus00179 00154
0 05415 0 01819 0 minus00179
minus02046 0071 02243 minus00449 00798 00119
0 minus02046 0 02243 0 00798
01781 02506 minus00252 minus00652 01555 00143
0 01781 0 minus00252 0 01555
)
)
(30)
4 Application to Fractional RiccatiDifferential Equation
In this section we use the generalized Legendre waveletoperational matrix to solve nonlinear Riccati differentialequation and we discuss the existence and uniqueness ofsolutions with initial conditions and convergence criteria ofthe proposed LWM approach
Consider the fractional-order Riccati differential equa-tion of the form
119863120572
119910 (119905) = 119875 (119905) 1199102
+ 119876 (119905) 119910 + 119877 (119905)
119905 gt 0 0 lt 120572 le 1
(31)
subject to the initial condition
119910 (0) = 119896 (32)
Let us suppose that the functions119863120572119910(119905) 119875(119905)119876(119905) and 119877(119905)are approximated using Legendre wavelet as follows
119863120572
119910 (119905) = 119880119879
Ψ (119905) 119875 (119905) = 119881119879
Ψ (119905)
119876 (119905) = 119882119879
Ψ (119905) 119877 (119905) = 119883119879
Ψ (119905)
(33)
where 119880 119881119882119883 and Ψ(119905) are given in (14)Using (6) we can write
119910 (119905) = 119868120572
(119863120572
119910 (119905)) minus 119910 (0) (34)
International Journal of Mathematics and Mathematical Sciences 5
By (25) and (32) (34) leads to
119910 (119905) asymp 119880119879
119875120572
Ψ (119905) + 119884119879
0Ψ (119905) = 119862
119879
Ψ (119905) (35)
where 119910(0) = 119896 asymp 119884119879
0Ψ(119905) 119862 = (119880
119879
119875120572
times+ 119884119879
0)119879
Substituting (33) and (35) into (31) we have
119880119879
Ψ (119905) = 119881119879
Ψ (119905) [119862119879
Ψ (119905)]2
+119882119879
Ψ (119905) 119862119879
Ψ (119905) + 119883119879
Ψ (119905)
(36)
Substituting (23) into (36) we have
119880119879
120601times
= 119881119879
[119862119879
120601times
]2
+119882119879
119862119879
120601times
+ 119883119879
(37)
where 119862 119881 119882 and 120601times
are known Equation (37) repre-sents a system of nonlinear equations with unknown vector119880 This system of nonlinear equations can be solved byNewton method for the unknown vector 119880 and we can getthe approximation solution by including 119880 into (35)
41 Existence and Uniqueness of Solutions Consider thefractional-order Riccati differential equation of the forms(31) and (32) The nonlinear term in (31) is 119910
2 and119875(119905) 119876(119905) and 119877(119905) are known functions For 120572 = 1 thefractional-order Riccati converts into the classical Riccatidifferential equation
Definition 1 Let 119868 = [0 119897] 119897 lt infin and 119862(119868) be the class of allcontinuous functions defined on 119868 with the norm
10038171003817100381710038171199101003817100381710038171003817 = sup119905isin119868
10038161003816100381610038161003816119890minusℎ119905
119910 (119905)10038161003816100381610038161003816 ℎ gt 0 (38)
which is equivalent to the sup norm of 119910 That is 119910 =
sup119905isin119868
|119890minusℎ119905
119910(119905)|
RemarkAssume that solution 119910(119905) of fractional-order Riccatidifferential equations (31) and (32) belongs to the space 119878 =
119910 isin 119877 |119910| le 119888 119888 is any constant in order to study theexistence and uniqueness of the initial value problem
Definition 2 The space of integrable functions 1198711[0 119897] in the
interval [0 119897] is defined as
1198711[0 119897] = 119906 (119905) int
119897
0
|119906 (119905)| 119889119905 lt infin (39)
Theorem 3 The initial value problem given by (31) and (32)has a unique solution
119910 isin 119862 (119868) 1199101015840
isin 119883 = 119910 isin 1198711[0 119897]
10038171003817100381710038171199101003817100381710038171003817 =
10038171003817100381710038171003817119890minusℎ119905
119910(119905)100381710038171003817100381710038171198711
(40)
Proof By (1) the fractional differential equation (31) can bewritten as
1198681minus120572
119889119910 (119905)
119889119905= 119875 (119905) 119910
2
+ 119876 (119905) 119910 + 119877 (119905) (41)
and becomes
119910 (119905) = 119868120572
(119875 (119905) 1199102
+ 119876 (119905) 119910 + 119877 (119905)) (42)
Now we define the operator Θ 119862(119868) rarr 119862(119868) by
Θ119910 (119905) = 119868120572
(119875 (119905) 1199102
+ 119876 (119905) 119910 + 119877 (119905)) (43)
and then
119890minusℎ119905
(Θ119910 minus Θ119908)
= 119890minusℎ119905
119868120572
[(1198751199102
(119905) + 119876119910 (119905) + 119877)
minus (1198751199082
(119905) + 119876119908 (119905) + 119877)]
le1
Γ (120572)int
119905
0
(119905 minus 119904)120572minus1
119890minusℎ(119905minus119904)
times [(119910 (119904) minus 119908 (119904)) (119910 (119904) + 119908 (119904))
minus119896 (119910 (119904) minus 119908 (119904))] 119890minusℎ119904
119889119904
le1003817100381710038171003817119910 minus 119908
1003817100381710038171003817
1
Γ (120572)int
119905
0
119904120572minus1
119890minusℎ119904
119889119904
(44)
hence we have1003817100381710038171003817Θ119910 minus Θ119908
1003817100381710038171003817 lt1003817100381710038171003817119910 minus 119908
1003817100381710038171003817 (45)
which implies that the operator given by (43) has a uniquefixed point and consequently the given integral equation hasa unique solution 119910(119905) isin 119862(119868) Also we can see that
119868120572
(119875(119905)1199102
+ 119876(119905)119910 + 119877(119905))10038161003816100381610038161003816119905=0
= 119896 (46)
Now from (42) we have
119910 (119905) = [119905120572
Γ (120572 + 1)(1198751199102
0+ 1198761199100+ 119877)
+ 119868120572+1
(1198751015840
1199102
+ 21199101015840
119875 + 1198761015840
119910 + 1198761199101015840
+ 1198771015840
) ]
119889119910
119889119905= [
119905120572minus1
Γ (120572)(1198751199102
0+ 1198761199100+ 119877)
+ 119868120572
(1198751015840
1199102
+ 21199101015840
119875 + 1198761015840
119910 + 1198761199101015840
+ 1198771015840
) ]
119890minusℎ119905
1199101015840
(119905) = 119890minusℎ119905
[119905120572minus1
Γ (120572)(1198751199102
0+ 1198761199100+ 119877)
+ 119868120572
(1198751015840
1199102
+ 21199101015840
119875 + 1198761015840
119910 + 1198761199101015840
+ 1198771015840
) ]
(47)
from which we can deduce that 1199101015840(119905) isin 119862(119868) and 1199101015840
(119905) isin 119878
6 International Journal of Mathematics and Mathematical Sciences
Now again from (42) (43) and (46) we get
119889119910
119889119905=
119889
119889119905119868120572
[1198751199102
(119905) + 119876119910 (119905) + 119877]
1198681minus120572
119889119910
119889119905= 1198681minus120572
119889
119889119905119868120572
[1198751199102
(119905) + 119876119910 (119905) + 119877]
=119889
1198891199051198681minus120572
119868120572
[1198751199102
(119905) + 119876119910 (119905) + 119877]
119863120572
119910 (119905) =119889
119889119905119868 [1198751199102
(119905) + 119876119910 (119905) + 119877]
= 1198751199102
(119905) + 119876119910 (119905) + 119877
119910 (0) = 119868120572
(1198751199102
(119905) + 119876119910 (119905) + 119877)10038161003816100381610038161003816119905=0
= 119896
(48)
which implies that the integral equation (46) is equivalent tothe initial value problem (32) and the theorem is proved
42 Convergence Analyses Let
120595119896119899
(119905) =10038161003816100381610038161198860
10038161003816100381610038161198962
120595 (119886119896
0119905 minus 119899119887
0) (49)
where 120595119896119899(119905) form a wavelet basis for 1198712(119877) In particular
when 1198860= 2 and 119887
0= 1 120595
119896119899(119905) form an orthonormal basis
[29]By (14) let 119910(119905) = sum
119872minus1
119894=111988811198941205951119894(119905) be the solution of (31)
where 1198881119894= ⟨119910(119905) 120595
1119894(119905)⟩ for 119896 = 1 in which ⟨sdot sdot⟩ denotes
the inner product
119910 (119905) =
119899
sum
119894=1
⟨119910 (119905) 1205951119894(119905)⟩ 120595
1119894(119905) (50)
Let 120573119895= ⟨119910(119905) 120595(119905)⟩ where 120595(119905) = 120595
1119894(119905)
Let119909119899= sum119899
119895=1120573119895120595(119905119895) be a sequence of partial sumsThen
⟨119910 (119905) 119909119899⟩ = ⟨119910 (119905)
119899
sum
119895=1
120573119895120595 (119905119895)⟩
=
119899
sum
119895=1
120573119895⟨119910 (119905) 120595 (119905
119895)⟩
=
119899
sum
119895=1
120573119895120573119895
=
119899
sum
119895=1
10038161003816100381610038161003816120573119895
10038161003816100381610038161003816
2
(51)
Further
1003817100381710038171003817119909119899 minus 119909119898
10038171003817100381710038172
=
10038171003817100381710038171003817100381710038171003817100381710038171003817
119899
sum
119895=119898+1
120573119895120595 (119905119895)
10038171003817100381710038171003817100381710038171003817100381710038171003817
2
= ⟨
119899
sum
119894=119898+1
120573119894120595 (119905119894)
119899
sum
119895=119898+1
120573119895120595 (119905119895)⟩
=
119899
sum
119894=119898+1
119899
sum
119895=119898+1
120573119894120573119895⟨120595 (119905119894) 120595 (119905
119895)⟩
=
119899
sum
119895=119898+1
10038161003816100381610038161003816120573119895
10038161003816100381610038161003816
2
(52)
As 119899 rarr infin from Besselrsquos inequality we have suminfin119895=1
|120573119895|2 is
convergentIt implies that 119909
119899 is a Cauchy sequence and it converges
to 119909 (say)Also
⟨119909 minus 119910 (119905) 120595 (119905119895)⟩ = ⟨119909 120595 (119905
119895)⟩ minus ⟨119910 (119905) 120595 (119905
119895)⟩
= ⟨ lim119899rarrinfin
119909119899 120595 (119905119895)⟩ minus 120573
119895
= lim119899rarrinfin
⟨119909119899 120595 (119905119895)⟩ minus 120573
119895
= lim119899rarrinfin
⟨
119899
sum
119895=1
120573119895120595 (119905119895) 120595 (119905
119895)⟩ minus 120573
119895
= 120573119895minus 120573119895= 0
(53)
which is possible only if 119910(119905) = 119909 That is both 119910(119905) and 119909119899
converge to the same value which indeed give the guaranteeof convergence of LWM
5 Numerical Examples
In order to show the effectiveness of the Legendre waveletsmethod (LWM) we implement LWM to the nonlinearfractional Riccati differential equations All the numericalexperiments were carried out on a personal computer withsomeMATLAB codes The specifications of PC are Intel corei5 processor and with Turbo boost up to 31 GHz and 4GB ofDDR3 memory The following problems of nonlinear Riccatidifferential equations are solved with real coefficients
Example 1 Consider the following nonlinear fractional Ric-cati differential equation
119863120572
119910 (119905) = 1 + 2119910 (119905) minus 1199102
(119905) 0 lt 120572 le 1 (54)
with initial condition
119910 (0) = 0 (55)
Exact solution for 120572 = 1 was found to be
119910 (119905) = 1 + radic2 tanh(radic2119905 +1
2log(
radic2 minus 1
radic2 + 1)) (56)
International Journal of Mathematics and Mathematical Sciences 7
01 02 03 04 05 06 07 08 09 10
02
04
06
08
1
12
14
16
18
Exact
LWM
t
y(t)
Figure 1 Numerical results of Example 1 by LWM for 120572 = 1
The integral representation of (54) and (55) is given by
119868120572
(119863120572
119910 (119905)) = 119868120572
(1 + 2119910 (119905) minus 1199102
(119905)) (57)
119910 (119905) = 119910 (0) +119905120572
Γ (120572 + 1)+ 2119868120572
119910 (119905) minus 119868120572
1199102
(119905) (58)
Let
119910 (119905) = 119862119879
Ψ (119905) (59)
and then
119868120572
119910 (119905) = 119862119879
119868120572
Ψ (119905)
= 119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(60)
By substituting (59) and (60) into (58) we get the followingsystem of algebraic equations
119862119879
Ψ (119905) =119905120572
Γ (120572 + 1)+ 2119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
minus 119862119879
1198752120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(61)
By solving the above system of linear equations we can findthe vector 119862 Numerical results are obtained for differentvalues of 119896 119872 and 120572 Solution obtained by the proposedLWM approach for 120572 = 1 119896 = 1 and 119872 = 3 is given inFigure 1 and for different values of 120572 = 06 07 08 and 09and for 119896 = 2 and 119872 = 5 is graphically given in Figure 2It can be seen from Figure 1 that the solution obtained bythe proposed LWM approach is more close to the exactsolution Table 1 describes the efficiency of the proposedmethod by comparing with the methods in [20 22] throughtheir absolute errorThe following is used for the errors of theapproximation119910(119905) of 119910(119905) that is 119910minus119910 = max |119910(119905)minus119910(119905)|Table 1 shows that very high accuracies are obtained for 119896= 3and119872= 5 by the present method
0204
0608
10
09 08 07 06 05
t
y(t)
120572
minus05
00
05
15
20
10
Figure 2 Numerical results of Example 1 by LWM for differentvalues of 120572
Example 2 Consider another fractional-order Riccati differ-ential equation
119863120572
119910 (119905) = 1 minus 1199102
(119905) 0 lt 120572 le 1 (62)
with initial condition
119910 (0) = 0 (63)
Exact solution for the above equation was found to be
119910 (119905) =1198902119905
minus 1
1198902119905 + 1 (64)
The integral representation of (62) and (63) is given by
119910 (119905) = 119910 (0) +119905120572
Γ (120572 + 1)minus 119868120572
1199102
(119905) (65)
Let
119910 (119905) = 119862119879
Ψ (119905) (66)
and then
119868120572
119910 (119905) = 119862119879
119868120572
Ψ (119905)
= 119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(67)
By substituting (66) and (67) in (62) we get the followingsystem of algebraic equations
119862119879
Ψ (119905) =119905120572
Γ (120572 + 1)minus 119862119879
1198752120572
2119896minus1119872times2119896minus1119872Ψ (119905) (68)
By solving the above system of linear equations we can findthe vector 119862 Numerical results are obtained for differentvalues of 119896119872 and 120572 Results obtained by LWM for 120572 = 1 119896 =2 and119872= 3 are shown in Figure 3 and it can be seen from thefigure that solution given by the LWMmerely coincides withthe exact solution Figure 4 shows the obtained results of (62)and (63) by LWM for different values of 120572 and for 119896 = 2 and
8 International Journal of Mathematics and Mathematical Sciences
Table 1 Numerical results of Example 1 for 120572 = 1
119905 Exact solution Absolute error in [22] Absolute error in [20] LWM LWM LWM119872 = 2 119896 = 2 119872 = 3 119896 = 2 119872 = 5 119896 = 3
01 0099667 751119864 minus 09 930119864 minus 05 0 0 002 0197375 152119864 minus 06 293119864 minus 02 191119864 minus 08 0 003 0291312 393119864 minus 05 378119864 minus 03 272119864 minus 08 145119864 minus 13 004 0379948 432119864 minus 04 281119864 minus 03 165119864 minus 08 117119864 minus 13 194119864 minus 16
05 0462117 841119864 minus 04 980119864 minus 04 131119864 minus 08 328119864 minus 13 320119864 minus 16
06 0537049 294119864 minus 05 793119864 minus 03 198119864 minus 08 497119864 minus 13 124119864 minus 16
07 0604367 335119864 minus 04 944119864 minus 03 252119864 minus 08 632119864 minus 13 158119864 minus 16
08 0664036 544119864 minus 04 117119864 minus 02 294119864 minus 08 736119864 minus 13 184119864 minus 16
09 0716297 656119864 minus 09 396119864 minus 02 323119864 minus 08 812119864 minus 13 194119864 minus 16
10 0761594 253119864 minus 06 295119864 minus 02 263119864 minus 08 462119864 minus 13 199119864 minus 16
Table 2 Numerical results of Example 2 for 120572 = 1
119905 Exact solution Absolute error in [22] Absolute error in [20] LWM LWM LWM119872 = 2 119896 = 2 119872 = 3 119896 = 2 119872 = 5 119896 = 3
01 0110295 123119864 minus 15 281119864 minus 03 0 0 002 0241976 524119864 minus 15 383119864 minus 04 0 0 003 0395104 816119864 minus 15 123119864 minus 04 0 0 004 0567812 115119864 minus 12 286119864 minus 03 168119864 minus 12 166119864 minus 15 143119864 minus 1605 0756014 617119864 minus 12 438119864 minus 04 222119864 minus 12 212119864 minus 15 166119864 minus 1606 0953566 455119864 minus 11 519119864 minus 02 115119864 minus 12 110119864 minus 15 154119864 minus 1607 1152946 757119864 minus 10 214119864 minus 02 127119864 minus 12 111119864 minus 15 133119864 minus 1608 1346363 633119864 minus 09 142119864 minus 02 187119864 minus 12 201119864 minus 15 175119864 minus 1609 1526911 367119864 minus 08 698119864 minus 03 193119864 minus 12 266119864 minus 15 187119864 minus 1610 1689498 164119864 minus 07 496119864 minus 03 156119864 minus 12 166119864 minus 15 164119864 minus 16
119872 = 5 Table 2 describes the efficiency of the proposedmethod by comparing with the methods in [20 22] throughtheir absolute error Table 1 shows that very high accuraciesare obtained for 119896 = 3 and 119872 = 5 by the present methodand from these results we can identify that guarantee ofconvergence of the proposed LWM approach is very high
Example 3 Let us consider another problem of nonlinearRiccati differential equation
119863120572
119910 (119905) = 1199052
+ 1199102
(119905) 0 lt 120572 le 1 119905 ge 0 (69)
with initial condition
119910 (0) = 1 (70)
When 120572 = 1 its exact solution is given by
119910 (119905) =119905 (119869minus34
(1199052
2) Γ (14) + 211986934
(1199052
2) Γ (34))
11986914
(11990522) Γ (14) minus 2119869minus14
(11990522) Γ (34)
(71)
where 119869119899(119905) is the Bessel function of first kind
0
01
02
03
04
05
06
07
08
LWM
01 02 03 04 05 06 07 08 09 1
Exact
t
y(t)
Figure 3 Numerical results of Example 2 by LWM for 120572 = 1
The integral representation of (69) and (70) is given by
119910 (119905) = 1 +2
Γ (120572 + 1) + 2Γ (120572)119905120572+2
+ 119868120572
1199102
(119905) (72)
International Journal of Mathematics and Mathematical Sciences 9
0204
060810
09 08 07 06 05
t
y(t)
120572
00
01
02
03
04
0605
0807
081
Figure 4 Numerical results of Example 2 by LWM for differentvalues of 120572
Let
119910 (119905) = 119862119879
Ψ (119905) (73)
and then
119868120572
119910 (119905) = 119862119879
119868120572
Ψ (119905)
= 119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(74)
By substituting (73) and (74) into (69) we get the followingsystem of algebraic equations
119862119879
Ψ (119905) = 1 +2
Γ (120572 + 1) + 2Γ (120572)(119862119879
Ψ (119905))120572+2
+ 119862119879
1198752120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(75)
By solving the above system of linear equations we can findthe vector 119862 Numerical results are obtained for differentvalues of 119896 119872 and 120572 Obtained results for (69) and (70)are shown in Figures 5 and 6 Figure 5 shows the solutionsobtained by LWM for different values of 120572 and for 119896 = 2 and119872 = 4 Figure 6 compares the solution obtained by LWMwith the exact solution of (69) and (70) when 120572 = 1 119896 = 1and 119872 = 2 So far there are no published results of absoluteerror for this problem and hence we are unable to compareabsolute error of ourmethodwith the existingmethods Fromthese results we can see that the proposed LWM approachgives the solution which is very close to the exact solutionand outperformed recently developed approaches for thenonlinear fractional-order Riccati differential equations interms of solution quality and convergence criteria
6 Conclusions
Nonlinear fractional-order Riccati differential equations playan important role in the modeling of many biologicalphysical chemical and real life problemsTherefore it is nec-essary to develop a method which would give more accuratesolutions to such type of problems with greater convergence
0204
0608
10
09 0807 06
05
t
y(t)
120572
10
0
10
20
30
40
0 406
08
0 9 t
Figure 5 Numerical results of Example 3 by LWM for differentvalues of 120572
1
15
2
25
3
35
4
45
5
55
6
LWM
01 02 03 04 05 06 07 08 09 1
Exact
t
y(t)
Figure 6 Numerical results of Example 3 by LWM for 120572 = 1
criteria In this work a Legendrersquos wavelet operational matrixmethod called LWM was proposed for solving nonlinearfractional-order Riccati differential equations Comparisonwas made for the solutions obtained by the proposedmethodand with the other recent approaches developed for thesame problem through their error analysis obtained resultsshow that the proposed LWM yields more accurate andreliable solutions even for small values of 119872 and 119896 whichassures the best approximate solution in less computationaleffort Further we have discussed the convergence criteriaof proposed scheme which indeed provides the guaranteeof consistency and stability of the proposed LWM schemefor the solutions of nonlinear fractional Riccati differentialequations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
10 International Journal of Mathematics and Mathematical Sciences
References
[1] N A Khan M Jamil A Ara and S Das ldquoExplicit solution fortime-fractional batch reactor systemrdquo International Journal ofChemical Reactor Engineering vol 9 article A91 2011
[2] V Feliu-Batlle R R Perez and L S Rodrıguez ldquoFractionalrobust control of main irrigation canals with variable dynamicparametersrdquoControl Engineering Practice vol 15 no 6 pp 673ndash686 2007
[3] I Podlubny ldquoFractional-order systems and 119875119868120582
119863120583-controllersrdquo
IEEE Transactions on Automatic Control vol 44 no 1 pp 208ndash214 1999
[4] R Garrappa ldquoOn some explicit Adams multistep methods forfractional differential equationsrdquo Journal of Computational andApplied Mathematics vol 229 no 2 pp 392ndash399 2009
[5] M Jamil and N A Khan ldquoSlip effects on fractional viscoelasticfluidsrdquo International Journal of Differential Equations vol 2011Article ID 193813 19 pages 2011
[6] F Mohammadi and M M Hosseini ldquoA comparative study ofnumerical methods for solving quadratic Riccati differentialequationsrdquo Journal of the Franklin Institute vol 348 no 2 pp156ndash164 2011
[7] R Shankar Principles of Quantum Mechanics Plenum PressNew York NY USA 1980
[8] S FragaM J Garcıa de la Vega and E S FragaTheSchrodingerand Riccati Equations vol 70 of Lect Notes Chem 1999
[9] L B Burrows and M Cohen ldquoSchrodingerrsquos wave equation-A lie algebra treatmentrdquo in Fundamental World of QuantumChemistry A Tribute to the Memory of Per-Olov Lowdin EJ Brandas and E S Kryachko Eds Kluwer Dordrecht TheNetherlands 2004
[10] S Abbasbandy ldquoHomotopy perturbation method for quadraticRiccati differential equation and comparison with Adomianrsquosdecomposition methodrdquo Applied Mathematics and Computa-tion vol 172 no 1 pp 485ndash490 2006
[11] Z Odibat and S Momani ldquoModified homotopy perturbationmethod application to quadratic Riccati differential equationof fractional orderrdquo Chaos Solitons amp Fractals vol 36 no 1 pp167ndash174 2008
[12] N A Khan A Ara and M Jamil ldquoAn efficient approach forsolving the Riccati equation with fractional ordersrdquo Computersamp Mathematics with Applications vol 61 no 9 pp 2683ndash26892011
[13] H Aminikhah and M Hemmatnezhad ldquoAn efficient methodfor quadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 4 pp835ndash839 2010
[14] S Abbasbandy ldquoIterated Hersquos homotopy perturbation methodfor quadratic Riccati differential equationrdquo Applied Mathemat-ics and Computation vol 175 no 1 pp 581ndash589 2006
[15] J Cang Y Tan H Xu and S Liao ldquoSeries solutions of non-lin-ear Riccati differential equations with fractional orderrdquo ChaosSolitons and Fractals vol 40 no 1 pp 1ndash9 2009
[16] Y Tan and S Abbasbandy ldquoHomotopy analysis method forquadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 3 pp539ndash546 2008
[17] M Gulsu and M Sezer ldquoOn the solution of the Riccati equa-tion by the Taylor matrix methodrdquo Applied Mathematics andComputation vol 176 no 2 pp 414ndash421 2006
[18] Y Li and LHu ldquoSolving fractional Riccati differential equationsusing Haar waveletrdquo in Proceedings of the 3rd InternationalConference on Information and Computing (ICIC 10) pp 314ndash317 Wuxi China June 2010
[19] N A Khan and A Ara ldquoFractional-order Riccati differentialequation analytical approximation and numerical resultsrdquoAdvances in Difference Equations vol 2013 article 185 2013
[20] M A Z Raja J A Khan and I M Qureshi ldquoA new stochasticapproach for solution of Riccati differential equation of frac-tional orderrdquo Annals of Mathematics and Artificial Intelligencevol 60 no 3-4 pp 229ndash250 2010
[21] M Merdan ldquoOn the solutions fractional Riccati differentialequation with modified RIEmann-Liouville derivativerdquo Inter-national Journal of Differential Equations vol 2012 Article ID346089 17 pages 2012
[22] N H Sweilam M M Khader and A M S Mahdy ldquoNumericalstudies for solving fractional Riccati differential equationrdquoApplications and AppliedMathematics vol 7 no 2 pp 595ndash6082012
[23] C K ChuiWavelets A Mathematical Tool for Signal ProcessingSIAM Philadelphia Pa USA 1997
[24] G Beylkin R Coifman and V Rokhlin ldquoFast wavelet trans-forms and numerical algorithms Irdquo Communications on Pureand Applied Mathematics vol 44 no 2 pp 141ndash183 1991
[25] M ur Rehman and R Ali Khan ldquoThe Legendre wavelet methodfor solving fractional differential equationsrdquoCommunications inNonlinear Science and Numerical Simulation vol 16 no 11 pp4163ndash4173 2011
[26] S Balaji ldquoA new approach for solving Duffing equationsinvolving both integral and non-integral forcing termsrdquo AinShams Engineering Journal 2014
[27] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent IIrdquo Geophysical Journal of the RoyalAstronomical Society vol 13 pp 529ndash539 1967
[28] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-Holland Mathematics Studies Elsevier 2006
[29] J S Gu and W S Jiang ldquoThe Haar wavelets operational matrixof integrationrdquo International Journal of Systems Science vol 27no 7 pp 623ndash628 1996
[30] M Razzaghi and S Yousefi ldquoLegendre wavelets direct methodfor variational problemsrdquo Mathematics and Computers in Sim-ulation vol 53 no 3 pp 185ndash192 2000
[31] M Razzaghi and S Yousefi ldquoLegendre wavelets method forconstrained optimal control problemsrdquo Mathematical Methodsin the Applied Sciences vol 25 no 7 pp 529ndash539 2002
[32] A Kilicman and Z A A Al Zhour ldquoKronecker operationalmatrices for fractional calculus and some applicationsrdquo AppliedMathematics and Computation vol 187 no 1 pp 250ndash265 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Mathematics and Mathematical Sciences 5
By (25) and (32) (34) leads to
119910 (119905) asymp 119880119879
119875120572
Ψ (119905) + 119884119879
0Ψ (119905) = 119862
119879
Ψ (119905) (35)
where 119910(0) = 119896 asymp 119884119879
0Ψ(119905) 119862 = (119880
119879
119875120572
times+ 119884119879
0)119879
Substituting (33) and (35) into (31) we have
119880119879
Ψ (119905) = 119881119879
Ψ (119905) [119862119879
Ψ (119905)]2
+119882119879
Ψ (119905) 119862119879
Ψ (119905) + 119883119879
Ψ (119905)
(36)
Substituting (23) into (36) we have
119880119879
120601times
= 119881119879
[119862119879
120601times
]2
+119882119879
119862119879
120601times
+ 119883119879
(37)
where 119862 119881 119882 and 120601times
are known Equation (37) repre-sents a system of nonlinear equations with unknown vector119880 This system of nonlinear equations can be solved byNewton method for the unknown vector 119880 and we can getthe approximation solution by including 119880 into (35)
41 Existence and Uniqueness of Solutions Consider thefractional-order Riccati differential equation of the forms(31) and (32) The nonlinear term in (31) is 119910
2 and119875(119905) 119876(119905) and 119877(119905) are known functions For 120572 = 1 thefractional-order Riccati converts into the classical Riccatidifferential equation
Definition 1 Let 119868 = [0 119897] 119897 lt infin and 119862(119868) be the class of allcontinuous functions defined on 119868 with the norm
10038171003817100381710038171199101003817100381710038171003817 = sup119905isin119868
10038161003816100381610038161003816119890minusℎ119905
119910 (119905)10038161003816100381610038161003816 ℎ gt 0 (38)
which is equivalent to the sup norm of 119910 That is 119910 =
sup119905isin119868
|119890minusℎ119905
119910(119905)|
RemarkAssume that solution 119910(119905) of fractional-order Riccatidifferential equations (31) and (32) belongs to the space 119878 =
119910 isin 119877 |119910| le 119888 119888 is any constant in order to study theexistence and uniqueness of the initial value problem
Definition 2 The space of integrable functions 1198711[0 119897] in the
interval [0 119897] is defined as
1198711[0 119897] = 119906 (119905) int
119897
0
|119906 (119905)| 119889119905 lt infin (39)
Theorem 3 The initial value problem given by (31) and (32)has a unique solution
119910 isin 119862 (119868) 1199101015840
isin 119883 = 119910 isin 1198711[0 119897]
10038171003817100381710038171199101003817100381710038171003817 =
10038171003817100381710038171003817119890minusℎ119905
119910(119905)100381710038171003817100381710038171198711
(40)
Proof By (1) the fractional differential equation (31) can bewritten as
1198681minus120572
119889119910 (119905)
119889119905= 119875 (119905) 119910
2
+ 119876 (119905) 119910 + 119877 (119905) (41)
and becomes
119910 (119905) = 119868120572
(119875 (119905) 1199102
+ 119876 (119905) 119910 + 119877 (119905)) (42)
Now we define the operator Θ 119862(119868) rarr 119862(119868) by
Θ119910 (119905) = 119868120572
(119875 (119905) 1199102
+ 119876 (119905) 119910 + 119877 (119905)) (43)
and then
119890minusℎ119905
(Θ119910 minus Θ119908)
= 119890minusℎ119905
119868120572
[(1198751199102
(119905) + 119876119910 (119905) + 119877)
minus (1198751199082
(119905) + 119876119908 (119905) + 119877)]
le1
Γ (120572)int
119905
0
(119905 minus 119904)120572minus1
119890minusℎ(119905minus119904)
times [(119910 (119904) minus 119908 (119904)) (119910 (119904) + 119908 (119904))
minus119896 (119910 (119904) minus 119908 (119904))] 119890minusℎ119904
119889119904
le1003817100381710038171003817119910 minus 119908
1003817100381710038171003817
1
Γ (120572)int
119905
0
119904120572minus1
119890minusℎ119904
119889119904
(44)
hence we have1003817100381710038171003817Θ119910 minus Θ119908
1003817100381710038171003817 lt1003817100381710038171003817119910 minus 119908
1003817100381710038171003817 (45)
which implies that the operator given by (43) has a uniquefixed point and consequently the given integral equation hasa unique solution 119910(119905) isin 119862(119868) Also we can see that
119868120572
(119875(119905)1199102
+ 119876(119905)119910 + 119877(119905))10038161003816100381610038161003816119905=0
= 119896 (46)
Now from (42) we have
119910 (119905) = [119905120572
Γ (120572 + 1)(1198751199102
0+ 1198761199100+ 119877)
+ 119868120572+1
(1198751015840
1199102
+ 21199101015840
119875 + 1198761015840
119910 + 1198761199101015840
+ 1198771015840
) ]
119889119910
119889119905= [
119905120572minus1
Γ (120572)(1198751199102
0+ 1198761199100+ 119877)
+ 119868120572
(1198751015840
1199102
+ 21199101015840
119875 + 1198761015840
119910 + 1198761199101015840
+ 1198771015840
) ]
119890minusℎ119905
1199101015840
(119905) = 119890minusℎ119905
[119905120572minus1
Γ (120572)(1198751199102
0+ 1198761199100+ 119877)
+ 119868120572
(1198751015840
1199102
+ 21199101015840
119875 + 1198761015840
119910 + 1198761199101015840
+ 1198771015840
) ]
(47)
from which we can deduce that 1199101015840(119905) isin 119862(119868) and 1199101015840
(119905) isin 119878
6 International Journal of Mathematics and Mathematical Sciences
Now again from (42) (43) and (46) we get
119889119910
119889119905=
119889
119889119905119868120572
[1198751199102
(119905) + 119876119910 (119905) + 119877]
1198681minus120572
119889119910
119889119905= 1198681minus120572
119889
119889119905119868120572
[1198751199102
(119905) + 119876119910 (119905) + 119877]
=119889
1198891199051198681minus120572
119868120572
[1198751199102
(119905) + 119876119910 (119905) + 119877]
119863120572
119910 (119905) =119889
119889119905119868 [1198751199102
(119905) + 119876119910 (119905) + 119877]
= 1198751199102
(119905) + 119876119910 (119905) + 119877
119910 (0) = 119868120572
(1198751199102
(119905) + 119876119910 (119905) + 119877)10038161003816100381610038161003816119905=0
= 119896
(48)
which implies that the integral equation (46) is equivalent tothe initial value problem (32) and the theorem is proved
42 Convergence Analyses Let
120595119896119899
(119905) =10038161003816100381610038161198860
10038161003816100381610038161198962
120595 (119886119896
0119905 minus 119899119887
0) (49)
where 120595119896119899(119905) form a wavelet basis for 1198712(119877) In particular
when 1198860= 2 and 119887
0= 1 120595
119896119899(119905) form an orthonormal basis
[29]By (14) let 119910(119905) = sum
119872minus1
119894=111988811198941205951119894(119905) be the solution of (31)
where 1198881119894= ⟨119910(119905) 120595
1119894(119905)⟩ for 119896 = 1 in which ⟨sdot sdot⟩ denotes
the inner product
119910 (119905) =
119899
sum
119894=1
⟨119910 (119905) 1205951119894(119905)⟩ 120595
1119894(119905) (50)
Let 120573119895= ⟨119910(119905) 120595(119905)⟩ where 120595(119905) = 120595
1119894(119905)
Let119909119899= sum119899
119895=1120573119895120595(119905119895) be a sequence of partial sumsThen
⟨119910 (119905) 119909119899⟩ = ⟨119910 (119905)
119899
sum
119895=1
120573119895120595 (119905119895)⟩
=
119899
sum
119895=1
120573119895⟨119910 (119905) 120595 (119905
119895)⟩
=
119899
sum
119895=1
120573119895120573119895
=
119899
sum
119895=1
10038161003816100381610038161003816120573119895
10038161003816100381610038161003816
2
(51)
Further
1003817100381710038171003817119909119899 minus 119909119898
10038171003817100381710038172
=
10038171003817100381710038171003817100381710038171003817100381710038171003817
119899
sum
119895=119898+1
120573119895120595 (119905119895)
10038171003817100381710038171003817100381710038171003817100381710038171003817
2
= ⟨
119899
sum
119894=119898+1
120573119894120595 (119905119894)
119899
sum
119895=119898+1
120573119895120595 (119905119895)⟩
=
119899
sum
119894=119898+1
119899
sum
119895=119898+1
120573119894120573119895⟨120595 (119905119894) 120595 (119905
119895)⟩
=
119899
sum
119895=119898+1
10038161003816100381610038161003816120573119895
10038161003816100381610038161003816
2
(52)
As 119899 rarr infin from Besselrsquos inequality we have suminfin119895=1
|120573119895|2 is
convergentIt implies that 119909
119899 is a Cauchy sequence and it converges
to 119909 (say)Also
⟨119909 minus 119910 (119905) 120595 (119905119895)⟩ = ⟨119909 120595 (119905
119895)⟩ minus ⟨119910 (119905) 120595 (119905
119895)⟩
= ⟨ lim119899rarrinfin
119909119899 120595 (119905119895)⟩ minus 120573
119895
= lim119899rarrinfin
⟨119909119899 120595 (119905119895)⟩ minus 120573
119895
= lim119899rarrinfin
⟨
119899
sum
119895=1
120573119895120595 (119905119895) 120595 (119905
119895)⟩ minus 120573
119895
= 120573119895minus 120573119895= 0
(53)
which is possible only if 119910(119905) = 119909 That is both 119910(119905) and 119909119899
converge to the same value which indeed give the guaranteeof convergence of LWM
5 Numerical Examples
In order to show the effectiveness of the Legendre waveletsmethod (LWM) we implement LWM to the nonlinearfractional Riccati differential equations All the numericalexperiments were carried out on a personal computer withsomeMATLAB codes The specifications of PC are Intel corei5 processor and with Turbo boost up to 31 GHz and 4GB ofDDR3 memory The following problems of nonlinear Riccatidifferential equations are solved with real coefficients
Example 1 Consider the following nonlinear fractional Ric-cati differential equation
119863120572
119910 (119905) = 1 + 2119910 (119905) minus 1199102
(119905) 0 lt 120572 le 1 (54)
with initial condition
119910 (0) = 0 (55)
Exact solution for 120572 = 1 was found to be
119910 (119905) = 1 + radic2 tanh(radic2119905 +1
2log(
radic2 minus 1
radic2 + 1)) (56)
International Journal of Mathematics and Mathematical Sciences 7
01 02 03 04 05 06 07 08 09 10
02
04
06
08
1
12
14
16
18
Exact
LWM
t
y(t)
Figure 1 Numerical results of Example 1 by LWM for 120572 = 1
The integral representation of (54) and (55) is given by
119868120572
(119863120572
119910 (119905)) = 119868120572
(1 + 2119910 (119905) minus 1199102
(119905)) (57)
119910 (119905) = 119910 (0) +119905120572
Γ (120572 + 1)+ 2119868120572
119910 (119905) minus 119868120572
1199102
(119905) (58)
Let
119910 (119905) = 119862119879
Ψ (119905) (59)
and then
119868120572
119910 (119905) = 119862119879
119868120572
Ψ (119905)
= 119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(60)
By substituting (59) and (60) into (58) we get the followingsystem of algebraic equations
119862119879
Ψ (119905) =119905120572
Γ (120572 + 1)+ 2119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
minus 119862119879
1198752120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(61)
By solving the above system of linear equations we can findthe vector 119862 Numerical results are obtained for differentvalues of 119896 119872 and 120572 Solution obtained by the proposedLWM approach for 120572 = 1 119896 = 1 and 119872 = 3 is given inFigure 1 and for different values of 120572 = 06 07 08 and 09and for 119896 = 2 and 119872 = 5 is graphically given in Figure 2It can be seen from Figure 1 that the solution obtained bythe proposed LWM approach is more close to the exactsolution Table 1 describes the efficiency of the proposedmethod by comparing with the methods in [20 22] throughtheir absolute errorThe following is used for the errors of theapproximation119910(119905) of 119910(119905) that is 119910minus119910 = max |119910(119905)minus119910(119905)|Table 1 shows that very high accuracies are obtained for 119896= 3and119872= 5 by the present method
0204
0608
10
09 08 07 06 05
t
y(t)
120572
minus05
00
05
15
20
10
Figure 2 Numerical results of Example 1 by LWM for differentvalues of 120572
Example 2 Consider another fractional-order Riccati differ-ential equation
119863120572
119910 (119905) = 1 minus 1199102
(119905) 0 lt 120572 le 1 (62)
with initial condition
119910 (0) = 0 (63)
Exact solution for the above equation was found to be
119910 (119905) =1198902119905
minus 1
1198902119905 + 1 (64)
The integral representation of (62) and (63) is given by
119910 (119905) = 119910 (0) +119905120572
Γ (120572 + 1)minus 119868120572
1199102
(119905) (65)
Let
119910 (119905) = 119862119879
Ψ (119905) (66)
and then
119868120572
119910 (119905) = 119862119879
119868120572
Ψ (119905)
= 119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(67)
By substituting (66) and (67) in (62) we get the followingsystem of algebraic equations
119862119879
Ψ (119905) =119905120572
Γ (120572 + 1)minus 119862119879
1198752120572
2119896minus1119872times2119896minus1119872Ψ (119905) (68)
By solving the above system of linear equations we can findthe vector 119862 Numerical results are obtained for differentvalues of 119896119872 and 120572 Results obtained by LWM for 120572 = 1 119896 =2 and119872= 3 are shown in Figure 3 and it can be seen from thefigure that solution given by the LWMmerely coincides withthe exact solution Figure 4 shows the obtained results of (62)and (63) by LWM for different values of 120572 and for 119896 = 2 and
8 International Journal of Mathematics and Mathematical Sciences
Table 1 Numerical results of Example 1 for 120572 = 1
119905 Exact solution Absolute error in [22] Absolute error in [20] LWM LWM LWM119872 = 2 119896 = 2 119872 = 3 119896 = 2 119872 = 5 119896 = 3
01 0099667 751119864 minus 09 930119864 minus 05 0 0 002 0197375 152119864 minus 06 293119864 minus 02 191119864 minus 08 0 003 0291312 393119864 minus 05 378119864 minus 03 272119864 minus 08 145119864 minus 13 004 0379948 432119864 minus 04 281119864 minus 03 165119864 minus 08 117119864 minus 13 194119864 minus 16
05 0462117 841119864 minus 04 980119864 minus 04 131119864 minus 08 328119864 minus 13 320119864 minus 16
06 0537049 294119864 minus 05 793119864 minus 03 198119864 minus 08 497119864 minus 13 124119864 minus 16
07 0604367 335119864 minus 04 944119864 minus 03 252119864 minus 08 632119864 minus 13 158119864 minus 16
08 0664036 544119864 minus 04 117119864 minus 02 294119864 minus 08 736119864 minus 13 184119864 minus 16
09 0716297 656119864 minus 09 396119864 minus 02 323119864 minus 08 812119864 minus 13 194119864 minus 16
10 0761594 253119864 minus 06 295119864 minus 02 263119864 minus 08 462119864 minus 13 199119864 minus 16
Table 2 Numerical results of Example 2 for 120572 = 1
119905 Exact solution Absolute error in [22] Absolute error in [20] LWM LWM LWM119872 = 2 119896 = 2 119872 = 3 119896 = 2 119872 = 5 119896 = 3
01 0110295 123119864 minus 15 281119864 minus 03 0 0 002 0241976 524119864 minus 15 383119864 minus 04 0 0 003 0395104 816119864 minus 15 123119864 minus 04 0 0 004 0567812 115119864 minus 12 286119864 minus 03 168119864 minus 12 166119864 minus 15 143119864 minus 1605 0756014 617119864 minus 12 438119864 minus 04 222119864 minus 12 212119864 minus 15 166119864 minus 1606 0953566 455119864 minus 11 519119864 minus 02 115119864 minus 12 110119864 minus 15 154119864 minus 1607 1152946 757119864 minus 10 214119864 minus 02 127119864 minus 12 111119864 minus 15 133119864 minus 1608 1346363 633119864 minus 09 142119864 minus 02 187119864 minus 12 201119864 minus 15 175119864 minus 1609 1526911 367119864 minus 08 698119864 minus 03 193119864 minus 12 266119864 minus 15 187119864 minus 1610 1689498 164119864 minus 07 496119864 minus 03 156119864 minus 12 166119864 minus 15 164119864 minus 16
119872 = 5 Table 2 describes the efficiency of the proposedmethod by comparing with the methods in [20 22] throughtheir absolute error Table 1 shows that very high accuraciesare obtained for 119896 = 3 and 119872 = 5 by the present methodand from these results we can identify that guarantee ofconvergence of the proposed LWM approach is very high
Example 3 Let us consider another problem of nonlinearRiccati differential equation
119863120572
119910 (119905) = 1199052
+ 1199102
(119905) 0 lt 120572 le 1 119905 ge 0 (69)
with initial condition
119910 (0) = 1 (70)
When 120572 = 1 its exact solution is given by
119910 (119905) =119905 (119869minus34
(1199052
2) Γ (14) + 211986934
(1199052
2) Γ (34))
11986914
(11990522) Γ (14) minus 2119869minus14
(11990522) Γ (34)
(71)
where 119869119899(119905) is the Bessel function of first kind
0
01
02
03
04
05
06
07
08
LWM
01 02 03 04 05 06 07 08 09 1
Exact
t
y(t)
Figure 3 Numerical results of Example 2 by LWM for 120572 = 1
The integral representation of (69) and (70) is given by
119910 (119905) = 1 +2
Γ (120572 + 1) + 2Γ (120572)119905120572+2
+ 119868120572
1199102
(119905) (72)
International Journal of Mathematics and Mathematical Sciences 9
0204
060810
09 08 07 06 05
t
y(t)
120572
00
01
02
03
04
0605
0807
081
Figure 4 Numerical results of Example 2 by LWM for differentvalues of 120572
Let
119910 (119905) = 119862119879
Ψ (119905) (73)
and then
119868120572
119910 (119905) = 119862119879
119868120572
Ψ (119905)
= 119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(74)
By substituting (73) and (74) into (69) we get the followingsystem of algebraic equations
119862119879
Ψ (119905) = 1 +2
Γ (120572 + 1) + 2Γ (120572)(119862119879
Ψ (119905))120572+2
+ 119862119879
1198752120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(75)
By solving the above system of linear equations we can findthe vector 119862 Numerical results are obtained for differentvalues of 119896 119872 and 120572 Obtained results for (69) and (70)are shown in Figures 5 and 6 Figure 5 shows the solutionsobtained by LWM for different values of 120572 and for 119896 = 2 and119872 = 4 Figure 6 compares the solution obtained by LWMwith the exact solution of (69) and (70) when 120572 = 1 119896 = 1and 119872 = 2 So far there are no published results of absoluteerror for this problem and hence we are unable to compareabsolute error of ourmethodwith the existingmethods Fromthese results we can see that the proposed LWM approachgives the solution which is very close to the exact solutionand outperformed recently developed approaches for thenonlinear fractional-order Riccati differential equations interms of solution quality and convergence criteria
6 Conclusions
Nonlinear fractional-order Riccati differential equations playan important role in the modeling of many biologicalphysical chemical and real life problemsTherefore it is nec-essary to develop a method which would give more accuratesolutions to such type of problems with greater convergence
0204
0608
10
09 0807 06
05
t
y(t)
120572
10
0
10
20
30
40
0 406
08
0 9 t
Figure 5 Numerical results of Example 3 by LWM for differentvalues of 120572
1
15
2
25
3
35
4
45
5
55
6
LWM
01 02 03 04 05 06 07 08 09 1
Exact
t
y(t)
Figure 6 Numerical results of Example 3 by LWM for 120572 = 1
criteria In this work a Legendrersquos wavelet operational matrixmethod called LWM was proposed for solving nonlinearfractional-order Riccati differential equations Comparisonwas made for the solutions obtained by the proposedmethodand with the other recent approaches developed for thesame problem through their error analysis obtained resultsshow that the proposed LWM yields more accurate andreliable solutions even for small values of 119872 and 119896 whichassures the best approximate solution in less computationaleffort Further we have discussed the convergence criteriaof proposed scheme which indeed provides the guaranteeof consistency and stability of the proposed LWM schemefor the solutions of nonlinear fractional Riccati differentialequations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
10 International Journal of Mathematics and Mathematical Sciences
References
[1] N A Khan M Jamil A Ara and S Das ldquoExplicit solution fortime-fractional batch reactor systemrdquo International Journal ofChemical Reactor Engineering vol 9 article A91 2011
[2] V Feliu-Batlle R R Perez and L S Rodrıguez ldquoFractionalrobust control of main irrigation canals with variable dynamicparametersrdquoControl Engineering Practice vol 15 no 6 pp 673ndash686 2007
[3] I Podlubny ldquoFractional-order systems and 119875119868120582
119863120583-controllersrdquo
IEEE Transactions on Automatic Control vol 44 no 1 pp 208ndash214 1999
[4] R Garrappa ldquoOn some explicit Adams multistep methods forfractional differential equationsrdquo Journal of Computational andApplied Mathematics vol 229 no 2 pp 392ndash399 2009
[5] M Jamil and N A Khan ldquoSlip effects on fractional viscoelasticfluidsrdquo International Journal of Differential Equations vol 2011Article ID 193813 19 pages 2011
[6] F Mohammadi and M M Hosseini ldquoA comparative study ofnumerical methods for solving quadratic Riccati differentialequationsrdquo Journal of the Franklin Institute vol 348 no 2 pp156ndash164 2011
[7] R Shankar Principles of Quantum Mechanics Plenum PressNew York NY USA 1980
[8] S FragaM J Garcıa de la Vega and E S FragaTheSchrodingerand Riccati Equations vol 70 of Lect Notes Chem 1999
[9] L B Burrows and M Cohen ldquoSchrodingerrsquos wave equation-A lie algebra treatmentrdquo in Fundamental World of QuantumChemistry A Tribute to the Memory of Per-Olov Lowdin EJ Brandas and E S Kryachko Eds Kluwer Dordrecht TheNetherlands 2004
[10] S Abbasbandy ldquoHomotopy perturbation method for quadraticRiccati differential equation and comparison with Adomianrsquosdecomposition methodrdquo Applied Mathematics and Computa-tion vol 172 no 1 pp 485ndash490 2006
[11] Z Odibat and S Momani ldquoModified homotopy perturbationmethod application to quadratic Riccati differential equationof fractional orderrdquo Chaos Solitons amp Fractals vol 36 no 1 pp167ndash174 2008
[12] N A Khan A Ara and M Jamil ldquoAn efficient approach forsolving the Riccati equation with fractional ordersrdquo Computersamp Mathematics with Applications vol 61 no 9 pp 2683ndash26892011
[13] H Aminikhah and M Hemmatnezhad ldquoAn efficient methodfor quadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 4 pp835ndash839 2010
[14] S Abbasbandy ldquoIterated Hersquos homotopy perturbation methodfor quadratic Riccati differential equationrdquo Applied Mathemat-ics and Computation vol 175 no 1 pp 581ndash589 2006
[15] J Cang Y Tan H Xu and S Liao ldquoSeries solutions of non-lin-ear Riccati differential equations with fractional orderrdquo ChaosSolitons and Fractals vol 40 no 1 pp 1ndash9 2009
[16] Y Tan and S Abbasbandy ldquoHomotopy analysis method forquadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 3 pp539ndash546 2008
[17] M Gulsu and M Sezer ldquoOn the solution of the Riccati equa-tion by the Taylor matrix methodrdquo Applied Mathematics andComputation vol 176 no 2 pp 414ndash421 2006
[18] Y Li and LHu ldquoSolving fractional Riccati differential equationsusing Haar waveletrdquo in Proceedings of the 3rd InternationalConference on Information and Computing (ICIC 10) pp 314ndash317 Wuxi China June 2010
[19] N A Khan and A Ara ldquoFractional-order Riccati differentialequation analytical approximation and numerical resultsrdquoAdvances in Difference Equations vol 2013 article 185 2013
[20] M A Z Raja J A Khan and I M Qureshi ldquoA new stochasticapproach for solution of Riccati differential equation of frac-tional orderrdquo Annals of Mathematics and Artificial Intelligencevol 60 no 3-4 pp 229ndash250 2010
[21] M Merdan ldquoOn the solutions fractional Riccati differentialequation with modified RIEmann-Liouville derivativerdquo Inter-national Journal of Differential Equations vol 2012 Article ID346089 17 pages 2012
[22] N H Sweilam M M Khader and A M S Mahdy ldquoNumericalstudies for solving fractional Riccati differential equationrdquoApplications and AppliedMathematics vol 7 no 2 pp 595ndash6082012
[23] C K ChuiWavelets A Mathematical Tool for Signal ProcessingSIAM Philadelphia Pa USA 1997
[24] G Beylkin R Coifman and V Rokhlin ldquoFast wavelet trans-forms and numerical algorithms Irdquo Communications on Pureand Applied Mathematics vol 44 no 2 pp 141ndash183 1991
[25] M ur Rehman and R Ali Khan ldquoThe Legendre wavelet methodfor solving fractional differential equationsrdquoCommunications inNonlinear Science and Numerical Simulation vol 16 no 11 pp4163ndash4173 2011
[26] S Balaji ldquoA new approach for solving Duffing equationsinvolving both integral and non-integral forcing termsrdquo AinShams Engineering Journal 2014
[27] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent IIrdquo Geophysical Journal of the RoyalAstronomical Society vol 13 pp 529ndash539 1967
[28] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-Holland Mathematics Studies Elsevier 2006
[29] J S Gu and W S Jiang ldquoThe Haar wavelets operational matrixof integrationrdquo International Journal of Systems Science vol 27no 7 pp 623ndash628 1996
[30] M Razzaghi and S Yousefi ldquoLegendre wavelets direct methodfor variational problemsrdquo Mathematics and Computers in Sim-ulation vol 53 no 3 pp 185ndash192 2000
[31] M Razzaghi and S Yousefi ldquoLegendre wavelets method forconstrained optimal control problemsrdquo Mathematical Methodsin the Applied Sciences vol 25 no 7 pp 529ndash539 2002
[32] A Kilicman and Z A A Al Zhour ldquoKronecker operationalmatrices for fractional calculus and some applicationsrdquo AppliedMathematics and Computation vol 187 no 1 pp 250ndash265 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 International Journal of Mathematics and Mathematical Sciences
Now again from (42) (43) and (46) we get
119889119910
119889119905=
119889
119889119905119868120572
[1198751199102
(119905) + 119876119910 (119905) + 119877]
1198681minus120572
119889119910
119889119905= 1198681minus120572
119889
119889119905119868120572
[1198751199102
(119905) + 119876119910 (119905) + 119877]
=119889
1198891199051198681minus120572
119868120572
[1198751199102
(119905) + 119876119910 (119905) + 119877]
119863120572
119910 (119905) =119889
119889119905119868 [1198751199102
(119905) + 119876119910 (119905) + 119877]
= 1198751199102
(119905) + 119876119910 (119905) + 119877
119910 (0) = 119868120572
(1198751199102
(119905) + 119876119910 (119905) + 119877)10038161003816100381610038161003816119905=0
= 119896
(48)
which implies that the integral equation (46) is equivalent tothe initial value problem (32) and the theorem is proved
42 Convergence Analyses Let
120595119896119899
(119905) =10038161003816100381610038161198860
10038161003816100381610038161198962
120595 (119886119896
0119905 minus 119899119887
0) (49)
where 120595119896119899(119905) form a wavelet basis for 1198712(119877) In particular
when 1198860= 2 and 119887
0= 1 120595
119896119899(119905) form an orthonormal basis
[29]By (14) let 119910(119905) = sum
119872minus1
119894=111988811198941205951119894(119905) be the solution of (31)
where 1198881119894= ⟨119910(119905) 120595
1119894(119905)⟩ for 119896 = 1 in which ⟨sdot sdot⟩ denotes
the inner product
119910 (119905) =
119899
sum
119894=1
⟨119910 (119905) 1205951119894(119905)⟩ 120595
1119894(119905) (50)
Let 120573119895= ⟨119910(119905) 120595(119905)⟩ where 120595(119905) = 120595
1119894(119905)
Let119909119899= sum119899
119895=1120573119895120595(119905119895) be a sequence of partial sumsThen
⟨119910 (119905) 119909119899⟩ = ⟨119910 (119905)
119899
sum
119895=1
120573119895120595 (119905119895)⟩
=
119899
sum
119895=1
120573119895⟨119910 (119905) 120595 (119905
119895)⟩
=
119899
sum
119895=1
120573119895120573119895
=
119899
sum
119895=1
10038161003816100381610038161003816120573119895
10038161003816100381610038161003816
2
(51)
Further
1003817100381710038171003817119909119899 minus 119909119898
10038171003817100381710038172
=
10038171003817100381710038171003817100381710038171003817100381710038171003817
119899
sum
119895=119898+1
120573119895120595 (119905119895)
10038171003817100381710038171003817100381710038171003817100381710038171003817
2
= ⟨
119899
sum
119894=119898+1
120573119894120595 (119905119894)
119899
sum
119895=119898+1
120573119895120595 (119905119895)⟩
=
119899
sum
119894=119898+1
119899
sum
119895=119898+1
120573119894120573119895⟨120595 (119905119894) 120595 (119905
119895)⟩
=
119899
sum
119895=119898+1
10038161003816100381610038161003816120573119895
10038161003816100381610038161003816
2
(52)
As 119899 rarr infin from Besselrsquos inequality we have suminfin119895=1
|120573119895|2 is
convergentIt implies that 119909
119899 is a Cauchy sequence and it converges
to 119909 (say)Also
⟨119909 minus 119910 (119905) 120595 (119905119895)⟩ = ⟨119909 120595 (119905
119895)⟩ minus ⟨119910 (119905) 120595 (119905
119895)⟩
= ⟨ lim119899rarrinfin
119909119899 120595 (119905119895)⟩ minus 120573
119895
= lim119899rarrinfin
⟨119909119899 120595 (119905119895)⟩ minus 120573
119895
= lim119899rarrinfin
⟨
119899
sum
119895=1
120573119895120595 (119905119895) 120595 (119905
119895)⟩ minus 120573
119895
= 120573119895minus 120573119895= 0
(53)
which is possible only if 119910(119905) = 119909 That is both 119910(119905) and 119909119899
converge to the same value which indeed give the guaranteeof convergence of LWM
5 Numerical Examples
In order to show the effectiveness of the Legendre waveletsmethod (LWM) we implement LWM to the nonlinearfractional Riccati differential equations All the numericalexperiments were carried out on a personal computer withsomeMATLAB codes The specifications of PC are Intel corei5 processor and with Turbo boost up to 31 GHz and 4GB ofDDR3 memory The following problems of nonlinear Riccatidifferential equations are solved with real coefficients
Example 1 Consider the following nonlinear fractional Ric-cati differential equation
119863120572
119910 (119905) = 1 + 2119910 (119905) minus 1199102
(119905) 0 lt 120572 le 1 (54)
with initial condition
119910 (0) = 0 (55)
Exact solution for 120572 = 1 was found to be
119910 (119905) = 1 + radic2 tanh(radic2119905 +1
2log(
radic2 minus 1
radic2 + 1)) (56)
International Journal of Mathematics and Mathematical Sciences 7
01 02 03 04 05 06 07 08 09 10
02
04
06
08
1
12
14
16
18
Exact
LWM
t
y(t)
Figure 1 Numerical results of Example 1 by LWM for 120572 = 1
The integral representation of (54) and (55) is given by
119868120572
(119863120572
119910 (119905)) = 119868120572
(1 + 2119910 (119905) minus 1199102
(119905)) (57)
119910 (119905) = 119910 (0) +119905120572
Γ (120572 + 1)+ 2119868120572
119910 (119905) minus 119868120572
1199102
(119905) (58)
Let
119910 (119905) = 119862119879
Ψ (119905) (59)
and then
119868120572
119910 (119905) = 119862119879
119868120572
Ψ (119905)
= 119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(60)
By substituting (59) and (60) into (58) we get the followingsystem of algebraic equations
119862119879
Ψ (119905) =119905120572
Γ (120572 + 1)+ 2119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
minus 119862119879
1198752120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(61)
By solving the above system of linear equations we can findthe vector 119862 Numerical results are obtained for differentvalues of 119896 119872 and 120572 Solution obtained by the proposedLWM approach for 120572 = 1 119896 = 1 and 119872 = 3 is given inFigure 1 and for different values of 120572 = 06 07 08 and 09and for 119896 = 2 and 119872 = 5 is graphically given in Figure 2It can be seen from Figure 1 that the solution obtained bythe proposed LWM approach is more close to the exactsolution Table 1 describes the efficiency of the proposedmethod by comparing with the methods in [20 22] throughtheir absolute errorThe following is used for the errors of theapproximation119910(119905) of 119910(119905) that is 119910minus119910 = max |119910(119905)minus119910(119905)|Table 1 shows that very high accuracies are obtained for 119896= 3and119872= 5 by the present method
0204
0608
10
09 08 07 06 05
t
y(t)
120572
minus05
00
05
15
20
10
Figure 2 Numerical results of Example 1 by LWM for differentvalues of 120572
Example 2 Consider another fractional-order Riccati differ-ential equation
119863120572
119910 (119905) = 1 minus 1199102
(119905) 0 lt 120572 le 1 (62)
with initial condition
119910 (0) = 0 (63)
Exact solution for the above equation was found to be
119910 (119905) =1198902119905
minus 1
1198902119905 + 1 (64)
The integral representation of (62) and (63) is given by
119910 (119905) = 119910 (0) +119905120572
Γ (120572 + 1)minus 119868120572
1199102
(119905) (65)
Let
119910 (119905) = 119862119879
Ψ (119905) (66)
and then
119868120572
119910 (119905) = 119862119879
119868120572
Ψ (119905)
= 119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(67)
By substituting (66) and (67) in (62) we get the followingsystem of algebraic equations
119862119879
Ψ (119905) =119905120572
Γ (120572 + 1)minus 119862119879
1198752120572
2119896minus1119872times2119896minus1119872Ψ (119905) (68)
By solving the above system of linear equations we can findthe vector 119862 Numerical results are obtained for differentvalues of 119896119872 and 120572 Results obtained by LWM for 120572 = 1 119896 =2 and119872= 3 are shown in Figure 3 and it can be seen from thefigure that solution given by the LWMmerely coincides withthe exact solution Figure 4 shows the obtained results of (62)and (63) by LWM for different values of 120572 and for 119896 = 2 and
8 International Journal of Mathematics and Mathematical Sciences
Table 1 Numerical results of Example 1 for 120572 = 1
119905 Exact solution Absolute error in [22] Absolute error in [20] LWM LWM LWM119872 = 2 119896 = 2 119872 = 3 119896 = 2 119872 = 5 119896 = 3
01 0099667 751119864 minus 09 930119864 minus 05 0 0 002 0197375 152119864 minus 06 293119864 minus 02 191119864 minus 08 0 003 0291312 393119864 minus 05 378119864 minus 03 272119864 minus 08 145119864 minus 13 004 0379948 432119864 minus 04 281119864 minus 03 165119864 minus 08 117119864 minus 13 194119864 minus 16
05 0462117 841119864 minus 04 980119864 minus 04 131119864 minus 08 328119864 minus 13 320119864 minus 16
06 0537049 294119864 minus 05 793119864 minus 03 198119864 minus 08 497119864 minus 13 124119864 minus 16
07 0604367 335119864 minus 04 944119864 minus 03 252119864 minus 08 632119864 minus 13 158119864 minus 16
08 0664036 544119864 minus 04 117119864 minus 02 294119864 minus 08 736119864 minus 13 184119864 minus 16
09 0716297 656119864 minus 09 396119864 minus 02 323119864 minus 08 812119864 minus 13 194119864 minus 16
10 0761594 253119864 minus 06 295119864 minus 02 263119864 minus 08 462119864 minus 13 199119864 minus 16
Table 2 Numerical results of Example 2 for 120572 = 1
119905 Exact solution Absolute error in [22] Absolute error in [20] LWM LWM LWM119872 = 2 119896 = 2 119872 = 3 119896 = 2 119872 = 5 119896 = 3
01 0110295 123119864 minus 15 281119864 minus 03 0 0 002 0241976 524119864 minus 15 383119864 minus 04 0 0 003 0395104 816119864 minus 15 123119864 minus 04 0 0 004 0567812 115119864 minus 12 286119864 minus 03 168119864 minus 12 166119864 minus 15 143119864 minus 1605 0756014 617119864 minus 12 438119864 minus 04 222119864 minus 12 212119864 minus 15 166119864 minus 1606 0953566 455119864 minus 11 519119864 minus 02 115119864 minus 12 110119864 minus 15 154119864 minus 1607 1152946 757119864 minus 10 214119864 minus 02 127119864 minus 12 111119864 minus 15 133119864 minus 1608 1346363 633119864 minus 09 142119864 minus 02 187119864 minus 12 201119864 minus 15 175119864 minus 1609 1526911 367119864 minus 08 698119864 minus 03 193119864 minus 12 266119864 minus 15 187119864 minus 1610 1689498 164119864 minus 07 496119864 minus 03 156119864 minus 12 166119864 minus 15 164119864 minus 16
119872 = 5 Table 2 describes the efficiency of the proposedmethod by comparing with the methods in [20 22] throughtheir absolute error Table 1 shows that very high accuraciesare obtained for 119896 = 3 and 119872 = 5 by the present methodand from these results we can identify that guarantee ofconvergence of the proposed LWM approach is very high
Example 3 Let us consider another problem of nonlinearRiccati differential equation
119863120572
119910 (119905) = 1199052
+ 1199102
(119905) 0 lt 120572 le 1 119905 ge 0 (69)
with initial condition
119910 (0) = 1 (70)
When 120572 = 1 its exact solution is given by
119910 (119905) =119905 (119869minus34
(1199052
2) Γ (14) + 211986934
(1199052
2) Γ (34))
11986914
(11990522) Γ (14) minus 2119869minus14
(11990522) Γ (34)
(71)
where 119869119899(119905) is the Bessel function of first kind
0
01
02
03
04
05
06
07
08
LWM
01 02 03 04 05 06 07 08 09 1
Exact
t
y(t)
Figure 3 Numerical results of Example 2 by LWM for 120572 = 1
The integral representation of (69) and (70) is given by
119910 (119905) = 1 +2
Γ (120572 + 1) + 2Γ (120572)119905120572+2
+ 119868120572
1199102
(119905) (72)
International Journal of Mathematics and Mathematical Sciences 9
0204
060810
09 08 07 06 05
t
y(t)
120572
00
01
02
03
04
0605
0807
081
Figure 4 Numerical results of Example 2 by LWM for differentvalues of 120572
Let
119910 (119905) = 119862119879
Ψ (119905) (73)
and then
119868120572
119910 (119905) = 119862119879
119868120572
Ψ (119905)
= 119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(74)
By substituting (73) and (74) into (69) we get the followingsystem of algebraic equations
119862119879
Ψ (119905) = 1 +2
Γ (120572 + 1) + 2Γ (120572)(119862119879
Ψ (119905))120572+2
+ 119862119879
1198752120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(75)
By solving the above system of linear equations we can findthe vector 119862 Numerical results are obtained for differentvalues of 119896 119872 and 120572 Obtained results for (69) and (70)are shown in Figures 5 and 6 Figure 5 shows the solutionsobtained by LWM for different values of 120572 and for 119896 = 2 and119872 = 4 Figure 6 compares the solution obtained by LWMwith the exact solution of (69) and (70) when 120572 = 1 119896 = 1and 119872 = 2 So far there are no published results of absoluteerror for this problem and hence we are unable to compareabsolute error of ourmethodwith the existingmethods Fromthese results we can see that the proposed LWM approachgives the solution which is very close to the exact solutionand outperformed recently developed approaches for thenonlinear fractional-order Riccati differential equations interms of solution quality and convergence criteria
6 Conclusions
Nonlinear fractional-order Riccati differential equations playan important role in the modeling of many biologicalphysical chemical and real life problemsTherefore it is nec-essary to develop a method which would give more accuratesolutions to such type of problems with greater convergence
0204
0608
10
09 0807 06
05
t
y(t)
120572
10
0
10
20
30
40
0 406
08
0 9 t
Figure 5 Numerical results of Example 3 by LWM for differentvalues of 120572
1
15
2
25
3
35
4
45
5
55
6
LWM
01 02 03 04 05 06 07 08 09 1
Exact
t
y(t)
Figure 6 Numerical results of Example 3 by LWM for 120572 = 1
criteria In this work a Legendrersquos wavelet operational matrixmethod called LWM was proposed for solving nonlinearfractional-order Riccati differential equations Comparisonwas made for the solutions obtained by the proposedmethodand with the other recent approaches developed for thesame problem through their error analysis obtained resultsshow that the proposed LWM yields more accurate andreliable solutions even for small values of 119872 and 119896 whichassures the best approximate solution in less computationaleffort Further we have discussed the convergence criteriaof proposed scheme which indeed provides the guaranteeof consistency and stability of the proposed LWM schemefor the solutions of nonlinear fractional Riccati differentialequations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
10 International Journal of Mathematics and Mathematical Sciences
References
[1] N A Khan M Jamil A Ara and S Das ldquoExplicit solution fortime-fractional batch reactor systemrdquo International Journal ofChemical Reactor Engineering vol 9 article A91 2011
[2] V Feliu-Batlle R R Perez and L S Rodrıguez ldquoFractionalrobust control of main irrigation canals with variable dynamicparametersrdquoControl Engineering Practice vol 15 no 6 pp 673ndash686 2007
[3] I Podlubny ldquoFractional-order systems and 119875119868120582
119863120583-controllersrdquo
IEEE Transactions on Automatic Control vol 44 no 1 pp 208ndash214 1999
[4] R Garrappa ldquoOn some explicit Adams multistep methods forfractional differential equationsrdquo Journal of Computational andApplied Mathematics vol 229 no 2 pp 392ndash399 2009
[5] M Jamil and N A Khan ldquoSlip effects on fractional viscoelasticfluidsrdquo International Journal of Differential Equations vol 2011Article ID 193813 19 pages 2011
[6] F Mohammadi and M M Hosseini ldquoA comparative study ofnumerical methods for solving quadratic Riccati differentialequationsrdquo Journal of the Franklin Institute vol 348 no 2 pp156ndash164 2011
[7] R Shankar Principles of Quantum Mechanics Plenum PressNew York NY USA 1980
[8] S FragaM J Garcıa de la Vega and E S FragaTheSchrodingerand Riccati Equations vol 70 of Lect Notes Chem 1999
[9] L B Burrows and M Cohen ldquoSchrodingerrsquos wave equation-A lie algebra treatmentrdquo in Fundamental World of QuantumChemistry A Tribute to the Memory of Per-Olov Lowdin EJ Brandas and E S Kryachko Eds Kluwer Dordrecht TheNetherlands 2004
[10] S Abbasbandy ldquoHomotopy perturbation method for quadraticRiccati differential equation and comparison with Adomianrsquosdecomposition methodrdquo Applied Mathematics and Computa-tion vol 172 no 1 pp 485ndash490 2006
[11] Z Odibat and S Momani ldquoModified homotopy perturbationmethod application to quadratic Riccati differential equationof fractional orderrdquo Chaos Solitons amp Fractals vol 36 no 1 pp167ndash174 2008
[12] N A Khan A Ara and M Jamil ldquoAn efficient approach forsolving the Riccati equation with fractional ordersrdquo Computersamp Mathematics with Applications vol 61 no 9 pp 2683ndash26892011
[13] H Aminikhah and M Hemmatnezhad ldquoAn efficient methodfor quadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 4 pp835ndash839 2010
[14] S Abbasbandy ldquoIterated Hersquos homotopy perturbation methodfor quadratic Riccati differential equationrdquo Applied Mathemat-ics and Computation vol 175 no 1 pp 581ndash589 2006
[15] J Cang Y Tan H Xu and S Liao ldquoSeries solutions of non-lin-ear Riccati differential equations with fractional orderrdquo ChaosSolitons and Fractals vol 40 no 1 pp 1ndash9 2009
[16] Y Tan and S Abbasbandy ldquoHomotopy analysis method forquadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 3 pp539ndash546 2008
[17] M Gulsu and M Sezer ldquoOn the solution of the Riccati equa-tion by the Taylor matrix methodrdquo Applied Mathematics andComputation vol 176 no 2 pp 414ndash421 2006
[18] Y Li and LHu ldquoSolving fractional Riccati differential equationsusing Haar waveletrdquo in Proceedings of the 3rd InternationalConference on Information and Computing (ICIC 10) pp 314ndash317 Wuxi China June 2010
[19] N A Khan and A Ara ldquoFractional-order Riccati differentialequation analytical approximation and numerical resultsrdquoAdvances in Difference Equations vol 2013 article 185 2013
[20] M A Z Raja J A Khan and I M Qureshi ldquoA new stochasticapproach for solution of Riccati differential equation of frac-tional orderrdquo Annals of Mathematics and Artificial Intelligencevol 60 no 3-4 pp 229ndash250 2010
[21] M Merdan ldquoOn the solutions fractional Riccati differentialequation with modified RIEmann-Liouville derivativerdquo Inter-national Journal of Differential Equations vol 2012 Article ID346089 17 pages 2012
[22] N H Sweilam M M Khader and A M S Mahdy ldquoNumericalstudies for solving fractional Riccati differential equationrdquoApplications and AppliedMathematics vol 7 no 2 pp 595ndash6082012
[23] C K ChuiWavelets A Mathematical Tool for Signal ProcessingSIAM Philadelphia Pa USA 1997
[24] G Beylkin R Coifman and V Rokhlin ldquoFast wavelet trans-forms and numerical algorithms Irdquo Communications on Pureand Applied Mathematics vol 44 no 2 pp 141ndash183 1991
[25] M ur Rehman and R Ali Khan ldquoThe Legendre wavelet methodfor solving fractional differential equationsrdquoCommunications inNonlinear Science and Numerical Simulation vol 16 no 11 pp4163ndash4173 2011
[26] S Balaji ldquoA new approach for solving Duffing equationsinvolving both integral and non-integral forcing termsrdquo AinShams Engineering Journal 2014
[27] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent IIrdquo Geophysical Journal of the RoyalAstronomical Society vol 13 pp 529ndash539 1967
[28] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-Holland Mathematics Studies Elsevier 2006
[29] J S Gu and W S Jiang ldquoThe Haar wavelets operational matrixof integrationrdquo International Journal of Systems Science vol 27no 7 pp 623ndash628 1996
[30] M Razzaghi and S Yousefi ldquoLegendre wavelets direct methodfor variational problemsrdquo Mathematics and Computers in Sim-ulation vol 53 no 3 pp 185ndash192 2000
[31] M Razzaghi and S Yousefi ldquoLegendre wavelets method forconstrained optimal control problemsrdquo Mathematical Methodsin the Applied Sciences vol 25 no 7 pp 529ndash539 2002
[32] A Kilicman and Z A A Al Zhour ldquoKronecker operationalmatrices for fractional calculus and some applicationsrdquo AppliedMathematics and Computation vol 187 no 1 pp 250ndash265 2007
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
Volume 2014
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Mathematics and Mathematical Sciences 7
01 02 03 04 05 06 07 08 09 10
02
04
06
08
1
12
14
16
18
Exact
LWM
t
y(t)
Figure 1 Numerical results of Example 1 by LWM for 120572 = 1
The integral representation of (54) and (55) is given by
119868120572
(119863120572
119910 (119905)) = 119868120572
(1 + 2119910 (119905) minus 1199102
(119905)) (57)
119910 (119905) = 119910 (0) +119905120572
Γ (120572 + 1)+ 2119868120572
119910 (119905) minus 119868120572
1199102
(119905) (58)
Let
119910 (119905) = 119862119879
Ψ (119905) (59)
and then
119868120572
119910 (119905) = 119862119879
119868120572
Ψ (119905)
= 119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(60)
By substituting (59) and (60) into (58) we get the followingsystem of algebraic equations
119862119879
Ψ (119905) =119905120572
Γ (120572 + 1)+ 2119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
minus 119862119879
1198752120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(61)
By solving the above system of linear equations we can findthe vector 119862 Numerical results are obtained for differentvalues of 119896 119872 and 120572 Solution obtained by the proposedLWM approach for 120572 = 1 119896 = 1 and 119872 = 3 is given inFigure 1 and for different values of 120572 = 06 07 08 and 09and for 119896 = 2 and 119872 = 5 is graphically given in Figure 2It can be seen from Figure 1 that the solution obtained bythe proposed LWM approach is more close to the exactsolution Table 1 describes the efficiency of the proposedmethod by comparing with the methods in [20 22] throughtheir absolute errorThe following is used for the errors of theapproximation119910(119905) of 119910(119905) that is 119910minus119910 = max |119910(119905)minus119910(119905)|Table 1 shows that very high accuracies are obtained for 119896= 3and119872= 5 by the present method
0204
0608
10
09 08 07 06 05
t
y(t)
120572
minus05
00
05
15
20
10
Figure 2 Numerical results of Example 1 by LWM for differentvalues of 120572
Example 2 Consider another fractional-order Riccati differ-ential equation
119863120572
119910 (119905) = 1 minus 1199102
(119905) 0 lt 120572 le 1 (62)
with initial condition
119910 (0) = 0 (63)
Exact solution for the above equation was found to be
119910 (119905) =1198902119905
minus 1
1198902119905 + 1 (64)
The integral representation of (62) and (63) is given by
119910 (119905) = 119910 (0) +119905120572
Γ (120572 + 1)minus 119868120572
1199102
(119905) (65)
Let
119910 (119905) = 119862119879
Ψ (119905) (66)
and then
119868120572
119910 (119905) = 119862119879
119868120572
Ψ (119905)
= 119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(67)
By substituting (66) and (67) in (62) we get the followingsystem of algebraic equations
119862119879
Ψ (119905) =119905120572
Γ (120572 + 1)minus 119862119879
1198752120572
2119896minus1119872times2119896minus1119872Ψ (119905) (68)
By solving the above system of linear equations we can findthe vector 119862 Numerical results are obtained for differentvalues of 119896119872 and 120572 Results obtained by LWM for 120572 = 1 119896 =2 and119872= 3 are shown in Figure 3 and it can be seen from thefigure that solution given by the LWMmerely coincides withthe exact solution Figure 4 shows the obtained results of (62)and (63) by LWM for different values of 120572 and for 119896 = 2 and
8 International Journal of Mathematics and Mathematical Sciences
Table 1 Numerical results of Example 1 for 120572 = 1
119905 Exact solution Absolute error in [22] Absolute error in [20] LWM LWM LWM119872 = 2 119896 = 2 119872 = 3 119896 = 2 119872 = 5 119896 = 3
01 0099667 751119864 minus 09 930119864 minus 05 0 0 002 0197375 152119864 minus 06 293119864 minus 02 191119864 minus 08 0 003 0291312 393119864 minus 05 378119864 minus 03 272119864 minus 08 145119864 minus 13 004 0379948 432119864 minus 04 281119864 minus 03 165119864 minus 08 117119864 minus 13 194119864 minus 16
05 0462117 841119864 minus 04 980119864 minus 04 131119864 minus 08 328119864 minus 13 320119864 minus 16
06 0537049 294119864 minus 05 793119864 minus 03 198119864 minus 08 497119864 minus 13 124119864 minus 16
07 0604367 335119864 minus 04 944119864 minus 03 252119864 minus 08 632119864 minus 13 158119864 minus 16
08 0664036 544119864 minus 04 117119864 minus 02 294119864 minus 08 736119864 minus 13 184119864 minus 16
09 0716297 656119864 minus 09 396119864 minus 02 323119864 minus 08 812119864 minus 13 194119864 minus 16
10 0761594 253119864 minus 06 295119864 minus 02 263119864 minus 08 462119864 minus 13 199119864 minus 16
Table 2 Numerical results of Example 2 for 120572 = 1
119905 Exact solution Absolute error in [22] Absolute error in [20] LWM LWM LWM119872 = 2 119896 = 2 119872 = 3 119896 = 2 119872 = 5 119896 = 3
01 0110295 123119864 minus 15 281119864 minus 03 0 0 002 0241976 524119864 minus 15 383119864 minus 04 0 0 003 0395104 816119864 minus 15 123119864 minus 04 0 0 004 0567812 115119864 minus 12 286119864 minus 03 168119864 minus 12 166119864 minus 15 143119864 minus 1605 0756014 617119864 minus 12 438119864 minus 04 222119864 minus 12 212119864 minus 15 166119864 minus 1606 0953566 455119864 minus 11 519119864 minus 02 115119864 minus 12 110119864 minus 15 154119864 minus 1607 1152946 757119864 minus 10 214119864 minus 02 127119864 minus 12 111119864 minus 15 133119864 minus 1608 1346363 633119864 minus 09 142119864 minus 02 187119864 minus 12 201119864 minus 15 175119864 minus 1609 1526911 367119864 minus 08 698119864 minus 03 193119864 minus 12 266119864 minus 15 187119864 minus 1610 1689498 164119864 minus 07 496119864 minus 03 156119864 minus 12 166119864 minus 15 164119864 minus 16
119872 = 5 Table 2 describes the efficiency of the proposedmethod by comparing with the methods in [20 22] throughtheir absolute error Table 1 shows that very high accuraciesare obtained for 119896 = 3 and 119872 = 5 by the present methodand from these results we can identify that guarantee ofconvergence of the proposed LWM approach is very high
Example 3 Let us consider another problem of nonlinearRiccati differential equation
119863120572
119910 (119905) = 1199052
+ 1199102
(119905) 0 lt 120572 le 1 119905 ge 0 (69)
with initial condition
119910 (0) = 1 (70)
When 120572 = 1 its exact solution is given by
119910 (119905) =119905 (119869minus34
(1199052
2) Γ (14) + 211986934
(1199052
2) Γ (34))
11986914
(11990522) Γ (14) minus 2119869minus14
(11990522) Γ (34)
(71)
where 119869119899(119905) is the Bessel function of first kind
0
01
02
03
04
05
06
07
08
LWM
01 02 03 04 05 06 07 08 09 1
Exact
t
y(t)
Figure 3 Numerical results of Example 2 by LWM for 120572 = 1
The integral representation of (69) and (70) is given by
119910 (119905) = 1 +2
Γ (120572 + 1) + 2Γ (120572)119905120572+2
+ 119868120572
1199102
(119905) (72)
International Journal of Mathematics and Mathematical Sciences 9
0204
060810
09 08 07 06 05
t
y(t)
120572
00
01
02
03
04
0605
0807
081
Figure 4 Numerical results of Example 2 by LWM for differentvalues of 120572
Let
119910 (119905) = 119862119879
Ψ (119905) (73)
and then
119868120572
119910 (119905) = 119862119879
119868120572
Ψ (119905)
= 119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(74)
By substituting (73) and (74) into (69) we get the followingsystem of algebraic equations
119862119879
Ψ (119905) = 1 +2
Γ (120572 + 1) + 2Γ (120572)(119862119879
Ψ (119905))120572+2
+ 119862119879
1198752120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(75)
By solving the above system of linear equations we can findthe vector 119862 Numerical results are obtained for differentvalues of 119896 119872 and 120572 Obtained results for (69) and (70)are shown in Figures 5 and 6 Figure 5 shows the solutionsobtained by LWM for different values of 120572 and for 119896 = 2 and119872 = 4 Figure 6 compares the solution obtained by LWMwith the exact solution of (69) and (70) when 120572 = 1 119896 = 1and 119872 = 2 So far there are no published results of absoluteerror for this problem and hence we are unable to compareabsolute error of ourmethodwith the existingmethods Fromthese results we can see that the proposed LWM approachgives the solution which is very close to the exact solutionand outperformed recently developed approaches for thenonlinear fractional-order Riccati differential equations interms of solution quality and convergence criteria
6 Conclusions
Nonlinear fractional-order Riccati differential equations playan important role in the modeling of many biologicalphysical chemical and real life problemsTherefore it is nec-essary to develop a method which would give more accuratesolutions to such type of problems with greater convergence
0204
0608
10
09 0807 06
05
t
y(t)
120572
10
0
10
20
30
40
0 406
08
0 9 t
Figure 5 Numerical results of Example 3 by LWM for differentvalues of 120572
1
15
2
25
3
35
4
45
5
55
6
LWM
01 02 03 04 05 06 07 08 09 1
Exact
t
y(t)
Figure 6 Numerical results of Example 3 by LWM for 120572 = 1
criteria In this work a Legendrersquos wavelet operational matrixmethod called LWM was proposed for solving nonlinearfractional-order Riccati differential equations Comparisonwas made for the solutions obtained by the proposedmethodand with the other recent approaches developed for thesame problem through their error analysis obtained resultsshow that the proposed LWM yields more accurate andreliable solutions even for small values of 119872 and 119896 whichassures the best approximate solution in less computationaleffort Further we have discussed the convergence criteriaof proposed scheme which indeed provides the guaranteeof consistency and stability of the proposed LWM schemefor the solutions of nonlinear fractional Riccati differentialequations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
10 International Journal of Mathematics and Mathematical Sciences
References
[1] N A Khan M Jamil A Ara and S Das ldquoExplicit solution fortime-fractional batch reactor systemrdquo International Journal ofChemical Reactor Engineering vol 9 article A91 2011
[2] V Feliu-Batlle R R Perez and L S Rodrıguez ldquoFractionalrobust control of main irrigation canals with variable dynamicparametersrdquoControl Engineering Practice vol 15 no 6 pp 673ndash686 2007
[3] I Podlubny ldquoFractional-order systems and 119875119868120582
119863120583-controllersrdquo
IEEE Transactions on Automatic Control vol 44 no 1 pp 208ndash214 1999
[4] R Garrappa ldquoOn some explicit Adams multistep methods forfractional differential equationsrdquo Journal of Computational andApplied Mathematics vol 229 no 2 pp 392ndash399 2009
[5] M Jamil and N A Khan ldquoSlip effects on fractional viscoelasticfluidsrdquo International Journal of Differential Equations vol 2011Article ID 193813 19 pages 2011
[6] F Mohammadi and M M Hosseini ldquoA comparative study ofnumerical methods for solving quadratic Riccati differentialequationsrdquo Journal of the Franklin Institute vol 348 no 2 pp156ndash164 2011
[7] R Shankar Principles of Quantum Mechanics Plenum PressNew York NY USA 1980
[8] S FragaM J Garcıa de la Vega and E S FragaTheSchrodingerand Riccati Equations vol 70 of Lect Notes Chem 1999
[9] L B Burrows and M Cohen ldquoSchrodingerrsquos wave equation-A lie algebra treatmentrdquo in Fundamental World of QuantumChemistry A Tribute to the Memory of Per-Olov Lowdin EJ Brandas and E S Kryachko Eds Kluwer Dordrecht TheNetherlands 2004
[10] S Abbasbandy ldquoHomotopy perturbation method for quadraticRiccati differential equation and comparison with Adomianrsquosdecomposition methodrdquo Applied Mathematics and Computa-tion vol 172 no 1 pp 485ndash490 2006
[11] Z Odibat and S Momani ldquoModified homotopy perturbationmethod application to quadratic Riccati differential equationof fractional orderrdquo Chaos Solitons amp Fractals vol 36 no 1 pp167ndash174 2008
[12] N A Khan A Ara and M Jamil ldquoAn efficient approach forsolving the Riccati equation with fractional ordersrdquo Computersamp Mathematics with Applications vol 61 no 9 pp 2683ndash26892011
[13] H Aminikhah and M Hemmatnezhad ldquoAn efficient methodfor quadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 4 pp835ndash839 2010
[14] S Abbasbandy ldquoIterated Hersquos homotopy perturbation methodfor quadratic Riccati differential equationrdquo Applied Mathemat-ics and Computation vol 175 no 1 pp 581ndash589 2006
[15] J Cang Y Tan H Xu and S Liao ldquoSeries solutions of non-lin-ear Riccati differential equations with fractional orderrdquo ChaosSolitons and Fractals vol 40 no 1 pp 1ndash9 2009
[16] Y Tan and S Abbasbandy ldquoHomotopy analysis method forquadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 3 pp539ndash546 2008
[17] M Gulsu and M Sezer ldquoOn the solution of the Riccati equa-tion by the Taylor matrix methodrdquo Applied Mathematics andComputation vol 176 no 2 pp 414ndash421 2006
[18] Y Li and LHu ldquoSolving fractional Riccati differential equationsusing Haar waveletrdquo in Proceedings of the 3rd InternationalConference on Information and Computing (ICIC 10) pp 314ndash317 Wuxi China June 2010
[19] N A Khan and A Ara ldquoFractional-order Riccati differentialequation analytical approximation and numerical resultsrdquoAdvances in Difference Equations vol 2013 article 185 2013
[20] M A Z Raja J A Khan and I M Qureshi ldquoA new stochasticapproach for solution of Riccati differential equation of frac-tional orderrdquo Annals of Mathematics and Artificial Intelligencevol 60 no 3-4 pp 229ndash250 2010
[21] M Merdan ldquoOn the solutions fractional Riccati differentialequation with modified RIEmann-Liouville derivativerdquo Inter-national Journal of Differential Equations vol 2012 Article ID346089 17 pages 2012
[22] N H Sweilam M M Khader and A M S Mahdy ldquoNumericalstudies for solving fractional Riccati differential equationrdquoApplications and AppliedMathematics vol 7 no 2 pp 595ndash6082012
[23] C K ChuiWavelets A Mathematical Tool for Signal ProcessingSIAM Philadelphia Pa USA 1997
[24] G Beylkin R Coifman and V Rokhlin ldquoFast wavelet trans-forms and numerical algorithms Irdquo Communications on Pureand Applied Mathematics vol 44 no 2 pp 141ndash183 1991
[25] M ur Rehman and R Ali Khan ldquoThe Legendre wavelet methodfor solving fractional differential equationsrdquoCommunications inNonlinear Science and Numerical Simulation vol 16 no 11 pp4163ndash4173 2011
[26] S Balaji ldquoA new approach for solving Duffing equationsinvolving both integral and non-integral forcing termsrdquo AinShams Engineering Journal 2014
[27] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent IIrdquo Geophysical Journal of the RoyalAstronomical Society vol 13 pp 529ndash539 1967
[28] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-Holland Mathematics Studies Elsevier 2006
[29] J S Gu and W S Jiang ldquoThe Haar wavelets operational matrixof integrationrdquo International Journal of Systems Science vol 27no 7 pp 623ndash628 1996
[30] M Razzaghi and S Yousefi ldquoLegendre wavelets direct methodfor variational problemsrdquo Mathematics and Computers in Sim-ulation vol 53 no 3 pp 185ndash192 2000
[31] M Razzaghi and S Yousefi ldquoLegendre wavelets method forconstrained optimal control problemsrdquo Mathematical Methodsin the Applied Sciences vol 25 no 7 pp 529ndash539 2002
[32] A Kilicman and Z A A Al Zhour ldquoKronecker operationalmatrices for fractional calculus and some applicationsrdquo AppliedMathematics and Computation vol 187 no 1 pp 250ndash265 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 International Journal of Mathematics and Mathematical Sciences
Table 1 Numerical results of Example 1 for 120572 = 1
119905 Exact solution Absolute error in [22] Absolute error in [20] LWM LWM LWM119872 = 2 119896 = 2 119872 = 3 119896 = 2 119872 = 5 119896 = 3
01 0099667 751119864 minus 09 930119864 minus 05 0 0 002 0197375 152119864 minus 06 293119864 minus 02 191119864 minus 08 0 003 0291312 393119864 minus 05 378119864 minus 03 272119864 minus 08 145119864 minus 13 004 0379948 432119864 minus 04 281119864 minus 03 165119864 minus 08 117119864 minus 13 194119864 minus 16
05 0462117 841119864 minus 04 980119864 minus 04 131119864 minus 08 328119864 minus 13 320119864 minus 16
06 0537049 294119864 minus 05 793119864 minus 03 198119864 minus 08 497119864 minus 13 124119864 minus 16
07 0604367 335119864 minus 04 944119864 minus 03 252119864 minus 08 632119864 minus 13 158119864 minus 16
08 0664036 544119864 minus 04 117119864 minus 02 294119864 minus 08 736119864 minus 13 184119864 minus 16
09 0716297 656119864 minus 09 396119864 minus 02 323119864 minus 08 812119864 minus 13 194119864 minus 16
10 0761594 253119864 minus 06 295119864 minus 02 263119864 minus 08 462119864 minus 13 199119864 minus 16
Table 2 Numerical results of Example 2 for 120572 = 1
119905 Exact solution Absolute error in [22] Absolute error in [20] LWM LWM LWM119872 = 2 119896 = 2 119872 = 3 119896 = 2 119872 = 5 119896 = 3
01 0110295 123119864 minus 15 281119864 minus 03 0 0 002 0241976 524119864 minus 15 383119864 minus 04 0 0 003 0395104 816119864 minus 15 123119864 minus 04 0 0 004 0567812 115119864 minus 12 286119864 minus 03 168119864 minus 12 166119864 minus 15 143119864 minus 1605 0756014 617119864 minus 12 438119864 minus 04 222119864 minus 12 212119864 minus 15 166119864 minus 1606 0953566 455119864 minus 11 519119864 minus 02 115119864 minus 12 110119864 minus 15 154119864 minus 1607 1152946 757119864 minus 10 214119864 minus 02 127119864 minus 12 111119864 minus 15 133119864 minus 1608 1346363 633119864 minus 09 142119864 minus 02 187119864 minus 12 201119864 minus 15 175119864 minus 1609 1526911 367119864 minus 08 698119864 minus 03 193119864 minus 12 266119864 minus 15 187119864 minus 1610 1689498 164119864 minus 07 496119864 minus 03 156119864 minus 12 166119864 minus 15 164119864 minus 16
119872 = 5 Table 2 describes the efficiency of the proposedmethod by comparing with the methods in [20 22] throughtheir absolute error Table 1 shows that very high accuraciesare obtained for 119896 = 3 and 119872 = 5 by the present methodand from these results we can identify that guarantee ofconvergence of the proposed LWM approach is very high
Example 3 Let us consider another problem of nonlinearRiccati differential equation
119863120572
119910 (119905) = 1199052
+ 1199102
(119905) 0 lt 120572 le 1 119905 ge 0 (69)
with initial condition
119910 (0) = 1 (70)
When 120572 = 1 its exact solution is given by
119910 (119905) =119905 (119869minus34
(1199052
2) Γ (14) + 211986934
(1199052
2) Γ (34))
11986914
(11990522) Γ (14) minus 2119869minus14
(11990522) Γ (34)
(71)
where 119869119899(119905) is the Bessel function of first kind
0
01
02
03
04
05
06
07
08
LWM
01 02 03 04 05 06 07 08 09 1
Exact
t
y(t)
Figure 3 Numerical results of Example 2 by LWM for 120572 = 1
The integral representation of (69) and (70) is given by
119910 (119905) = 1 +2
Γ (120572 + 1) + 2Γ (120572)119905120572+2
+ 119868120572
1199102
(119905) (72)
International Journal of Mathematics and Mathematical Sciences 9
0204
060810
09 08 07 06 05
t
y(t)
120572
00
01
02
03
04
0605
0807
081
Figure 4 Numerical results of Example 2 by LWM for differentvalues of 120572
Let
119910 (119905) = 119862119879
Ψ (119905) (73)
and then
119868120572
119910 (119905) = 119862119879
119868120572
Ψ (119905)
= 119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(74)
By substituting (73) and (74) into (69) we get the followingsystem of algebraic equations
119862119879
Ψ (119905) = 1 +2
Γ (120572 + 1) + 2Γ (120572)(119862119879
Ψ (119905))120572+2
+ 119862119879
1198752120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(75)
By solving the above system of linear equations we can findthe vector 119862 Numerical results are obtained for differentvalues of 119896 119872 and 120572 Obtained results for (69) and (70)are shown in Figures 5 and 6 Figure 5 shows the solutionsobtained by LWM for different values of 120572 and for 119896 = 2 and119872 = 4 Figure 6 compares the solution obtained by LWMwith the exact solution of (69) and (70) when 120572 = 1 119896 = 1and 119872 = 2 So far there are no published results of absoluteerror for this problem and hence we are unable to compareabsolute error of ourmethodwith the existingmethods Fromthese results we can see that the proposed LWM approachgives the solution which is very close to the exact solutionand outperformed recently developed approaches for thenonlinear fractional-order Riccati differential equations interms of solution quality and convergence criteria
6 Conclusions
Nonlinear fractional-order Riccati differential equations playan important role in the modeling of many biologicalphysical chemical and real life problemsTherefore it is nec-essary to develop a method which would give more accuratesolutions to such type of problems with greater convergence
0204
0608
10
09 0807 06
05
t
y(t)
120572
10
0
10
20
30
40
0 406
08
0 9 t
Figure 5 Numerical results of Example 3 by LWM for differentvalues of 120572
1
15
2
25
3
35
4
45
5
55
6
LWM
01 02 03 04 05 06 07 08 09 1
Exact
t
y(t)
Figure 6 Numerical results of Example 3 by LWM for 120572 = 1
criteria In this work a Legendrersquos wavelet operational matrixmethod called LWM was proposed for solving nonlinearfractional-order Riccati differential equations Comparisonwas made for the solutions obtained by the proposedmethodand with the other recent approaches developed for thesame problem through their error analysis obtained resultsshow that the proposed LWM yields more accurate andreliable solutions even for small values of 119872 and 119896 whichassures the best approximate solution in less computationaleffort Further we have discussed the convergence criteriaof proposed scheme which indeed provides the guaranteeof consistency and stability of the proposed LWM schemefor the solutions of nonlinear fractional Riccati differentialequations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
10 International Journal of Mathematics and Mathematical Sciences
References
[1] N A Khan M Jamil A Ara and S Das ldquoExplicit solution fortime-fractional batch reactor systemrdquo International Journal ofChemical Reactor Engineering vol 9 article A91 2011
[2] V Feliu-Batlle R R Perez and L S Rodrıguez ldquoFractionalrobust control of main irrigation canals with variable dynamicparametersrdquoControl Engineering Practice vol 15 no 6 pp 673ndash686 2007
[3] I Podlubny ldquoFractional-order systems and 119875119868120582
119863120583-controllersrdquo
IEEE Transactions on Automatic Control vol 44 no 1 pp 208ndash214 1999
[4] R Garrappa ldquoOn some explicit Adams multistep methods forfractional differential equationsrdquo Journal of Computational andApplied Mathematics vol 229 no 2 pp 392ndash399 2009
[5] M Jamil and N A Khan ldquoSlip effects on fractional viscoelasticfluidsrdquo International Journal of Differential Equations vol 2011Article ID 193813 19 pages 2011
[6] F Mohammadi and M M Hosseini ldquoA comparative study ofnumerical methods for solving quadratic Riccati differentialequationsrdquo Journal of the Franklin Institute vol 348 no 2 pp156ndash164 2011
[7] R Shankar Principles of Quantum Mechanics Plenum PressNew York NY USA 1980
[8] S FragaM J Garcıa de la Vega and E S FragaTheSchrodingerand Riccati Equations vol 70 of Lect Notes Chem 1999
[9] L B Burrows and M Cohen ldquoSchrodingerrsquos wave equation-A lie algebra treatmentrdquo in Fundamental World of QuantumChemistry A Tribute to the Memory of Per-Olov Lowdin EJ Brandas and E S Kryachko Eds Kluwer Dordrecht TheNetherlands 2004
[10] S Abbasbandy ldquoHomotopy perturbation method for quadraticRiccati differential equation and comparison with Adomianrsquosdecomposition methodrdquo Applied Mathematics and Computa-tion vol 172 no 1 pp 485ndash490 2006
[11] Z Odibat and S Momani ldquoModified homotopy perturbationmethod application to quadratic Riccati differential equationof fractional orderrdquo Chaos Solitons amp Fractals vol 36 no 1 pp167ndash174 2008
[12] N A Khan A Ara and M Jamil ldquoAn efficient approach forsolving the Riccati equation with fractional ordersrdquo Computersamp Mathematics with Applications vol 61 no 9 pp 2683ndash26892011
[13] H Aminikhah and M Hemmatnezhad ldquoAn efficient methodfor quadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 4 pp835ndash839 2010
[14] S Abbasbandy ldquoIterated Hersquos homotopy perturbation methodfor quadratic Riccati differential equationrdquo Applied Mathemat-ics and Computation vol 175 no 1 pp 581ndash589 2006
[15] J Cang Y Tan H Xu and S Liao ldquoSeries solutions of non-lin-ear Riccati differential equations with fractional orderrdquo ChaosSolitons and Fractals vol 40 no 1 pp 1ndash9 2009
[16] Y Tan and S Abbasbandy ldquoHomotopy analysis method forquadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 3 pp539ndash546 2008
[17] M Gulsu and M Sezer ldquoOn the solution of the Riccati equa-tion by the Taylor matrix methodrdquo Applied Mathematics andComputation vol 176 no 2 pp 414ndash421 2006
[18] Y Li and LHu ldquoSolving fractional Riccati differential equationsusing Haar waveletrdquo in Proceedings of the 3rd InternationalConference on Information and Computing (ICIC 10) pp 314ndash317 Wuxi China June 2010
[19] N A Khan and A Ara ldquoFractional-order Riccati differentialequation analytical approximation and numerical resultsrdquoAdvances in Difference Equations vol 2013 article 185 2013
[20] M A Z Raja J A Khan and I M Qureshi ldquoA new stochasticapproach for solution of Riccati differential equation of frac-tional orderrdquo Annals of Mathematics and Artificial Intelligencevol 60 no 3-4 pp 229ndash250 2010
[21] M Merdan ldquoOn the solutions fractional Riccati differentialequation with modified RIEmann-Liouville derivativerdquo Inter-national Journal of Differential Equations vol 2012 Article ID346089 17 pages 2012
[22] N H Sweilam M M Khader and A M S Mahdy ldquoNumericalstudies for solving fractional Riccati differential equationrdquoApplications and AppliedMathematics vol 7 no 2 pp 595ndash6082012
[23] C K ChuiWavelets A Mathematical Tool for Signal ProcessingSIAM Philadelphia Pa USA 1997
[24] G Beylkin R Coifman and V Rokhlin ldquoFast wavelet trans-forms and numerical algorithms Irdquo Communications on Pureand Applied Mathematics vol 44 no 2 pp 141ndash183 1991
[25] M ur Rehman and R Ali Khan ldquoThe Legendre wavelet methodfor solving fractional differential equationsrdquoCommunications inNonlinear Science and Numerical Simulation vol 16 no 11 pp4163ndash4173 2011
[26] S Balaji ldquoA new approach for solving Duffing equationsinvolving both integral and non-integral forcing termsrdquo AinShams Engineering Journal 2014
[27] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent IIrdquo Geophysical Journal of the RoyalAstronomical Society vol 13 pp 529ndash539 1967
[28] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-Holland Mathematics Studies Elsevier 2006
[29] J S Gu and W S Jiang ldquoThe Haar wavelets operational matrixof integrationrdquo International Journal of Systems Science vol 27no 7 pp 623ndash628 1996
[30] M Razzaghi and S Yousefi ldquoLegendre wavelets direct methodfor variational problemsrdquo Mathematics and Computers in Sim-ulation vol 53 no 3 pp 185ndash192 2000
[31] M Razzaghi and S Yousefi ldquoLegendre wavelets method forconstrained optimal control problemsrdquo Mathematical Methodsin the Applied Sciences vol 25 no 7 pp 529ndash539 2002
[32] A Kilicman and Z A A Al Zhour ldquoKronecker operationalmatrices for fractional calculus and some applicationsrdquo AppliedMathematics and Computation vol 187 no 1 pp 250ndash265 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Mathematics and Mathematical Sciences 9
0204
060810
09 08 07 06 05
t
y(t)
120572
00
01
02
03
04
0605
0807
081
Figure 4 Numerical results of Example 2 by LWM for differentvalues of 120572
Let
119910 (119905) = 119862119879
Ψ (119905) (73)
and then
119868120572
119910 (119905) = 119862119879
119868120572
Ψ (119905)
= 119862119879
119875120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(74)
By substituting (73) and (74) into (69) we get the followingsystem of algebraic equations
119862119879
Ψ (119905) = 1 +2
Γ (120572 + 1) + 2Γ (120572)(119862119879
Ψ (119905))120572+2
+ 119862119879
1198752120572
2119896minus1119872times2119896minus1119872Ψ (119905)
(75)
By solving the above system of linear equations we can findthe vector 119862 Numerical results are obtained for differentvalues of 119896 119872 and 120572 Obtained results for (69) and (70)are shown in Figures 5 and 6 Figure 5 shows the solutionsobtained by LWM for different values of 120572 and for 119896 = 2 and119872 = 4 Figure 6 compares the solution obtained by LWMwith the exact solution of (69) and (70) when 120572 = 1 119896 = 1and 119872 = 2 So far there are no published results of absoluteerror for this problem and hence we are unable to compareabsolute error of ourmethodwith the existingmethods Fromthese results we can see that the proposed LWM approachgives the solution which is very close to the exact solutionand outperformed recently developed approaches for thenonlinear fractional-order Riccati differential equations interms of solution quality and convergence criteria
6 Conclusions
Nonlinear fractional-order Riccati differential equations playan important role in the modeling of many biologicalphysical chemical and real life problemsTherefore it is nec-essary to develop a method which would give more accuratesolutions to such type of problems with greater convergence
0204
0608
10
09 0807 06
05
t
y(t)
120572
10
0
10
20
30
40
0 406
08
0 9 t
Figure 5 Numerical results of Example 3 by LWM for differentvalues of 120572
1
15
2
25
3
35
4
45
5
55
6
LWM
01 02 03 04 05 06 07 08 09 1
Exact
t
y(t)
Figure 6 Numerical results of Example 3 by LWM for 120572 = 1
criteria In this work a Legendrersquos wavelet operational matrixmethod called LWM was proposed for solving nonlinearfractional-order Riccati differential equations Comparisonwas made for the solutions obtained by the proposedmethodand with the other recent approaches developed for thesame problem through their error analysis obtained resultsshow that the proposed LWM yields more accurate andreliable solutions even for small values of 119872 and 119896 whichassures the best approximate solution in less computationaleffort Further we have discussed the convergence criteriaof proposed scheme which indeed provides the guaranteeof consistency and stability of the proposed LWM schemefor the solutions of nonlinear fractional Riccati differentialequations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
10 International Journal of Mathematics and Mathematical Sciences
References
[1] N A Khan M Jamil A Ara and S Das ldquoExplicit solution fortime-fractional batch reactor systemrdquo International Journal ofChemical Reactor Engineering vol 9 article A91 2011
[2] V Feliu-Batlle R R Perez and L S Rodrıguez ldquoFractionalrobust control of main irrigation canals with variable dynamicparametersrdquoControl Engineering Practice vol 15 no 6 pp 673ndash686 2007
[3] I Podlubny ldquoFractional-order systems and 119875119868120582
119863120583-controllersrdquo
IEEE Transactions on Automatic Control vol 44 no 1 pp 208ndash214 1999
[4] R Garrappa ldquoOn some explicit Adams multistep methods forfractional differential equationsrdquo Journal of Computational andApplied Mathematics vol 229 no 2 pp 392ndash399 2009
[5] M Jamil and N A Khan ldquoSlip effects on fractional viscoelasticfluidsrdquo International Journal of Differential Equations vol 2011Article ID 193813 19 pages 2011
[6] F Mohammadi and M M Hosseini ldquoA comparative study ofnumerical methods for solving quadratic Riccati differentialequationsrdquo Journal of the Franklin Institute vol 348 no 2 pp156ndash164 2011
[7] R Shankar Principles of Quantum Mechanics Plenum PressNew York NY USA 1980
[8] S FragaM J Garcıa de la Vega and E S FragaTheSchrodingerand Riccati Equations vol 70 of Lect Notes Chem 1999
[9] L B Burrows and M Cohen ldquoSchrodingerrsquos wave equation-A lie algebra treatmentrdquo in Fundamental World of QuantumChemistry A Tribute to the Memory of Per-Olov Lowdin EJ Brandas and E S Kryachko Eds Kluwer Dordrecht TheNetherlands 2004
[10] S Abbasbandy ldquoHomotopy perturbation method for quadraticRiccati differential equation and comparison with Adomianrsquosdecomposition methodrdquo Applied Mathematics and Computa-tion vol 172 no 1 pp 485ndash490 2006
[11] Z Odibat and S Momani ldquoModified homotopy perturbationmethod application to quadratic Riccati differential equationof fractional orderrdquo Chaos Solitons amp Fractals vol 36 no 1 pp167ndash174 2008
[12] N A Khan A Ara and M Jamil ldquoAn efficient approach forsolving the Riccati equation with fractional ordersrdquo Computersamp Mathematics with Applications vol 61 no 9 pp 2683ndash26892011
[13] H Aminikhah and M Hemmatnezhad ldquoAn efficient methodfor quadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 4 pp835ndash839 2010
[14] S Abbasbandy ldquoIterated Hersquos homotopy perturbation methodfor quadratic Riccati differential equationrdquo Applied Mathemat-ics and Computation vol 175 no 1 pp 581ndash589 2006
[15] J Cang Y Tan H Xu and S Liao ldquoSeries solutions of non-lin-ear Riccati differential equations with fractional orderrdquo ChaosSolitons and Fractals vol 40 no 1 pp 1ndash9 2009
[16] Y Tan and S Abbasbandy ldquoHomotopy analysis method forquadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 3 pp539ndash546 2008
[17] M Gulsu and M Sezer ldquoOn the solution of the Riccati equa-tion by the Taylor matrix methodrdquo Applied Mathematics andComputation vol 176 no 2 pp 414ndash421 2006
[18] Y Li and LHu ldquoSolving fractional Riccati differential equationsusing Haar waveletrdquo in Proceedings of the 3rd InternationalConference on Information and Computing (ICIC 10) pp 314ndash317 Wuxi China June 2010
[19] N A Khan and A Ara ldquoFractional-order Riccati differentialequation analytical approximation and numerical resultsrdquoAdvances in Difference Equations vol 2013 article 185 2013
[20] M A Z Raja J A Khan and I M Qureshi ldquoA new stochasticapproach for solution of Riccati differential equation of frac-tional orderrdquo Annals of Mathematics and Artificial Intelligencevol 60 no 3-4 pp 229ndash250 2010
[21] M Merdan ldquoOn the solutions fractional Riccati differentialequation with modified RIEmann-Liouville derivativerdquo Inter-national Journal of Differential Equations vol 2012 Article ID346089 17 pages 2012
[22] N H Sweilam M M Khader and A M S Mahdy ldquoNumericalstudies for solving fractional Riccati differential equationrdquoApplications and AppliedMathematics vol 7 no 2 pp 595ndash6082012
[23] C K ChuiWavelets A Mathematical Tool for Signal ProcessingSIAM Philadelphia Pa USA 1997
[24] G Beylkin R Coifman and V Rokhlin ldquoFast wavelet trans-forms and numerical algorithms Irdquo Communications on Pureand Applied Mathematics vol 44 no 2 pp 141ndash183 1991
[25] M ur Rehman and R Ali Khan ldquoThe Legendre wavelet methodfor solving fractional differential equationsrdquoCommunications inNonlinear Science and Numerical Simulation vol 16 no 11 pp4163ndash4173 2011
[26] S Balaji ldquoA new approach for solving Duffing equationsinvolving both integral and non-integral forcing termsrdquo AinShams Engineering Journal 2014
[27] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent IIrdquo Geophysical Journal of the RoyalAstronomical Society vol 13 pp 529ndash539 1967
[28] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-Holland Mathematics Studies Elsevier 2006
[29] J S Gu and W S Jiang ldquoThe Haar wavelets operational matrixof integrationrdquo International Journal of Systems Science vol 27no 7 pp 623ndash628 1996
[30] M Razzaghi and S Yousefi ldquoLegendre wavelets direct methodfor variational problemsrdquo Mathematics and Computers in Sim-ulation vol 53 no 3 pp 185ndash192 2000
[31] M Razzaghi and S Yousefi ldquoLegendre wavelets method forconstrained optimal control problemsrdquo Mathematical Methodsin the Applied Sciences vol 25 no 7 pp 529ndash539 2002
[32] A Kilicman and Z A A Al Zhour ldquoKronecker operationalmatrices for fractional calculus and some applicationsrdquo AppliedMathematics and Computation vol 187 no 1 pp 250ndash265 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 International Journal of Mathematics and Mathematical Sciences
References
[1] N A Khan M Jamil A Ara and S Das ldquoExplicit solution fortime-fractional batch reactor systemrdquo International Journal ofChemical Reactor Engineering vol 9 article A91 2011
[2] V Feliu-Batlle R R Perez and L S Rodrıguez ldquoFractionalrobust control of main irrigation canals with variable dynamicparametersrdquoControl Engineering Practice vol 15 no 6 pp 673ndash686 2007
[3] I Podlubny ldquoFractional-order systems and 119875119868120582
119863120583-controllersrdquo
IEEE Transactions on Automatic Control vol 44 no 1 pp 208ndash214 1999
[4] R Garrappa ldquoOn some explicit Adams multistep methods forfractional differential equationsrdquo Journal of Computational andApplied Mathematics vol 229 no 2 pp 392ndash399 2009
[5] M Jamil and N A Khan ldquoSlip effects on fractional viscoelasticfluidsrdquo International Journal of Differential Equations vol 2011Article ID 193813 19 pages 2011
[6] F Mohammadi and M M Hosseini ldquoA comparative study ofnumerical methods for solving quadratic Riccati differentialequationsrdquo Journal of the Franklin Institute vol 348 no 2 pp156ndash164 2011
[7] R Shankar Principles of Quantum Mechanics Plenum PressNew York NY USA 1980
[8] S FragaM J Garcıa de la Vega and E S FragaTheSchrodingerand Riccati Equations vol 70 of Lect Notes Chem 1999
[9] L B Burrows and M Cohen ldquoSchrodingerrsquos wave equation-A lie algebra treatmentrdquo in Fundamental World of QuantumChemistry A Tribute to the Memory of Per-Olov Lowdin EJ Brandas and E S Kryachko Eds Kluwer Dordrecht TheNetherlands 2004
[10] S Abbasbandy ldquoHomotopy perturbation method for quadraticRiccati differential equation and comparison with Adomianrsquosdecomposition methodrdquo Applied Mathematics and Computa-tion vol 172 no 1 pp 485ndash490 2006
[11] Z Odibat and S Momani ldquoModified homotopy perturbationmethod application to quadratic Riccati differential equationof fractional orderrdquo Chaos Solitons amp Fractals vol 36 no 1 pp167ndash174 2008
[12] N A Khan A Ara and M Jamil ldquoAn efficient approach forsolving the Riccati equation with fractional ordersrdquo Computersamp Mathematics with Applications vol 61 no 9 pp 2683ndash26892011
[13] H Aminikhah and M Hemmatnezhad ldquoAn efficient methodfor quadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 4 pp835ndash839 2010
[14] S Abbasbandy ldquoIterated Hersquos homotopy perturbation methodfor quadratic Riccati differential equationrdquo Applied Mathemat-ics and Computation vol 175 no 1 pp 581ndash589 2006
[15] J Cang Y Tan H Xu and S Liao ldquoSeries solutions of non-lin-ear Riccati differential equations with fractional orderrdquo ChaosSolitons and Fractals vol 40 no 1 pp 1ndash9 2009
[16] Y Tan and S Abbasbandy ldquoHomotopy analysis method forquadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 3 pp539ndash546 2008
[17] M Gulsu and M Sezer ldquoOn the solution of the Riccati equa-tion by the Taylor matrix methodrdquo Applied Mathematics andComputation vol 176 no 2 pp 414ndash421 2006
[18] Y Li and LHu ldquoSolving fractional Riccati differential equationsusing Haar waveletrdquo in Proceedings of the 3rd InternationalConference on Information and Computing (ICIC 10) pp 314ndash317 Wuxi China June 2010
[19] N A Khan and A Ara ldquoFractional-order Riccati differentialequation analytical approximation and numerical resultsrdquoAdvances in Difference Equations vol 2013 article 185 2013
[20] M A Z Raja J A Khan and I M Qureshi ldquoA new stochasticapproach for solution of Riccati differential equation of frac-tional orderrdquo Annals of Mathematics and Artificial Intelligencevol 60 no 3-4 pp 229ndash250 2010
[21] M Merdan ldquoOn the solutions fractional Riccati differentialequation with modified RIEmann-Liouville derivativerdquo Inter-national Journal of Differential Equations vol 2012 Article ID346089 17 pages 2012
[22] N H Sweilam M M Khader and A M S Mahdy ldquoNumericalstudies for solving fractional Riccati differential equationrdquoApplications and AppliedMathematics vol 7 no 2 pp 595ndash6082012
[23] C K ChuiWavelets A Mathematical Tool for Signal ProcessingSIAM Philadelphia Pa USA 1997
[24] G Beylkin R Coifman and V Rokhlin ldquoFast wavelet trans-forms and numerical algorithms Irdquo Communications on Pureand Applied Mathematics vol 44 no 2 pp 141ndash183 1991
[25] M ur Rehman and R Ali Khan ldquoThe Legendre wavelet methodfor solving fractional differential equationsrdquoCommunications inNonlinear Science and Numerical Simulation vol 16 no 11 pp4163ndash4173 2011
[26] S Balaji ldquoA new approach for solving Duffing equationsinvolving both integral and non-integral forcing termsrdquo AinShams Engineering Journal 2014
[27] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent IIrdquo Geophysical Journal of the RoyalAstronomical Society vol 13 pp 529ndash539 1967
[28] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-Holland Mathematics Studies Elsevier 2006
[29] J S Gu and W S Jiang ldquoThe Haar wavelets operational matrixof integrationrdquo International Journal of Systems Science vol 27no 7 pp 623ndash628 1996
[30] M Razzaghi and S Yousefi ldquoLegendre wavelets direct methodfor variational problemsrdquo Mathematics and Computers in Sim-ulation vol 53 no 3 pp 185ndash192 2000
[31] M Razzaghi and S Yousefi ldquoLegendre wavelets method forconstrained optimal control problemsrdquo Mathematical Methodsin the Applied Sciences vol 25 no 7 pp 529ndash539 2002
[32] A Kilicman and Z A A Al Zhour ldquoKronecker operationalmatrices for fractional calculus and some applicationsrdquo AppliedMathematics and Computation vol 187 no 1 pp 250ndash265 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
