Research ArticleSpectral Collocation Method for Fractional DifferentialIntegralEquations with Generalized Fractional Operator

Qinwu Xu 1 and Zhoushun Zheng2

1Department of Mathematical Science Nanjing University 210093 Nanjing China2School of Mathematics and Statistics Central South University 410083 Changsha China

Correspondence should be addressed to Qinwu Xu xuqinwupkueducn

Received 30 September 2018 Accepted 13 November 2018 Published 1 January 2019

Guest Editor Ram Jiwari

Copyright copy 2019 Qinwu Xu and Zhoushun Zheng This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Generalized fractional operators are generalization of the Riemann-Liouville and Caputo fractional derivatives which includeErdelyi-Kober and Hadamard operators as their special cases Due to the complicated form of the kernel and weight function inthe convolution it is even harder to design high order numerical methods for differential equations with generalized fractionaloperators In this paper we first derive analytical formulas for 120572 minus 119905ℎ (120572 gt 0) order fractional derivative of Jacobi polynomialsSpectral approximation method is proposed for generalized fractional operators through a variable transform technique Thenoperational matrices for generalized fractional operators are derived and spectral collocation methods are proposed for differentialand integral equations with different fractional operators At last the method is applied to generalized fractional ordinarydifferential equation and Hadamard-type integral equations and exponential convergence of the method is confirmed Furtherbased on the proposed method a kind of generalized grey Brownian motion is simulated and properties of the model are analyzed

1 Introduction

During the last decade fractional calculus has emerged asa model for a broad range of nonclassical phenomena inthe applied sciences and engineering [1ndash5] Along with theexpansion of numerous and even unexpected recent appli-cations of the operators of the classical fractional calculusthe generalized fractional calculus is another powerful toolstimulating the development of this field [6ndash8] The notionldquogeneralized operator of fractional integrationrdquo appeared firstin the papers of the jubilarian professor S L Kalla in the years1969-1979 [9 10] A notable reference is the book by professorKiryakova [11] Later generalized fractional operator wasused successfully in papers of Mura and Mainardi [12]to model a class of self-similar stochastic processes withstationary increments which providedmodels for both slow-and fast-anomalous diffusion A brief review of generalizedfractional calculus is surveyed in [8] and further generaliza-tions of fractional integrals and derivatives are presented byAgrawal in [13] In the definition of generalized fractionaloperator more peculiar kernels are used which include the

classical power kernel function as a special case Due tothe special formulation all known fractional integrals andderivatives and other generalized integration and differentialoperators such as Hadamard operator and the Erdelyi-Koberoperator in various areas of analysis happened to fall inthe framework of this generalized fractional calculus It isshown in [13] that many integral equations can be writtenand solved in an elegant way using the generalized fractionaloperator

Differential and integral equations with generalized frac-tional operators have been investigated bymany authors fromtheoretical and application aspects [6 11 12 14ndash16] In [1217 18] analytical form solutions of some of these differentialand integral equations are given based on transmutationmethod Mittag-Leffler function Mainardi function and HFox functions However the analytical form solutions arevery complicated with infinity serials or integrations whichmakes them not suitable for fast computing while numericalmethods are more practical for these equations in applica-tions In [19] Xu proposes a finite difference scheme for time-fractional advection-diffusion equations with generalized

HindawiInternational Journal of Differential EquationsVolume 2019 Article ID 3734617 14 pageshttpsdoiorg10115520193734617

2 International Journal of Differential Equations

fractional derivative [19] Later a finite difference schemeand an analytical solution are studied for generalized time-fractional diffusion equation by Xu and Argrawal [20 21]These schemes are based on finite difference discretizationfor first order derivative which converge with order no morethan 2

Numerical methods with high order convergence havealso been developed for fractional differential equations egspectral method [22] discontinuous Galerkin method [23]and wavelet method [24] However they have not yet beenapplied to fractional differential equations with generalizedfractional operator Among these spectral method has beenconfirmed to be efficient and of high accuracy for somefractional differential equations To name a few in [25ndash27]Liu and his cooperators proposed a spectral approxima-tion method to fractional derivatives and studied spectralmethods for time-fractional Fokker-Planck equations Rieszspace fractional nonlinear reaction-diffusion equations Liand Xu proposed a spectral method for time-space fractionaldiffusion equation [22] Doha studied a spectral collocationmethod for multiterm fractional differential equations [28]Zayernouri and Karniadakis proposed a spectral method forfractional ODEs based on polyfractonomials [29] Zhao andZhang studied the super-convergence property of spectralmethod for fractional differential equations [30] Xu Hes-thaven and Chen proposed multidomain spectral methodsfor space and time-fractional differential equations separately[23 31] Recently efficient spectral-Galerkin algorithms aredeveloped by Mao and Shen to solve multidimensionalfractional elliptic equations with variable coefficients inconserved form as well as nonconserved form [32] Theclassical fractional derivative of any power function can beexpressed analyticallyThis indicates that fractional derivativeof any function in polynomial spaces can be evaluated exactlyHowever the definition of generalized fractional deriva-tive is more complicated than classical fractional deriva-tive It is far more difficult to construct a direct spectralapproximation to the generalized fractional derivative Inthis paper we firstly study fractional derivative of Jacobipolynomials and derive a general formula for fractionalderivative of Jacobi polynomial of any order Through a suit-able variable transform technique spectral approximationformulas are proposed for generalized fractional operatorsbased on Jacobi polynomials Then operational matricesare constructed and efficient spectral collocation meth-ods are proposed for the generalized fractional differentialand integral equations appearing in the references andapplications

The rest of the current paper is organized as follows InSection 2 we introduce the definitions of different general-ized fractional operators and someof their properties are alsogiven In Section 3 a general formula for fractional derivativeof Jacobi polynomial of any order is derived A spectralapproximation method for generalized fractional operatorsis proposed and operational matrices are constructed InSection 4 collocation methods are proposed for severaldifferential and integral equations Numerical experimentsare carried out to verify the accuracy and efficiency of themethods Finally we draw our conclusions in Section 5

2 Notations and Definitions

We first introduce some definitions and properties of frac-tional operators

Definition 1 (fractional integral [33]) The left fractionalintegral of order 120572 of a given function 119906(119905) in [119886 119887] is definedas

119886119868120572119905 119906 (119905) fl 1Γ (120572) int119905

0(119905 minus 119904)120572minus1 119906 (119904) d119904 (1)

The right fractional integral of order 120572 of 119906(119905) in [119886 119887] isdefined as

119905119868120572119887119906 (119905) fl 1Γ (120572) int119887

119905(119904 minus 119905)120572minus1 119906 (119904) d119904 (2)

Here Γ(sdot) denotes the Gamma function

Based on the fractional integral Riemann-Liouville andCaputo derivatives can be defined in the following way

Definition 2 (Riemann-Liouville derivative [33]) The leftRiemann-Liouville derivative of order 120572 of function 119906(119905) in[119886 119887] is defined as

119886119863120572119905 119906 (119905) fl 119889119899119889119905119899 119886119868119899minus120572119905 119906 (119905)

= 1Γ (119899 minus 120572) d119899

d119905119899 int119905

119886(119905 minus 119904)119899minus1minus120572 119906 (119904) d119904

(3)

The right Riemann-Liouville derivative of order 120572 of 119906(119905) in[119886 119887] is defined as

119905119863120572119887119906 (119905) fl (minus 119889119889119905)

119899

119905119868119899minus120572119887 119906 (119905)= 1Γ (119899 minus 120572) (minus 119889119889119905)

119899 int119887

119905(119904 minus 119905)119899minus1minus120572 119906 (119904) d119904

(4)

where 119899 = lceil120572rceil is an integer

Definition 3 (Caputo derivative [33]) The left Caputo deriva-tive of order 120572 of function 119906(119905) in [119886 119887] is given by

1198880119863120572

119905 119906 (119905) fl 0119868119899minus120572119905

119889119899119906 (119905)119889119905119899= 1Γ (119899 minus 120572) int

119905

0(119905 minus 119904)119899minus1minus120572 d119899119906 (119904)

d119904119899 d119904 (5)

The right Caputo derivative of order 120572 function 119906(119905) in [119886 119887]is given by

119888119905119863120572

119887119906 (119905) fl (minus1)119899 119905119868119899minus120572119887

119889119899119906 (119905)119889119905119899= (minus1)119899Γ (119899 minus 120572) int

119887

119905(119904 minus 119905)119899minus1minus120572 d119899119906 (119904)

d119904119899 d119904(6)

where 119899 = lceil120572rceil is an integer

International Journal of Differential Equations 3

In investigations of dual integral equations in some appli-cations the modifications of Riemann-Liouville fractionalintegrals and derivatives are widely usedThe important casesinclude Hadamard fractional operators and Erdelyi-Koberfractional operators

Riemann-Liouville fractional integro-differentiation isformally a fractional power (119889119889119909)120572 of the differentiationoperator 119889119889119909 and is invariant relative to translation if con-sidered on the whole axis Hadamard suggested a construc-tion of fractional integro-differentiation which is a fractionalpower of the type (119909(119889119889119909))120572This construction is well suitedto the case of the half-axis and invariant relative to dilation [7sect183]Thus Hadamard introduced fractional integrals of thefollowing form

Definition 4 (Hadamard fractional integral [34])

(119867120572119886+0119906) (119905) fl 1Γ (120572) int

119905

119886(log 119905119904)

120572minus1 119906 (119904) 119889119904119904 119905 gt 119886 gt 0

(7)

In 1993 Kilbas studied a weighted Hadamard fractionalintegral also calledHadamard-type fractional integral whichextended the application of Hadamard operators

Definition 5 (Hadamard-type fractional integral [34])

(119867120572119886+120583119906) (119905) fl 1Γ (120572) int

119905

119886(119904119905)

120583 (log 119905119904)120572minus1 119906 (119904) 119889119904119904

119905 gt 119886 gt 0(8)

In investigation of Hankel transform Erdelyi and Koberproposed the Erdelyi-Kober (E-K) operators They are gen-eralizations of the classical Riemann-Liouville fractionaloperators The left-sided E-K fractional integral of order 120572 isdefined by the following formula

Definition 6 (Erdelyi-Kober fractional integral [35])

119868120574120572120573 119906 (119905) fl 120573119905minus120573(120574+120572)Γ (120572) int119905

0(119905120573 minus 119904120573)120572minus1 119904120573(120574+1)minus1119906 (119904) d119904

120572 gt 0 120573 = 1 2 120574 isin R(9)

In order to introduce the definition of E-K fractionalderivative and its properties we define a special space offunctions that was first introduced in [36]

Definition 7 (see [36]) The function space119862120578 120578 isin R consistsof all functions 119891(119909) 119909 gt 0 that can be represented in theform 119891(119909) = 1199091199011198911(119909) with 119901 gt 120578 and 1198911 isin 119862([0infin])

The E-K fractional derivative of order 120572 is defined in thefollowing form

Definition 8 (Erdelyi-Kober fractional derivative [11 37])

(119863120574120572120573 119906) (119905) fl 119899prod

119895=1

(120574 + 119895 + 1120573119905 119889119889119905) (119868120574+120572119899minus120572120573 119906) (119905) (10)

where 119899 = lceil120572rceil is an integer

For the functions from the space 119862120578 120578 ge minus120573(120574 + 1) theleft-sidedE-K fractional derivative is a left-inverse operator tothe left-sidedE-K fractional integral (9) [38] then the relation

(119863120574120572120573 119868120574120572120573 119891) (119905) = 119891 (119905) (11)

holds true for every 119891 isin 119862120578In order to unify these definitions Agrawal [13] proposed

a new definitionwhich includesmost of them as special cases

Definition 9 (generalized fractional integral [13]) Theleftforward weightedscaled fractional integral of order120572 gt 0 of a function 119891(119905) with respect to another function 119911(119905)and weight 119908(119905) is defined as

(119868120572119886+[119911119908]119891) (119909) = [119908 (119909)]minus1Γ (120572) int119909

119886

119908 (119905) 1199111015840 (119905) 119891 (119905)[119911 (119909) minus 119911 (119905)]1minus120572 119889119905 (12)

In this definition if we set 119911(119904) = 119904 119908(119904) = 1 it reduces tothe classical Riemann-Liouville fractional integral Similarlysetting 119911(119904) = log(119904)119908(119904) = 119904120583 will lead toHadamard integraland 119911(119904) = 119904120573 119908(119904) = 119904120573120574 will lead to E-K fractional integralwith a factor 119905120573120572Definition 10 (see [13]) The leftforward weightedscaledderivative of integer order 119898 ⩾ 1 of a function 119891(119905) withrespect to another function 119911(119905) and weight 119908(119905) is definedas

(119863119898[119911119908119871]119891) (119909)= [119908 (119909)]minus1 [( 11199111015840 (119909)119863119909)119898 (119908 (119909) 119891 (119909))] (119909) (13)

Definition 11 (generalized Riemann-Liouville derivative [13])The leftforward weighted generalized Riemann-Liouvillefractional derivative of order 120572 gt 0 of a function 119891(119905) withrespect to another function 119911(119905) and weight119908(119905) is defined as

(119863120572119886+[1199111199081]119891) (119909) = 119863119898

119911119908119871 (119868119898minus120572119886+[119911119908]119891) (119909) (14)

Definition 12 (generalized Caputo derivative [13]) Theleftforward weighted generalized Caputo fractional deriva-tive of order 120572 gt 0 of a function 119891(119905) with respect to anotherfunction 119911(119905) and weight 119908(119905) is defined as

(119863120572119886+[1199111199082]119891) (119909) = (119868119898minus120572

119886+[119911119908]119863119898119911119908119871119891) (119909) (15)

Remark 13 In the definitions of generalized fractional oper-ators more general kernels and weight functions are used Itgeneralized nearly all the existing fractional operators in onespace dimension such as the Riemann-Liouville derivativethe Grunwald-Letnikov derivative the Caputo derivative theErdelyi-Kober-type fractional operator and the Hadamard-type fractional operator

3 Spectral Approximation of GeneralizedFractional Operator

In this section we will first study fractional deriva-tiveintegral of Jacobi polynomials and then derive a spectral

4 International Journal of Differential Equations

approximation for generalized fractional operators basedon Agrawalrsquos definitions Fractional derivativesintegrals ofothers type can be obtained as special cases

31 Fractional Derivative of Orthogonal Polynomials Denoteby 119869120573120574119895 (119909) the 119895-th order Jacobi polynomial with index (120573 120574)defined on [minus1 1]

As a set of orthogonal polynomials 119869120573120574119895 (119909)119873119895=0 satisfiesthe following three-term-recurrence relation [39]

1198691205731205740 (119909) = 11198691205731205741 (119909) = (120573 + 120574 + 2) 119909 + (120573 minus 120574) 119869120573120574119895+1 (119909) = (119860120573120574

119895 119909 minus 119861120573120574119895 ) 119869120573120574119895 (119909) minus 119862120573120574

119895 119869120573120574119895minus1 (119909) 1 ⩽ 119895 ⩽ 119873 minus 1

(16)

where the recursive coefficients are defined as

119860120573120574119895 = (2119895 + 120573 + 120574 + 1) (2119895 + 120573 + 120574 + 2)2 (119895 + 1) (119895 + 120573 + 120574 + 1)

119861120573120574119895 = (1205742 minus 1205732) (2119895 + 120573 + 120574 + 1)

2 (119895 + 1) (119895 + 120573 + 120574 + 1) (2119895 + 120573 + 120574) 119862120573120574119895 = (119895 + 120573) (119895 + 120574) (2119895 + 120573 + 120574 + 2)(119895 + 1) (119895 + 120573 + 120574 + 1) (2119895 + 120573 + 120574)

(17)

In order to derive fractional derivative of Jacobi polyno-mials we introduce some useful lemmas first

Lemma 14 For any 119899 isin N+ 120573 120574 isin R 120572 gt 0 the followingrelation holds

119869(minus1minus120572120574+120572+1)119899 (119909)= minus(119899 + 120574 + 1) (1 minus 119909)2 (119899 minus 120572) 119869(1minus120572120574+120572+1)119899minus1 (119909)minus 120572119869(minus120572120574+120572)119899 (119909)119899 minus 120572

(18)

119869(120573+120572+1minus1minus120572)119899 (119909)= (119899 + 120573 + 1) (1 + 119909)2 (119899 minus 120572) 119869(120573+120572+11minus120572)119899minus1 (119909)minus 120572119869(120573+120572minus120572)119899 (119909)119899 minus 120572

(19)

Proof According to [39 421] Jacobi polynomials withparameters 120573 120574 isin R are defined by

119869(120573120574)119899 (119909)= 119899sum

119895=0

1119895 (119899 minus 119895) Γ (119899 + 119895 + 120573 + 120574 + 1)Γ (119899 + 120573 + 120574 + 1) Γ (119899 + 120573 + 1)Γ (119895 + 120573 + 1) (119909 minus 12 )119895 (20)

Considering property of Jacobi polynomials

119869(120573120574)119899 (119909) = (minus1)119899 119869(120574120573)119899 (minus119909) (21)

Jacobi polynomials can be rewritten in the following form

119869(120573120574)119899 (119909)= 119899sum

119895=0

(minus1)119895+119899119895 (119899 minus 119895) Γ (119899 + 119895 + 120573 + 120574 + 1)Γ (119899 + 120573 + 120574 + 1) Γ (119899 + 120574 + 1)Γ (119895 + 120574 + 1) (119909 + 12 )119895 (22)

Define the following symbols

119875119860 = (119899 + 120573 + 1) (1 + 119909) 119869(120573+120572+11minus120572)119899minus1 (119909) 119875119861 = 2120572119869(120573+120572minus120572)119899 (119909) 119875119862 = 2 (119899 minus 120572) 119869(120573+120572+1minus1minus120572)119899 (119909)

(23)

Through a series of calculation we have

119875119861 + 119875119862= 119899sum

119895=0

2120572 (minus1)119895+119899119895 (119899 minus 119895) Γ (119899 + 119895 + 120573 + 1)Γ (119899 + 120573 + 1) Γ (119899 minus 120572 + 1)Γ (119895 minus 120572 + 1) (119909 + 12 )119895

+ 119899sum119895=0

2 (119899 minus 120572) (minus1)119895+119899119895 (119899 minus 119895) Γ (119899 + 119895 + 120573 + 1)Γ (119899 + 120573 + 1) Γ (119899 minus 120572)Γ (119895 minus 120572) (119909 + 12 )119895

= 119899sum119895=0

(minus1)119895+119899119895 (119899 minus 119895) Γ (119899 + 119895 + 120573 + 1)Γ (119899 + 120573 + 1) Γ (119899 minus 120572 + 1)Γ (119895 minus 120572 + 1) (119909 + 12 )119895 (2119895) 119875119860= 119899sum

119895=1

2 (minus1)119895+119899(119895 minus 1) (119899 minus 119895) Γ (119899 + 119895 + 120573 + 1)Γ (119899 + 120573 + 1) Γ (119899 minus 120572 + 1)Γ (119895 minus 120572 + 1) (119909 + 12 )119895

= 119875119861 + 119875119862

(24)

The equality (19) is proved Equality (18) can be provedsimilarly

Lemma 15 (see [39 P96]) For120572 gt 0 minus1 lt 119909 lt 1 120573 gt minus1 120574 isinR

(1 minus 119909)120573+120572 119869(120573+120572120574minus120572)119899 (119909)119869(120573+120572120574minus120572)119899 (1)

= Γ (120573 + 120572 + 1)Γ (120573 + 1) Γ (120572) int1

119909

(1 minus 119910)120573(119910 minus 119909)1minus120572

119869(120573120574)119899 (119910)119869(120573120574)119899 (1) 119889119910

(25)

For 120572 gt 0 minus1 lt 119909 lt 1 120574 gt minus1 120573 isin R

(1 + 119909)120574+120572 119869(120573minus120572120574+120572)119899 (119909)119869(120573minus120572120574+120572)119899 (1)

= Γ (120574 + 120572 + 1)Γ (120574 + 1) Γ (120572) int119909

minus1

(1 + 119910)120574(119909 minus 119910)1minus120572

119869(120573120574)119899 (119910)119869(120573120574)119899 (1) 119889119910

(26)

Lemma 16 (see [29 40]) For 120572 gt 0 minus1 lt 119909 lt 1 120573 120574 isin R

1199091198681205721119869(0120574)119899 (119909) = 1Γ (120572) int1

119909

119869(0120574)119899 (119910)(119910 minus 119909)1minus120572 119889119910

= 119899 (1 minus 119909)120572Γ (119899 + 120572 + 1)119869(120572120574minus120572)119899 (119909) (27)

International Journal of Differential Equations 5

minus1119868120572119909119869(1205730)119899 (119909) = 1Γ (120572) int119909

minus1

119869(1205730)119899 (119910)(119909 minus 119910)1minus120572 119889119910

= 119899 (1 + 119909)120572Γ (119899 + 120572 + 1)119869(120573minus120572120572)119899 (119909) (28)

Lemma 17 forall120572 gt 0 120572 notin N minus1 lt 119909 lt 1 120573 120574 isin R

1199091198631205721119869(0120574)119899 (119909) = 119899 (1 minus 119909)minus120572Γ (119899 + 1 minus 120572)119869(minus120572120574+120572)119899 (119909) (29)

minus1119863120572119909119869(1205730)119899 (119909) = 119899 (1 + 119909)minus120572Γ (119899 + 1 minus 120572)119869(120573+120572minus120572)119899 (119909) (30)

Proof First for the case 0 lt 120572 lt 1 formulas (29) and (30) canbe obtained by setting 120573 = minus120572 in formula (25) setting 120574 = minus120572in formula (26) and applying Riemann-Liouville fractionalderivative operator to both sides of themWe refer to [22] fordetailed discussion

For the case 120572 gt 1 120572 notin N the formulas cannot beobtained from Lemma 15 due to the constrains 120573 gt minus1 and120574 gt minus1 in Lemma 15 Here we prove the lemma inductively

For left Riemann-Liouville derivative from case 1 equal-ity (30) holds for lfloor120572rfloor = 0 Assume the equality holds forlfloor120572rfloor = 119896 minus 1 119896 isin N+ when lfloor120572rfloor = 119896 and let = 120572 minus 1from the assumption it holds that

minus1119863119909119869(1205730)119899 (119909) = 119899 (1 + 119909)minusΓ (119899 + 1 minus )119869(120573+minus)119899 (119909) (31)

We apply 119889119889119909 to both sides of (31) and then

minus1119863120572119909119869(1205730)119899 (119909) = minus1119863+1

119909 119869(1205730)119899 (119909)= 119899Γ (119899 + 1 minus ) (minus (1 + 119909)minusminus1 119869(120573+minus)119899 (119909)+ (1 + 119909)minus 119889119889119909119869(120573+minus)119899 (119909))= 119899 (1 + 119909)minusminus1Γ (119899 + 1 minus ) (minus119869(120573+minus)119899 (119909)+ (119899 + 120573 + 1) (1 + 119909)2 119869(120573++11minus)119899minus1 (119909))

(32)

Applying Lemma 14 we have

minus1119863120572119909119869(1205730)119899 (119909)= 119899 (1 + 119909)minusminus1Γ (119899 + 1 minus ) (minus (119899 minus 120572) 119869(120573++1minusminus1)119899 (119909))= 119899 (1 + 119909)minusminus1Γ (119899 + 1 minus ) ((119899 minus ) 119869(120573++1minusminus1)119899 (119909))= 119899 (1 + 119909)minus120572Γ (119899 + 1 minus 120572)119869(120573+120572minus120572)119899 (119909)

(33)

Equality (30) is proved Equality (29) can be proved similarly

Through the relationship between Caputo derivative andRiemann-Liouville derivative we immediately obtain thefractional derivative for Caputo derivative

Lemma 18 For 120572 gt 0 minus1 lt 119909 lt 1119888119909119863120572

1119869(0120574)119899 (119909) = 119899 (1 minus 119909)minus120572Γ (119899 + 1 minus 120572)119869(minus120572120574+120572)119899 (119909)minus 119898minus1sum

119895=0

120597119895119909119869(0120574)119899 (1) (1 minus 119909)119895minus120572(minus1)119895 Γ (1 + 119895 minus 120572)

119888minus1119863120572

119909119869(1205730)119899 (119909) = 119899 (1 + 119909)minus120572Γ (119899 + 1 minus 120572)119869(120573+120572minus120572)119899 (119909)minus 119898minus1sum

119895=0

120597119895119909119869(1205730)119899 (minus1) (1 + 119909)119895minus120572Γ (1 + 119895 minus 120572)

(34)

where119898 = lceil120572rceilGiven a function 119906(119909) isin 119867119898

120596 [minus1 1] and polynomialsspace P119873 the projection of 119906(119909) in space P119873 120587119873119906(119909)satisfies the following relation

(119906 minus 120587119873119906 V)120596 = 0 forallV isin P119873 (35)

According properties of space P119873 and Jacobi polynomials119906119873 can be expressed as

120587119873119906 (119909) = 119873sum119895=0

119895119869(120573120574)119895 (119909) (36)

where 119895 = (119906(119909) 119869(120573120574)(119909))120596119869(120573120574)(119909)2120596 120596 = 120596(119909) is theweight function

Then fractional derivative of 119906(119909) can be approximatedas

minus1119863120572119909119906 (119909) asymp 119873sum

119895=0

119895 ( minus1119863120572119909119869(120573120574)119895 (119909)) (37)

And we have the following lemma

Lemma 19 (see [41]) forall119906 isin 119867119898+1120596 [minus1 1] 0 le 119897 le 119898 119897 119898 isin

N 0 lt 120572 lt 1 119906119873 is a spectral approximation to 119906 in polynomialspace such that (119906 minus 120587119873119906 V)120596 = 0 forallV isin P119873 and then thereexists a constant 119862 such that10038171003817100381710038171003817 minus1119863119897+120572

119909 (119906 (119909) minus 120587119873119906 (119909))10038171003817100381710038171003817120596(120573+120572+119897120572+119897)le 119862119873119897minus119898 1003817100381710038171003817 minus1119863119898+120572

119909 1199061003817100381710038171003817120596(120573+120572+119898120572+119898) (38)

where 120596(120573120574) = (1 minus 119909)120573(1 + 119909)120574Combining Lemmas 18 and 19 and (37) approximation

method can be obtained immediately for Caputo derivativeand we have the following corollary

Corollary 20 forall119906 isin 119867119898+1120596 [minus1 1] 0 le 119897 le 119898 119897 119898 isin N 0 lt120572 lt 1 119906119873 is a spectral approximation to 119906 in polynomial space

6 International Journal of Differential Equations

such that (119906 minus 120587119873119906 V)120596 = 0 forallV isin P119873 and then there exists aconstant 119862 such that100381710038171003817100381710038171003817 119888

minus1119863119897+120572

119909 (119906 (119909) minus 120587119873119906 (119909))100381710038171003817100381710038171003817120596 le 119862119873119897minus119898 1003817100381710038171003817 minus1119863119898+120572119909 1199061003817100381710038171003817120596 (39)

Based on above lemmas fractional integral and derivativeof Jacobi polynomials in the standard interval [minus1 1] can beexpressed explicitly Fractional integrals and derivatives inshifted interval [119886 119887] can be obtained through proper variabletransforms

For any 119910 isin [119886 119887] we assume 119869120573120574119899lowast(119910) is defined in [119886 119887]Let ℎ = 119887 minus 119886 119909 = minus1 + 2((119910 minus 119886)ℎ) then 119909 isin [minus1 1]Substituting 119909 = minus1 + 2((119910 minus 119886)(119887 minus 119886)) into equation (27)-(28) and (29)-(30) we obtained

119910119868120572119887119869(0120574)119899lowast (119910) = (ℎ2)120572

1199091198681205721119869(0120574)119899 (119909) 119886119868120572119910119869(1205730)119899lowast (119910) = (ℎ2)

120572

minus1119868120572119909119869(1205730)119899 (119909) 119888119910119863120572

119887119869(0120574)119899lowast (119910) = (2ℎ)

120572119888119909119863120572

1119869(0120574)119899 (119909) 119888119886119863120572

119910119869(1205730)119899lowast (119910) = (2ℎ)120572

119888minus1119863120572

119909119869(1205730)119899 (119909)

(40)

32 Spectral Approximation to Generalized Fractional Opera-tors We assume 119911(119909) and 119908(119909) are positive monotone func-tions and 119911 119908 isin 119862[119886 119887] Obviously 119911(119909) and119908(119909) are invert-ible The following lemmas can be derived for generalizedfractional operators

Lemma 21 Let 119892(120577) = 119908(119911minus1(120577))119891(119911minus1(120577)) 0 lt 120572 lt 1 thenthe generalized fractional integral operator is equivalent to thefollowing classical fractional integral

(119868120572119886+[119911119908]119891) (119909) = [119908 (119909)]minus1 1205770119868120572120577119892 (119911 (119909)) (41)

Here 1205770 = 119911(119886)Proof From the definition of generalized fractional integralwe have

(119868120572119886+[119911119908]119891) (119909) = [119908 (119909)]minus1Γ (120572) int119909

119886

119908 (119905) 1199111015840 (119905) 119891 (119905)[119911 (119909) minus 119911 (119905)]1minus120572 119889119905 (42)

Since 120577 = 119911(119905) is positive monotone function then 120577 = 119911(119905) isinvertible and we have 119905 = 119911minus1(120577)(119868120572119886+[119911119908]119891) (119909) = [119908 (119909)]minus1Γ (120572)sdot int119911(119909)

119911(119886)

119908(119911minus1 (120577)) 1199111015840 (119911minus1 (120577)) 119891 (119911minus1 (120577))[119911 (119909) minus 120577]1minus120572 119889119911minus1 (120577)

= [119908 (119909)]minus1Γ (120572) int119911(119909)

119911(119886)

119908(119911minus1 (120577)) 119891 (119911minus1 (120577))[119911 (119909119896) minus 120577]1minus120572 119889120577

(43)

Let 119892(120577) = 119908(119911minus1(120577))119891(119911minus1(120577)) then the generalized frac-tional integral of 119891(119909) is converted to classical fractionalintegral of 119892(120577) in the following form

(119868120572119886+[119911119908]119891) (119909) = [119908 (119909)]minus1 1205770119868120572120577119892 (119911 (119909)) (44)

Lemma is proved

Lemma 22 Let 119892(120577) = 119908(119911minus1(120577))119891(119911minus1(120577)) 0 lt 120572 lt 1 thenthe generalized fractional derivative of order 120572 is equivalent tothe following classical fractional derivative

(119863120572119886+[1199111199082]119891) (119909) = [119908 (119909)]minus1 119888

1205770119863120572

120577119892 (119911 (119909)) (45)

Here 1205770 = 119911(119886)Proof Similar to the proof of Lemma 21 we have

(119863120572119886+[1199111199082]119891) (119909) = [119908 (119909)]minus1Γ (1 minus 120572)sdot int119911(119909)

119911(119886)

(120597120597120577) (119908 (119911minus1 (120577)) 119891 (119911minus1 (120577))) (120597120577120597119905)[119911 (119909) minus 120577]120572 119889119911minus1 (120577)

= [119908 (119909)]minus1Γ (1 minus 120572) int119911(119909)

119911(119886)

(120597120597120577) (119908 (119911minus1 (120577)) 119891 (119911minus1 (120577)))[119911 (119909) minus 120577]120572 119889120577

(46)

Let 119892(120577) = 119908(119911minus1(120577))119891(119911minus1(120577)) then the generalized frac-tional derivative of 119891(119909) is expressed through the classicalfractional derivative of 119892(120577)

(119863120572119886+[1199111199082]119891) (119909) = [119908 (119909)]minus1 119888

1205770119863120572

120577119892 (119911 (119909)) (47)

Lemma is proved

Remark 23 These two lemmas establish important relationbetween classical and generalized fractional operators Withthese lemmas generalized fractional differential equationscan be solved via classical fractional differential equationsand vice versa Which kind of transform should be takendepends on characters of the problem to be solved

In order to design a high order numerical approximationof the generalized fractional operator we define a scaled spaceP119899

[119908119911] such that

P119899[119908119911] = V (119909) = [119908 (119909)]minus1 119892 (119911 (119909)) 119892 (119909) isin P

119899 119909isin Ω = [119886 119887] (48)

whereP119899 is a polynomial space of up to order 119899Define a inner product and norm for space P119899

[119908119911] suchthat

(119906 (119909) V (119909))120596 = int119887

119886119906 (119909) V (119909) 120596 (119909) 119889119909

and V120596 = radic(V V)120596(49)

International Journal of Differential Equations 7

Define projection 119876119873 into space P119873119908119911 such that for any

function 119906(119909)(119906 minus 119876119873119906 V)120596 = 0 forallV isin P

119873119908119911 (50)

SupposeΦ119895(119909) 119895 = 0 119873 are a set of orthogonal basisfunctions in spaceP119873

119911119908 satisfying

(Φ119894 (119909) Φ119895 (119909))120596 = 1 119894 = 1198950 119894 = 119895 (51)

Let119873 997888rarr infin then Φ119895(119909) 119895 = 0 1 119873 form a 1198712120596(Ω)space and for any 119906(119909) isin 1198712120596(Ω) the projection 119876119873119906(119909) canbe written as

119876119873119906 (119909) = 119873sum119895=0

119895Φ119895 (119909) (52)

Here 119895 (119895 = 0 119873) are expansion coefficients such that

119895 = (119906 (119909) Φ119895 (119909))12059610038171003817100381710038171003817Φ119895 (119909)100381710038171003817100381710038172120596 (53)

The weight function 120596(119909) plays an important role in thecomputational process and analysis of the method Herewe choose a proper weight function to use properties oforthogonal polynomials and make the computation moreefficient Let 119901119895(119911(119909)) = 119908(119909)Φ119895(119909) and note that

(119876119873119906 (119909) Φ119894 (119909))120596 = int119887

119886( 119873sum

119895=0

119895119908minus1 (119909) 119901119895 (119911 (119909)))sdot 119908minus1 (119909) 119901119894 (119911 (119909)) 120596 (119909) 119889119909

(54)

Take 120596(119909) = 1199082(119909)1199111015840(119909) then(119876119873119906 (119909) Φ119894 (119909))120596 = int119911(119887)

119911(119886)( 119873sum

119895=0

119895119901119895 (119911))119901119894 (119911) 119889119911 (55)

Since 119901119895(119911) is polynomial of order 119895 the computation can becarried out easily through properties of orthogonal polyno-mials

Suppose 119875119895lowast(119909) is shifted Legendre polynomial definedon [119911(119886) 119911(119887)] Then given any function 119906(119909) isin 1198712120596 withweight 120596(119909) = 1199082(119909)1199111015840(119909) we have

119876119873119906 (119909) = [119908 (119909)]minus1 119873sum119895=0

119895119875119895lowast (119911 (119909)) (56)

Recalling Lemmas 21 and 22 the generalized fractionalintegral and derivative can be obtained in the form

(119868120572119886+[119911119908]119876119873119906) (119909)= [119908 (119909)]minus1 119873sum

119895=0

119895 ( 1205770119868120572120577119875119895lowast (120577)1003816100381610038161003816120577=119911(119909)) (57)

(119863120572119886+[1199111199082]119876119873119906) (119909)= [119908 (119909)]minus1 119873sum

119895=0

119895 ( 1198881205770119863120572

120577119875119895lowast (120577)1003816100381610038161003816120577=119911(119909)) (58)

From Lemma 19 the following corollary can be obtainedimmediately

Corollary 24 120577(119909) = 119911(119909) is a monotone increasing function119908(119909) gt 0 0 lt 120572 lt 1 119876119873 is a projection into space P119873119908119911119906(119911minus1(120577)) isin 119862119898(Ω) 119898 isin N then there exists a constant 119862120572

such that 10038171003817100381710038171003817119863120572119886+[1199111199082] (119906 (119909) minus 119876119873119906 (119909))100381710038171003817100381710038171205962le 1198621205721198731minus119898 10038171003817100381710038171003817 1205770119863119898+120572

119911119906 (120577minus1 (119911))100381710038171003817100381710038171205961

(59)

Here the weight function 1205961(120577) = (120577 minus 119886)minus120572(119887 minus 120577)120572 1205962(119909) =1205961(119911(119909))1199082(119909)1199111015840(119909)Remark 25 Unlike the convergence theory in classical poly-nomial space the convergence order in space P119873

119908119911 dependson regularity of the function 119906(119909) with respect to 119911(119909) Wewill illustrate this through numerical examples

Most of the time it is more convenient to consider theproblems in nodal form We assume the given interpolationpoints are 120577119895 = 119911(119909119895) (119895 = 0 1 119873) then the Lagrangebasis functions L119895(119909) (119895 = 0 1 119873) can be defined asfollows

L119894 (120577) = prod119895=0119873119895 =119894

(120577 minus 120577119895)(120577119894 minus 120577119895) (60)

The function 119906(119909) can be expressed using both Jacobipolynomials and Lagrange polynomials The following rela-tion is derived

119906 (119909) = [119908 (119909)]minus1 119873sum119895=0

119892119895119875119895lowast (119911 (119909))

= [119908 (119909)]minus1 119873sum119895=0

119908119895119906119895L119895 (119911 (119909)) (61)

Considering the equivalence between Legendre basis andLagrange basis the following equality holds

L119895 (119911) = 119873sum119896=0

119897119896119895119875119896lowast (119911) 119896 = 0 1 119873 (62)

where 119897119894119895 = (119871119895(119911) 119875119894lowast(119911))119875119894lowast(119911)21198712 Then the nodal form expansion of 119906(119909) is obtained

119906 (119909) = [119908 (119909)]minus1 119873sum119895=0

119873sum119896=0

119908119895119906119895119897119896119895119875119896lowast (119911 (119909)) (63)

8 International Journal of Differential Equations

From (57) and (58) the corresponding nodal form of gener-alized fractional integral and derivative are obtained

(119868120572119886+[119911119908]119906) (119909)= [119908 (119909)]minus1 119873sum

119895=0

119873sum119896=0

119908119895119906119895119897119896119895 ( 1205770119868120572120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119909)) (64)

(119863120572119886+[1199111199082]119906) (119909)= [119908 (119909)]minus1 119873sum

119895=0

119873sum119896=0

119908119895119906119895119897119896119895 ( 1198881205770119863120572

120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119909)) (65)

Example 26 Now we give an example to show the effec-tiveness and accuracy of the method Assuming 119911(119909) =radic1199093 119908(119909) = 1 we consider generalized fractional derivativeof 119910(119909) on the interval [0 1] with the following form

119910 (119909) = 5sum119895=1

(minus1)119895+1 1199093119895(2119895) (66)

The exact generalized fractional derivative of 119910(119909) is119863120572

0+[1199111199082]119910 (119909) = 5sum119895=1

(minus1)119895+1 1199093119895minus31205722Γ (2119895 + 1 minus 120572) (67)

Numerical approximation to generalized fractionalderivative of 119910(119909) can be obtained using (58) We considerthe maximum absolute error

119890 = max119909isin[01]

100381610038161003816100381610038161198631205720+[1199111199082]119910 (119909) minus 119863120572

0+[1199111199082]119876119873119910 (119909)10038161003816100381610038161003816 (68)

of the numerical derivative Results for 120572 = 02 05 08 areshown in Figure 1

For this example it is easy to check that 119910 isin P10119908119911 From

theory of spectral approximation the error would decreaseexponentially when 119873 lt 10 and the numerical fractionalderivative would be exact when119873 ge 10 Our numerical resultcoincides with the theory exactly

Example 27 In this example we test the spectral approxi-mation of Hadamard integral Considering Hadamard-typefractional integral of 119910(119909) = sin(120587119909) on the interval [1 2]when 119908(119909) = 1 and 120572 = 1 the Hadamard integral of 119910(119909) isSine integral function Si(120587119909) minus Si(120587) for more general 119908(119909)and 120572 the exact Hadamard-type integral of 119910(119909) is unknownHere we consider several pairs of 119908(119909) and 120572 NumericalHadamard-type integral of 119910(119909) would be computed using(57) with 119911(119909) = log(119909) To evaluate the approximationaccuracy for the case 119908(119909) = 1 120572 = 1 the exact Hada-mard fractional integral is computed usingMATLAB built-infunction 119904119894119899119894119899119905 for other cases ldquoexactrdquo Hadamard fractionalintegral is computed using (57) with large 119873 (eg 119873 =50) which is treated as reference solution The results fornumerical Hadamard integral and approximation error areshown in Figures 2 and 3

For this example 119906 is not in the spaceP119899119908119911 for any 119899 Max-

imum error of approximated fractional integral convergesexponentially until reaching machine accuracy

10minus16

10minus14

10minus12

10minus10

10minus8

10minus6

10minus4

10minus2

max

imum

erro

r

5 6 7 8 94 11 1210

N

alpha=02alpha=05alpha=08

Figure 1 Error of numerical approximation to generalized frac-tional derivative of 119910(119909)

11 12 13 14 15 16 17 18 19 21x

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Had

amar

d fra

ctio

nal i

nteg

ral

w(x)=1=1w(x)=1=06

w(x)=x=1w(x)=x=06

Figure 2 Hadamard fractional integral for different weight 119908(119909)and 120572

33 Fractional IntegralDifferential Matrices Suppose 119906(119909) isinP119873

119911119908 119909119894 (119894 = 0 1 119873) are the interpolation points and119910119895 (119895 = 0 1 119873) are the collocation points we define thefollowing symbols

U = [119906 (1199090) 1199061199091 119906 (119909119873)]119879 (69)

U(120572) = [119863120572

119886+[119911119908]119906 (1199100) 119863120572119886+[119911119908]119906 (1199101)

119863120572119886+[119911119908]119906 (119910119873)]119879 120572 gt 0 (70)

International Journal of Differential Equations 9

10minus16

10minus14

10minus12

10minus10

10minus8

10minus6

10minus4

max

imum

erro

r

w(x)=1=1w(x)=1=06

w(x)=x=1w(x)=x=06

10 12 14 16 18 20 228x

Figure 3 Approximation error of Hadamard fractional integral fordifferent weight 119908 and 120572

U(120572) = [119868minus120572119886+[119911119908]119906 (1199100) 119868minus120572119886+[119911119908]119906 (1199101) 119868minus120572119886+[119911119908]119906 (119910119873)]119879 120572 lt 0 (71)

We define a generalized fractional differentialintegralmatrixM120572 such that

M120572119894119895 = 119873sum

119896=0

119908(119909119895) 119897119896119895119908 (119910119894) ( 1198881205770119863120572

120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119910119894))

for 120572 gt 0(72)

M120572119894119895 = 119873sum

119896=0

119908(119909119895) 119897119896119895119908 (119910119894) ( 1205770119868120572120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119910119894))

for 120572 lt 0(73)

In order to compute the fractional matrix M120572 moreefficiently we define a few more matrices L and L120572 are(119873 + 1) times (119873 + 1) matrices such thatL119894119895 = L119895(119911(119910119894))L120572

119894119895 =1205770119863120572

120577L119895(119911)|119911=119911(119910119894) 119894 119895 = 0 1 119873 V is defined based on

values of Legendre polynomials at interpolation points 119911(119909119894)such that

V119894119895 = 119875119895lowast (119911 (119909119894)) 119894 119895 = 0 1 119873 (74)

D120572 is defined as fractional derivativeintegral of Legendrepolynomials at collocation points 120577119894 = 119911(119910119894)

D120572119894119895 = 119888

1205770119863120572

120577119875119895lowast (120577119894) 120572 gt 0 (75)

D120572119894119895 = 1205770

119868120572120577119875119895lowast (120577119894) 120572 lt 0 (76)

W119897 andW119903 are the weight matrices defined as follows

W119903119894119895 =

119908(119909119895) 119894 = 1198950 otherwise (77)

W119897119894119895 =

1119908(119910119895) 119894 = 119895 gt 10 otherwise (78)

Theorem 28 For 119906(119909) isin P119873119911119908 vectors defined in (69)ndash(71)

and matrices defined in (72)ndash(78) the following relation holds

U(120572) = M

120572U = (W119897D120572V

minus1W

119903)U (79)

Proof From the definition (69)ndash(71) (72)ndash(78) and (63)-(64) it is easy to obtain

U(120572) = M

120572U (80)

Next we prove thatM120572 = W119897D120572Vminus1W119903From the definition of the interpolation function L119895(119911)

119875119895lowast (119911) = 119873sum119894=0

119875119895lowast (119911 (119909119894)) L119894 (119911) (81)

Suppose 997888rarr119875(119911) = (1198750lowast(119911) 1198751lowast(119911) 119875119873lowast(119911)) 997888rarrL (119911) =(L0(119911) L1(119911) L119873(119911)) for 120572 gt 0 we have997888rarr119875 (119911) = 997888rarrL (119911)V

1205770119863120572

120577

997888rarr119875 (119911) = ( 1205770119863120572

120577

997888rarrL (119911))V (82)

Evaluating the matrices multiplication for each elementthe following relation is derived

(W119897D120572V

minus1W119903)119894119895 = (W119897L

120572W119903)119894119895

= 119908 (119909119895)119908 (119910119894) L(120572)119895 (119911 (119910119894))(83)

= 119873sum119896=0

119908(119909119895) 119897119896119895119908 (119910119894) ( 1205770119863120572

120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119910119894)) = M

120572119894119895 (84)

The case 120572 lt 0 can be proved similarly

Remark 29 Collocation points 119910119895 and the interpolationpoints 119909119894 are not necessary the same To obtain a goodapproximation interpolation points ofGauss-type are usuallyused At the same time collocation points should be chosenproperly to guarantee stability properties of the methodIn the following numerical examples for computation andstability aim both interpolation and collocation points arechosen based on Gauss-type points with respect to 119911(119909)

10 International Journal of Differential Equations

4 Collocation Methods for FractionalDifferential and Integral Equations

41 Fractional Ordinary Differential Equations In this sub-section we consider collocation method for the generalizedfractional ordinary differential equation of the form

1198631205720+[1199111199082]119906 (119909) = 120582 (119909) 119906 (119909) + 119891 (119909)

119909 isin Ω = (0 119887] (85)

119906 (0) = 1199060 (86)

Here 0 lt 120572 lt 1 119908(119909) gt 0 119911(119909) gt 0 and 119911(119909) is a monotonefunction inΩ

We assume 119906119873(119909) isin P119873119911119908 is a numerical solution of the

equation 119909119894(119894 = 0 1 119873) are the chosen interpolationpoints 119906119895119873 = 119906119873(119909119895) The following discretized equation isobtained

[119908 (119909)]minus1 119873sum119895=0

119873sum119896=0

119908(119909119895) 119906119895119873119897119896119895 ( 1198881205770119863120572

120577119875119896lowast (119911 (119909)))

= 120582 (119909) 119906119873 (119909) + 119891 (119909) (87)

Let (87) hold on collocation points 119910119895 (119895 = 1 119873) thematrix form is obtained

(M120572 minus Λ)U = 119865 (88)

Here Λ is a diagonal matrix with Λ 119894119894 = 120582(119910119894) 119865 = (1199060 119891(1199101) 119891(119910119873))119879Considering the initial condition we set S = M120572 minus Λ

with the first row replaced by (1 0 0) 1198651015840 = 119865with its firstelement replaced by 1199060 Then the solution U is obtained bysolving the matrix equation SU = 1198651015840

Example 30 Consider the following example

1198631205720+[1199111199082]119906 (119909) = (1 + 119909) 119906 (119909)

+ Γ (15119903 + 25) 119909119903minus(23)120572Γ (15119903 + 25 minus 120572)minus 119909119903 (1 + 119909) 119909 isin (0 1]

(89)

119906 (0) = 0 (90)

Here 119911(119909) = 11990923119908(119909) = 119909 119903 is an arbitrary positive numberThe exact solution of the ordinary differential equation is119906(119909) = 119909119903 Maximum absolute errors of numerical solutions

for 119903 = 6 7 and 120572 = 03 06 09 are shown in Figures 4 and 5When 119911(119909) = 11990923 119908(119909) = 119909 the scaled polynomial

spaceP119873119911119908 becomes

P119873119911119908 = 119904119901119886119899 11990921198993minus1 119899 = 1 2 119873 (91)

For 119903 = 7 the error converges exponentially and reachesmachine accuracy at 119873 = 12 It is faster than any finitedifference method while for 119903 = 6 solution convergesalgebraically as119873 increases however it still reaches machineaccuracy at 119873 = 23 The major reason for this is that 119906(119909) =1199097 isin P12

119911119908 and 119906(119909) = 1199096 notin P119873119911119908 for any119873 isin N

6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

N

10minus15

10minus10

10minus5

100

max

erro

r

=03=06=09

Figure 4 Log-log plot of the maximum error for 119903 = 6

=03=06=09

10minus15

10minus10

10minus5

100

max

erro

r

15 20 25 30 3510N

Figure 5 Semilog plot of the maximum error for 119903 = 7

Remark 31 Since space P119873119908119911 is transformed from classical

polynomial space with respect to 119911(119909) the convergenceof spectral collocation method for ODEs with generalizedfractional operators depends not only on the smoothness ofthe solution itself but also on the scale function 119911(119905)42 Hadamard-Type Integral Equations We consider thefollowing Hadamard-type boundary value problem

1Γ (120572) int119909

119886( 119904119909)

120583 (log 119909119904 )120572minus1 119891 (119904) 119889119904119904 = 119892 (119909)

119909 isin Ω = (119886 119887] (92)

International Journal of Differential Equations 11

In [42] Kilbas discussed the existence of the solution of(92) Explicit formulas for the solution 119891(119905) were establishedin the following theorem

Theorem 32 (see [42]) If 119909120583119892(119909) isin 119860119862[119886 119887] then theHadamard-type integral equation (92) with 0 lt 120572 lt 1 issolvable in 119883120583(119886 119887) and its solution may be represented in theform

119891 (119909) = 119909minus120583Γ (1 minus 120572) [119886120583119892 (119886) (log 119909119886)minus120572

+ int119909

119886(log 119909119904 )

minus120572 (119904120583119892 (119904))1015840 119889119904] (93)

Here 119860119862[119886 119887] is the set of absolutely continuous functions on[119886 119887] and119883120583(119886 119887) is space of those Lebesgue measurable func-tions 119891 on [119886 119887] for which 119909120583minus1119891(119909) is absolutely integrable

Solution of the Hadamard-type integral equation isexactly the same as generalized fractional derivative ofRiemann-Liouville type with 119911(119904) = log(119904) 119908(119904) = 119904120583 Therelationship between (93) and Caputo type generalized frac-tional derivative is

119891 (119909) = 119886120583119892 (119886) 119909minus120583Γ (1 minus 120572) (log 119909119886)minus120572 + 119863120572

119886+[1199111199082]119892 (119909) (94)

Suppose 119909119895 119895 = 0 1 119873 are interpolation points thediscretized solution of equation (92) is

119891119873 (119909) = 119886120583119892 (119886) 119909minus120583Γ (1 minus 120572) (log 119909119886)minus120572

+ 119909minus120583 119873sum119895=0

119873sum119896=0

119909120583119895119892 (119909119895) 119897119896119895( 119888120577119886119863120572

120577119875119896lowast (log (119909))

(95)

Rewriting (95) in matrix form we have

119865 = 119866119886 +M120572119866 (96)

where M120572 is generalized fractional differential matrix119866 = (119892(1199090) 119892(1199091) 119892(119909119873))119879 119865 = (119891119873(1199090) 119891119873(1199091) 119891119873(119909119873))119879 119866119886 is a vector about the initial condition of theintegral equation defined by

119866119886 = 119886120583119892 (119886)Γ (1 minus 120572) (119909minus1205830 (log 1199090119886 )minus120572 119909minus1205831 (log 1199091119886 )

minus120572 119909minus120583119873 (log 119909119873119886 )minus120572)119879

(97)

Example 33 Assume 119892(119909) = sin(119909 minus 1) 120583 = 13 Ω = [110] Solutions of (92) for 120572 = 03 06 09 are shown inFigure 6

2 4 6 8 10 120X

minus8

minus6

minus4

minus2

0

2

4

6

8

f(X)

=03=06=09

Figure 6 Solution of (92)

43 Erdelyi-Kober Fractional Diffusion Equation In this sub-section we consider the following Erdelyi-Kober fractionaldiffusion equation [12]

119906 (119909 119905) = 1199060 (119909) + 1Γ (120573)sdot int119905

0

120572120573119904120572120573minus1 (119905120572120573 minus 119904120572120573)120573minus1 12059721205971199092 119906 (119909 119904) d119904(98)

Erdelyi-Kober fractional diffusion equation which is alsocalled stretched time-fractional diffusion equation is themaster equation of a kind of generalized grey Brownianmotion (ggBm) The ggBm is a parametric class of stochasticprocesses that provides models for both fast and slow anoma-lous diffusion This class is made up of self-similar processes119861120572120573(119905)with stationary increments and it depends on two realparameters 0 lt 120572 le 2 and 0 lt 120573 le 1 It includes thefractional Brownian motion when 0 lt 120572 le 2 and 120573 = 1 thetime-fractional diffusion stochastic processes when 0 lt 120572 =120573 lt 1 and the standard Brownian motion when 120572 = 120573 = 1About the relationship between stochastic process119861120572120573(119905) andstretched time-fractional diffusion equation the followingproposition is presented in [12]

Proposition 34 The marginal probability density function119891120572120573(119909 119905) of the process 119861120572120573(119905) 119905 ge 0 is the fundamentalsolution of the stretched time-fractional diffusion equation

119906 (119909 119905) = 1199060 (119909) + 1Γ (120573)sdot int119905

0

120572120573119904120572120573minus1 (119905120572120573 minus 119904120572120573)120573minus1 12059721205971199092 119906 (119909 119904) d119904(99)

12 International Journal of Differential Equations

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 7 Standard Brownian motion 120572 = 1 120573 = 1

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 8 Time-fractional diffusion with 120572 = 06 120573 = 06

Recalling the definition of generalized fractional integraland setting 119911(119905) = 119905120572120573 119908(119905) = 1 the equation can be rewrit-ten as

119906 (119909 119905) = 1199060 (119909) + 1198681205730+[119911119908]119906119909119909 (119909 119905) (100)

We use collocation method for both space and timediscretization We choose Legendre-Gauss-Lobatto (L-G-L)points 119909119894 (119894 = 0 1 119872) as the space collocation points andchoose 119905119895 (119895 = 0 1 119873) such that 119911(119905119895) are L-G-L pointsas the time collocation points

Define space collocation matrix M2 such that M2119894119895 =(11988921198891199092)L119895(119909119894) and generalized fractional integral matrix

M120573 Matrix M120573 is computed through Theorem 28 and thespace-time collocation matrices are obtained using Kro-necker product 1015840otimes1015840 Suppose A and B are space-timecollocation matrices with dimension (119872 + 1)(119873 + 1) times (119872 +1)(119873 + 1) for the second order derivative and fractionalintegral of order 120573 separately Then

A = M2 otimesI119873+1

B = I119872+1 otimesM120573 (101)

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 9 Fractional Brownian motion with 120572 = 15 120573 = 1

Suppose 119906119873(119909 119905) is the numerical solution of (98) defin-ing 119906119873119894119895 = 119906119873(119909119894 119905119895) solution vectorU and initial vector 1198800such that

U = [11990611987300 11990611987301 1199061198730119873 11990611987310 1199061198731119873 1199061198731198720 119906119873119872119873]119879

1198800 = [1199060 (1199090) 1199060 (1199090) 1199060 (1199091) 1199060 (1199091) 1199060 (119909119872) 1199060 (119909119872)]119879

(102)

where in the definition of 1198800 each 1199060(119909119895) is repeated 119873 + 1times

The matrix form discretized equation of (98) is obtainedas

U = 1198800 +BAU (103)

In the discretized equation initial condition is explicitlyinvolved After boundary condition added properly numeri-cal solution can be obtained by solving the matrix equation

Example 35 Assume 119909 isin Ω = (minus1 1) 119905 isin (0 05] 1199060(119909) =119890minus101199092 minus 119890minus10 119906(sdot 119905)|120597Ω = 0 Numerical solutions with 119872 =119873 = 50 are shown in Figures 7ndash10

Erdelyi-Kober diffusion equation characterizes the mar-ginal density function of the process 119861120572120573(119905) 119905 ge 0 When120572 = 120573 = 1 we recover the standard diffusion equationWhen0 lt 120572 = 120573 lt 1 we get the time-fractional diffusion equationof order 120573 When 120573 = 1 and 0 lt 120572 lt 2 we have the equationof the fractional Brownian motion marginal density

As shown in Figures 7 and 8 when 1 lt 120572 lt 2 the diffu-sion is fast and the increments exhibit long-range depen-dence when 0 lt 120572 lt 1 the diffusion is slow and theincrements form a stationary process which does not exhibitlong-range dependenceThe results coincide with theoreticalanalysis in [12 14]

International Journal of Differential Equations 13

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 10 Fractional Brownian motion with 120572 = 05 120573 = 1

5 Conclusion

In this paper we propose a spectral collocation method fordifferential and integral equations with generalized fractionaloperators To deal with the difficulty in designing spectralapproximation scheme due to complexity of integral kerneland weight a variable transform technique is applied to thegeneralized fractional operator and a spectral approximationmethod is proposed for the generalized fractional operatorOperational matrices for generalized fractional operatorsare derived Spectral collocation methods are designed forfractional ordinary differential equations Hadamard-typeintegral equations and Erdelyi-Kober diffusion equationsseparately Numerical experiments are carried out to verifythe accuracy and efficiency of the method and characteristicsof the Erdelyi-Kober diffusion equation are analyzed basedon numerical results

Data Availability

(i) The programs used to support the findings of this studyhave been deposited in the GitHub repository (httpsgithubcomqinwuxuSpectralGFPDE ) (ii) No data were used tosupport this study

Disclosure

An earlier version of this workwas presented at the ldquo8th Inter-national Congress on Industrial and Applied Mathematics(ICIAM 2015)rdquo

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The first author is supported by the National Key RampD Pro-gram of China (No 2017YFC0209804) the National ScienceFoundation for Young Scientists of China (No 11701273) andYouth Foundation of Jiangsu Province (No BK20170628)

The second author is supported by the National Key RampDProgram of China (No 2017YFB0305601)

References

[1] B BaeumerDA BensonMMMeerschaert and SWWheat-craft ldquoSubordinated advection-dispersion equation for contam-inant transportrdquo Water Resources Research vol 37 no 6 pp1543ndash1550 2001

[2] E Barkai RMetzler and J Klafter ldquoFrom continuous time ran-dom walks to the fractional Fokker-Planck equationrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 61no 1 pp 132ndash138 2000

[3] A Blumen G Zumofen and J Klafter ldquoTransport aspects inanomalous diffusion Levy walksrdquo Physical Review A AtomicMolecular and Optical Physics vol 40 no 7 pp 3964ndash39731989

[4] J P Bouchaud and A Georges ldquoAnomalous diffusion in dis-ordered media statistical mechanisms models and physicalapplicationsrdquoPhysics Reports vol 195 no 4-5 pp 127ndash293 1990

[5] M Raberto E Scalas and F Mainardi ldquoWaiting-times andreturns in high-frequency financial data an empirical studyrdquoPhysica A Statistical Mechanics and its Applications vol 314 no1ndash4 pp 749ndash755 2002

[6] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations New York NYUSA Elsevier 2006

[7] S G Samko A A Kilbas and O I Marichev Fractional Inte-grals and Derivatives Theory and Applications Gordon andBreach Yverdon Switzerland 1993

[8] V Kiryakova ldquoA brief story about the operators of the general-ized fractional calculusrdquo Fractional CalculusampAppliedAnalysisAn International Journal forTheory and Applications vol 11 no2 pp 203ndash220 2008

[9] S L Kalla ldquoOn operators of fractional integration Irdquo Mathe-maticae Notae vol 22 pp 89ndash93 197071

[10] S L Kalla ldquoOn operators of fractional integration IIrdquo Mathe-maticae Notae vol 25 pp 29ndash35 1976

[11] V S Kiryakova Generalized Fractional Calculus and Applica-tions Long-man amp J Wiley Harlow New York NY USA 1994

[12] A Mura and F Mainardi ldquoA class of self-similar stochasticprocesses with stationary increments to model anomalousdiffusion in physicsrdquo Integral Transforms and Special Functionsvol 20 no 3-4 pp 185ndash198 2009

[13] O P Agrawal ldquoSome generalized fractional calculus operatorsand their applications in integral equationsrdquo Fractional Calculusand Applied Analysis An International Journal for Theory andApplications vol 15 no 4 pp 700ndash711 2012

[14] G Pagnini ldquoErdelyi-Kober fractional diffusionrdquo FractionalCalculus and Applied Analysis An International Journal forTheory and Applications vol 15 no 1 pp 117ndash127 2012

[15] A Mura and G Pagnini ldquoCharacterizations and simulations ofa class of stochastic processes to model anomalous diffusionrdquoJournal of Physics A Mathematical and General vol 41 no 28285003 22 pages 2008

[16] E K Lenzi L R Evangelista M K Lenzi H V Ribeiro and EC de Oliveira ldquoSolutions for a non-Markovian diffusion equa-tionrdquo Physics Letters A vol 374 no 41 pp 4193ndash4198 2010

[17] B Al-Saqabi and V S Kiryakova ldquoExplicit solutions of frac-tional integral and differential equations involving Erderyi-Kober operatorsrdquo Applied Mathematics and Computation vol95 no 1 pp 1ndash13 1998

14 International Journal of Differential Equations

[18] L A Hanna and Y F Luchko ldquoOperational calculus for theCaputo-type fractional Erdelyi-Kober derivative and its appli-cationsrdquo Integral Transforms and Special Functions vol 25 no5 pp 359ndash373 2014

[19] Y Xu Z He and Q Xu ldquoNumerical solutions of fractionaladvection-diffusion equations with a kind of new generalizedfractional derivativerdquo International Journal of Computer Math-ematics vol 91 no 3 pp 588ndash600 2014

[20] Y Xu Z He and O P Agrawal ldquoNumerical and analytical solu-tions of new generalized fractional diffusion equationrdquo Com-puters amp Mathematics with Applications vol 66 no 10 pp2019ndash2029 2013

[21] Y Xu and O P Agrawal ldquoNumerical solutions and analysisof diffusion for new generalized fractional Burgers equationrdquoFractional Calculus and Applied Analysis An International Jour-nal forTheory and Applications vol 16 no 3 pp 709ndash736 2013

[22] X Li and C Xu ldquoA space-time spectral method for the timefractional diffusion equationrdquo SIAM Journal on NumericalAnalysis vol 47 no 3 pp 2108ndash2131 2009

[23] Q Xu and J S Hesthaven ldquoStable multi-domain spectralpenalty methods for fractional partial differential equationsrdquoJournal of Computational Physics vol 257 pp 241ndash258 2014

[24] RMittal and S Pandit ldquoQuasilinearized Scale-3 Haar wavelets-based algorithm for numerical simulation of fractional dynam-ical systemsrdquo Engineering Computations vol 35 no 5 pp 1907ndash1931 2018

[25] C Li F Zeng and F Liu ldquoSpectral approximations to the frac-tional integral and derivativerdquo Fractional Calculus and AppliedAnalysis vol 15 no 3 pp 383ndash406 2012

[26] M Zheng F Liu I Turner and V Anh ldquoA novel high orderspace-time spectral method for the time fractional Fokker-Planck equationrdquo SIAM Journal on Scientific Computing vol 37no 2 pp A701ndashA724 2015

[27] F Zeng F Liu C Li K Burrage I Turner and V Anh ldquoACrank-Nicolson ADI spectral method for a two-dimensionalRiesz space fractional nonlinear reaction-diffusion equationrdquoSIAM Journal on Numerical Analysis vol 52 no 6 pp 2599ndash2622 2014

[28] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical ModellingSimulation andComputation for Engineering and EnvironmentalSystems vol 35 no 12 pp 5662ndash5672 2011

[29] M Zayernouri and G E Karniadakis ldquoExponentially accuratespectral and spectral element methods for fractional ODEsrdquoJournal of Computational Physics vol 257 pp 460ndash480 2014

[30] X Zhao and Z Zhang ldquoSuperconvergence points of fractionalspectral interpolationrdquo SIAM Journal on Scientific Computingvol 38 no 1 pp A598ndashA613 2016

[31] F Chen Q Xu and J S Hesthaven ldquoA multi-domain spectralmethod for time-fractional differential equationsrdquo Journal ofComputational Physics vol 293 pp 157ndash172 2015

[32] Z Mao and J Shen ldquoEfficient spectral-Galerkin methods forfractional partial differential equations with variable coeffi-cientsrdquo Journal of Computational Physics vol 307 pp 243ndash2612016

[33] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[34] J Hadamard ldquoEssai sur lrsquoetude des fonctions donnees par leurdeveloppement de Taylorrdquo Journal de Mathematiques Pures etAppliquees vol 4 pp 101ndash186 1892

[35] A Erdelyi andHKober ldquoSome remarks onHankel transformsrdquoQuarterly Journal of Mathematics vol 11 pp 212ndash221 1940

[36] I Dimovski ldquoOperational calculus for a class of differentialoperatorsrdquo Comptes Rendus De L Academie Bulgare Des Sci-ences vol 19 pp 1111ndash1114 1966

[37] S B Yakubovich and Y F LuchkoThe hypergeometric approachto integral transforms and convolutions vol 287 ofMathematicsand its Applications Kluwer Academic Publishers Dordrecht-Boston-London 1994

[38] Y Luchko ldquoOperational rules for a mixed operator of theErdelyi-Kober typerdquo Fractional Calculus and Applied Analysisvol 7 no 3 pp 339ndash364 2004

[39] G Szego Orthogonal polynomials American MathematicalSociety Providence 1992

[40] M Zayernouri and G E Karniadakis ldquoFractional Sturm-Liou-ville eigen-problems theory and numerical approximationrdquoJournal of Computational Physics vol 252 pp 495ndash517 2013

[41] S Chen J Shen and L-L Wang ldquoGeneralized Jacobi functionsand their applications to fractional differential equationsrdquoMathematics of Computation vol 85 no 300 pp 1603ndash16382016

[42] A A Kilbas ldquoHadamard-type integral equations and fractionalcalculus operatorsrdquo in Singular integral operators factorizationand applications vol 142 ofOperTheory Adv Appl pp 175ndash188Birkhauser Basel 2003

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Submit your manuscripts atwwwhindawicom

International Journal of Differential Equations 3

In investigations of dual integral equations in some appli-cations the modifications of Riemann-Liouville fractionalintegrals and derivatives are widely usedThe important casesinclude Hadamard fractional operators and Erdelyi-Koberfractional operators

Riemann-Liouville fractional integro-differentiation isformally a fractional power (119889119889119909)120572 of the differentiationoperator 119889119889119909 and is invariant relative to translation if con-sidered on the whole axis Hadamard suggested a construc-tion of fractional integro-differentiation which is a fractionalpower of the type (119909(119889119889119909))120572This construction is well suitedto the case of the half-axis and invariant relative to dilation [7sect183]Thus Hadamard introduced fractional integrals of thefollowing form

Definition 4 (Hadamard fractional integral [34])

(119867120572119886+0119906) (119905) fl 1Γ (120572) int

119905

119886(log 119905119904)

120572minus1 119906 (119904) 119889119904119904 119905 gt 119886 gt 0

(7)

In 1993 Kilbas studied a weighted Hadamard fractionalintegral also calledHadamard-type fractional integral whichextended the application of Hadamard operators

Definition 5 (Hadamard-type fractional integral [34])

(119867120572119886+120583119906) (119905) fl 1Γ (120572) int

119905

119886(119904119905)

120583 (log 119905119904)120572minus1 119906 (119904) 119889119904119904

119905 gt 119886 gt 0(8)

In investigation of Hankel transform Erdelyi and Koberproposed the Erdelyi-Kober (E-K) operators They are gen-eralizations of the classical Riemann-Liouville fractionaloperators The left-sided E-K fractional integral of order 120572 isdefined by the following formula

Definition 6 (Erdelyi-Kober fractional integral [35])

119868120574120572120573 119906 (119905) fl 120573119905minus120573(120574+120572)Γ (120572) int119905

0(119905120573 minus 119904120573)120572minus1 119904120573(120574+1)minus1119906 (119904) d119904

120572 gt 0 120573 = 1 2 120574 isin R(9)

In order to introduce the definition of E-K fractionalderivative and its properties we define a special space offunctions that was first introduced in [36]

Definition 7 (see [36]) The function space119862120578 120578 isin R consistsof all functions 119891(119909) 119909 gt 0 that can be represented in theform 119891(119909) = 1199091199011198911(119909) with 119901 gt 120578 and 1198911 isin 119862([0infin])

The E-K fractional derivative of order 120572 is defined in thefollowing form

Definition 8 (Erdelyi-Kober fractional derivative [11 37])

(119863120574120572120573 119906) (119905) fl 119899prod

119895=1

(120574 + 119895 + 1120573119905 119889119889119905) (119868120574+120572119899minus120572120573 119906) (119905) (10)

where 119899 = lceil120572rceil is an integer

For the functions from the space 119862120578 120578 ge minus120573(120574 + 1) theleft-sidedE-K fractional derivative is a left-inverse operator tothe left-sidedE-K fractional integral (9) [38] then the relation

(119863120574120572120573 119868120574120572120573 119891) (119905) = 119891 (119905) (11)

holds true for every 119891 isin 119862120578In order to unify these definitions Agrawal [13] proposed

a new definitionwhich includesmost of them as special cases

Definition 9 (generalized fractional integral [13]) Theleftforward weightedscaled fractional integral of order120572 gt 0 of a function 119891(119905) with respect to another function 119911(119905)and weight 119908(119905) is defined as

(119868120572119886+[119911119908]119891) (119909) = [119908 (119909)]minus1Γ (120572) int119909

119886

119908 (119905) 1199111015840 (119905) 119891 (119905)[119911 (119909) minus 119911 (119905)]1minus120572 119889119905 (12)

In this definition if we set 119911(119904) = 119904 119908(119904) = 1 it reduces tothe classical Riemann-Liouville fractional integral Similarlysetting 119911(119904) = log(119904)119908(119904) = 119904120583 will lead toHadamard integraland 119911(119904) = 119904120573 119908(119904) = 119904120573120574 will lead to E-K fractional integralwith a factor 119905120573120572Definition 10 (see [13]) The leftforward weightedscaledderivative of integer order 119898 ⩾ 1 of a function 119891(119905) withrespect to another function 119911(119905) and weight 119908(119905) is definedas

(119863119898[119911119908119871]119891) (119909)= [119908 (119909)]minus1 [( 11199111015840 (119909)119863119909)119898 (119908 (119909) 119891 (119909))] (119909) (13)

Definition 11 (generalized Riemann-Liouville derivative [13])The leftforward weighted generalized Riemann-Liouvillefractional derivative of order 120572 gt 0 of a function 119891(119905) withrespect to another function 119911(119905) and weight119908(119905) is defined as

(119863120572119886+[1199111199081]119891) (119909) = 119863119898

119911119908119871 (119868119898minus120572119886+[119911119908]119891) (119909) (14)

Definition 12 (generalized Caputo derivative [13]) Theleftforward weighted generalized Caputo fractional deriva-tive of order 120572 gt 0 of a function 119891(119905) with respect to anotherfunction 119911(119905) and weight 119908(119905) is defined as

(119863120572119886+[1199111199082]119891) (119909) = (119868119898minus120572

119886+[119911119908]119863119898119911119908119871119891) (119909) (15)

Remark 13 In the definitions of generalized fractional oper-ators more general kernels and weight functions are used Itgeneralized nearly all the existing fractional operators in onespace dimension such as the Riemann-Liouville derivativethe Grunwald-Letnikov derivative the Caputo derivative theErdelyi-Kober-type fractional operator and the Hadamard-type fractional operator

3 Spectral Approximation of GeneralizedFractional Operator

In this section we will first study fractional deriva-tiveintegral of Jacobi polynomials and then derive a spectral

4 International Journal of Differential Equations

approximation for generalized fractional operators basedon Agrawalrsquos definitions Fractional derivativesintegrals ofothers type can be obtained as special cases

31 Fractional Derivative of Orthogonal Polynomials Denoteby 119869120573120574119895 (119909) the 119895-th order Jacobi polynomial with index (120573 120574)defined on [minus1 1]

As a set of orthogonal polynomials 119869120573120574119895 (119909)119873119895=0 satisfiesthe following three-term-recurrence relation [39]

1198691205731205740 (119909) = 11198691205731205741 (119909) = (120573 + 120574 + 2) 119909 + (120573 minus 120574) 119869120573120574119895+1 (119909) = (119860120573120574

119895 119909 minus 119861120573120574119895 ) 119869120573120574119895 (119909) minus 119862120573120574

119895 119869120573120574119895minus1 (119909) 1 ⩽ 119895 ⩽ 119873 minus 1

(16)

where the recursive coefficients are defined as

119860120573120574119895 = (2119895 + 120573 + 120574 + 1) (2119895 + 120573 + 120574 + 2)2 (119895 + 1) (119895 + 120573 + 120574 + 1)

119861120573120574119895 = (1205742 minus 1205732) (2119895 + 120573 + 120574 + 1)

2 (119895 + 1) (119895 + 120573 + 120574 + 1) (2119895 + 120573 + 120574) 119862120573120574119895 = (119895 + 120573) (119895 + 120574) (2119895 + 120573 + 120574 + 2)(119895 + 1) (119895 + 120573 + 120574 + 1) (2119895 + 120573 + 120574)

(17)

In order to derive fractional derivative of Jacobi polyno-mials we introduce some useful lemmas first

Lemma 14 For any 119899 isin N+ 120573 120574 isin R 120572 gt 0 the followingrelation holds

119869(minus1minus120572120574+120572+1)119899 (119909)= minus(119899 + 120574 + 1) (1 minus 119909)2 (119899 minus 120572) 119869(1minus120572120574+120572+1)119899minus1 (119909)minus 120572119869(minus120572120574+120572)119899 (119909)119899 minus 120572

(18)

119869(120573+120572+1minus1minus120572)119899 (119909)= (119899 + 120573 + 1) (1 + 119909)2 (119899 minus 120572) 119869(120573+120572+11minus120572)119899minus1 (119909)minus 120572119869(120573+120572minus120572)119899 (119909)119899 minus 120572

(19)

Proof According to [39 421] Jacobi polynomials withparameters 120573 120574 isin R are defined by

119869(120573120574)119899 (119909)= 119899sum

119895=0

1119895 (119899 minus 119895) Γ (119899 + 119895 + 120573 + 120574 + 1)Γ (119899 + 120573 + 120574 + 1) Γ (119899 + 120573 + 1)Γ (119895 + 120573 + 1) (119909 minus 12 )119895 (20)

Considering property of Jacobi polynomials

119869(120573120574)119899 (119909) = (minus1)119899 119869(120574120573)119899 (minus119909) (21)

Jacobi polynomials can be rewritten in the following form

119869(120573120574)119899 (119909)= 119899sum

119895=0

(minus1)119895+119899119895 (119899 minus 119895) Γ (119899 + 119895 + 120573 + 120574 + 1)Γ (119899 + 120573 + 120574 + 1) Γ (119899 + 120574 + 1)Γ (119895 + 120574 + 1) (119909 + 12 )119895 (22)

Define the following symbols

119875119860 = (119899 + 120573 + 1) (1 + 119909) 119869(120573+120572+11minus120572)119899minus1 (119909) 119875119861 = 2120572119869(120573+120572minus120572)119899 (119909) 119875119862 = 2 (119899 minus 120572) 119869(120573+120572+1minus1minus120572)119899 (119909)

(23)

Through a series of calculation we have

119875119861 + 119875119862= 119899sum

119895=0

2120572 (minus1)119895+119899119895 (119899 minus 119895) Γ (119899 + 119895 + 120573 + 1)Γ (119899 + 120573 + 1) Γ (119899 minus 120572 + 1)Γ (119895 minus 120572 + 1) (119909 + 12 )119895

+ 119899sum119895=0

2 (119899 minus 120572) (minus1)119895+119899119895 (119899 minus 119895) Γ (119899 + 119895 + 120573 + 1)Γ (119899 + 120573 + 1) Γ (119899 minus 120572)Γ (119895 minus 120572) (119909 + 12 )119895

= 119899sum119895=0

(minus1)119895+119899119895 (119899 minus 119895) Γ (119899 + 119895 + 120573 + 1)Γ (119899 + 120573 + 1) Γ (119899 minus 120572 + 1)Γ (119895 minus 120572 + 1) (119909 + 12 )119895 (2119895) 119875119860= 119899sum

119895=1

2 (minus1)119895+119899(119895 minus 1) (119899 minus 119895) Γ (119899 + 119895 + 120573 + 1)Γ (119899 + 120573 + 1) Γ (119899 minus 120572 + 1)Γ (119895 minus 120572 + 1) (119909 + 12 )119895

= 119875119861 + 119875119862

(24)

The equality (19) is proved Equality (18) can be provedsimilarly

Lemma 15 (see [39 P96]) For120572 gt 0 minus1 lt 119909 lt 1 120573 gt minus1 120574 isinR

(1 minus 119909)120573+120572 119869(120573+120572120574minus120572)119899 (119909)119869(120573+120572120574minus120572)119899 (1)

= Γ (120573 + 120572 + 1)Γ (120573 + 1) Γ (120572) int1

119909

(1 minus 119910)120573(119910 minus 119909)1minus120572

119869(120573120574)119899 (119910)119869(120573120574)119899 (1) 119889119910

(25)

For 120572 gt 0 minus1 lt 119909 lt 1 120574 gt minus1 120573 isin R

(1 + 119909)120574+120572 119869(120573minus120572120574+120572)119899 (119909)119869(120573minus120572120574+120572)119899 (1)

= Γ (120574 + 120572 + 1)Γ (120574 + 1) Γ (120572) int119909

minus1

(1 + 119910)120574(119909 minus 119910)1minus120572

119869(120573120574)119899 (119910)119869(120573120574)119899 (1) 119889119910

(26)

Lemma 16 (see [29 40]) For 120572 gt 0 minus1 lt 119909 lt 1 120573 120574 isin R

1199091198681205721119869(0120574)119899 (119909) = 1Γ (120572) int1

119909

119869(0120574)119899 (119910)(119910 minus 119909)1minus120572 119889119910

= 119899 (1 minus 119909)120572Γ (119899 + 120572 + 1)119869(120572120574minus120572)119899 (119909) (27)

International Journal of Differential Equations 5

minus1119868120572119909119869(1205730)119899 (119909) = 1Γ (120572) int119909

minus1

119869(1205730)119899 (119910)(119909 minus 119910)1minus120572 119889119910

= 119899 (1 + 119909)120572Γ (119899 + 120572 + 1)119869(120573minus120572120572)119899 (119909) (28)

Lemma 17 forall120572 gt 0 120572 notin N minus1 lt 119909 lt 1 120573 120574 isin R

1199091198631205721119869(0120574)119899 (119909) = 119899 (1 minus 119909)minus120572Γ (119899 + 1 minus 120572)119869(minus120572120574+120572)119899 (119909) (29)

minus1119863120572119909119869(1205730)119899 (119909) = 119899 (1 + 119909)minus120572Γ (119899 + 1 minus 120572)119869(120573+120572minus120572)119899 (119909) (30)

Proof First for the case 0 lt 120572 lt 1 formulas (29) and (30) canbe obtained by setting 120573 = minus120572 in formula (25) setting 120574 = minus120572in formula (26) and applying Riemann-Liouville fractionalderivative operator to both sides of themWe refer to [22] fordetailed discussion

For the case 120572 gt 1 120572 notin N the formulas cannot beobtained from Lemma 15 due to the constrains 120573 gt minus1 and120574 gt minus1 in Lemma 15 Here we prove the lemma inductively

For left Riemann-Liouville derivative from case 1 equal-ity (30) holds for lfloor120572rfloor = 0 Assume the equality holds forlfloor120572rfloor = 119896 minus 1 119896 isin N+ when lfloor120572rfloor = 119896 and let = 120572 minus 1from the assumption it holds that

minus1119863119909119869(1205730)119899 (119909) = 119899 (1 + 119909)minusΓ (119899 + 1 minus )119869(120573+minus)119899 (119909) (31)

We apply 119889119889119909 to both sides of (31) and then

minus1119863120572119909119869(1205730)119899 (119909) = minus1119863+1

119909 119869(1205730)119899 (119909)= 119899Γ (119899 + 1 minus ) (minus (1 + 119909)minusminus1 119869(120573+minus)119899 (119909)+ (1 + 119909)minus 119889119889119909119869(120573+minus)119899 (119909))= 119899 (1 + 119909)minusminus1Γ (119899 + 1 minus ) (minus119869(120573+minus)119899 (119909)+ (119899 + 120573 + 1) (1 + 119909)2 119869(120573++11minus)119899minus1 (119909))

(32)

Applying Lemma 14 we have

minus1119863120572119909119869(1205730)119899 (119909)= 119899 (1 + 119909)minusminus1Γ (119899 + 1 minus ) (minus (119899 minus 120572) 119869(120573++1minusminus1)119899 (119909))= 119899 (1 + 119909)minusminus1Γ (119899 + 1 minus ) ((119899 minus ) 119869(120573++1minusminus1)119899 (119909))= 119899 (1 + 119909)minus120572Γ (119899 + 1 minus 120572)119869(120573+120572minus120572)119899 (119909)

(33)

Equality (30) is proved Equality (29) can be proved similarly

Through the relationship between Caputo derivative andRiemann-Liouville derivative we immediately obtain thefractional derivative for Caputo derivative

Lemma 18 For 120572 gt 0 minus1 lt 119909 lt 1119888119909119863120572

1119869(0120574)119899 (119909) = 119899 (1 minus 119909)minus120572Γ (119899 + 1 minus 120572)119869(minus120572120574+120572)119899 (119909)minus 119898minus1sum

119895=0

120597119895119909119869(0120574)119899 (1) (1 minus 119909)119895minus120572(minus1)119895 Γ (1 + 119895 minus 120572)

119888minus1119863120572

119909119869(1205730)119899 (119909) = 119899 (1 + 119909)minus120572Γ (119899 + 1 minus 120572)119869(120573+120572minus120572)119899 (119909)minus 119898minus1sum

119895=0

120597119895119909119869(1205730)119899 (minus1) (1 + 119909)119895minus120572Γ (1 + 119895 minus 120572)

(34)

where119898 = lceil120572rceilGiven a function 119906(119909) isin 119867119898

120596 [minus1 1] and polynomialsspace P119873 the projection of 119906(119909) in space P119873 120587119873119906(119909)satisfies the following relation

(119906 minus 120587119873119906 V)120596 = 0 forallV isin P119873 (35)

According properties of space P119873 and Jacobi polynomials119906119873 can be expressed as

120587119873119906 (119909) = 119873sum119895=0

119895119869(120573120574)119895 (119909) (36)

where 119895 = (119906(119909) 119869(120573120574)(119909))120596119869(120573120574)(119909)2120596 120596 = 120596(119909) is theweight function

Then fractional derivative of 119906(119909) can be approximatedas

minus1119863120572119909119906 (119909) asymp 119873sum

119895=0

119895 ( minus1119863120572119909119869(120573120574)119895 (119909)) (37)

And we have the following lemma

Lemma 19 (see [41]) forall119906 isin 119867119898+1120596 [minus1 1] 0 le 119897 le 119898 119897 119898 isin

N 0 lt 120572 lt 1 119906119873 is a spectral approximation to 119906 in polynomialspace such that (119906 minus 120587119873119906 V)120596 = 0 forallV isin P119873 and then thereexists a constant 119862 such that10038171003817100381710038171003817 minus1119863119897+120572

119909 (119906 (119909) minus 120587119873119906 (119909))10038171003817100381710038171003817120596(120573+120572+119897120572+119897)le 119862119873119897minus119898 1003817100381710038171003817 minus1119863119898+120572

119909 1199061003817100381710038171003817120596(120573+120572+119898120572+119898) (38)

where 120596(120573120574) = (1 minus 119909)120573(1 + 119909)120574Combining Lemmas 18 and 19 and (37) approximation

method can be obtained immediately for Caputo derivativeand we have the following corollary

Corollary 20 forall119906 isin 119867119898+1120596 [minus1 1] 0 le 119897 le 119898 119897 119898 isin N 0 lt120572 lt 1 119906119873 is a spectral approximation to 119906 in polynomial space

6 International Journal of Differential Equations

such that (119906 minus 120587119873119906 V)120596 = 0 forallV isin P119873 and then there exists aconstant 119862 such that100381710038171003817100381710038171003817 119888

minus1119863119897+120572

119909 (119906 (119909) minus 120587119873119906 (119909))100381710038171003817100381710038171003817120596 le 119862119873119897minus119898 1003817100381710038171003817 minus1119863119898+120572119909 1199061003817100381710038171003817120596 (39)

Based on above lemmas fractional integral and derivativeof Jacobi polynomials in the standard interval [minus1 1] can beexpressed explicitly Fractional integrals and derivatives inshifted interval [119886 119887] can be obtained through proper variabletransforms

For any 119910 isin [119886 119887] we assume 119869120573120574119899lowast(119910) is defined in [119886 119887]Let ℎ = 119887 minus 119886 119909 = minus1 + 2((119910 minus 119886)ℎ) then 119909 isin [minus1 1]Substituting 119909 = minus1 + 2((119910 minus 119886)(119887 minus 119886)) into equation (27)-(28) and (29)-(30) we obtained

119910119868120572119887119869(0120574)119899lowast (119910) = (ℎ2)120572

1199091198681205721119869(0120574)119899 (119909) 119886119868120572119910119869(1205730)119899lowast (119910) = (ℎ2)

120572

minus1119868120572119909119869(1205730)119899 (119909) 119888119910119863120572

119887119869(0120574)119899lowast (119910) = (2ℎ)

120572119888119909119863120572

1119869(0120574)119899 (119909) 119888119886119863120572

119910119869(1205730)119899lowast (119910) = (2ℎ)120572

119888minus1119863120572

119909119869(1205730)119899 (119909)

(40)

32 Spectral Approximation to Generalized Fractional Opera-tors We assume 119911(119909) and 119908(119909) are positive monotone func-tions and 119911 119908 isin 119862[119886 119887] Obviously 119911(119909) and119908(119909) are invert-ible The following lemmas can be derived for generalizedfractional operators

Lemma 21 Let 119892(120577) = 119908(119911minus1(120577))119891(119911minus1(120577)) 0 lt 120572 lt 1 thenthe generalized fractional integral operator is equivalent to thefollowing classical fractional integral

(119868120572119886+[119911119908]119891) (119909) = [119908 (119909)]minus1 1205770119868120572120577119892 (119911 (119909)) (41)

Here 1205770 = 119911(119886)Proof From the definition of generalized fractional integralwe have

(119868120572119886+[119911119908]119891) (119909) = [119908 (119909)]minus1Γ (120572) int119909

119886

119908 (119905) 1199111015840 (119905) 119891 (119905)[119911 (119909) minus 119911 (119905)]1minus120572 119889119905 (42)

Since 120577 = 119911(119905) is positive monotone function then 120577 = 119911(119905) isinvertible and we have 119905 = 119911minus1(120577)(119868120572119886+[119911119908]119891) (119909) = [119908 (119909)]minus1Γ (120572)sdot int119911(119909)

119911(119886)

119908(119911minus1 (120577)) 1199111015840 (119911minus1 (120577)) 119891 (119911minus1 (120577))[119911 (119909) minus 120577]1minus120572 119889119911minus1 (120577)

= [119908 (119909)]minus1Γ (120572) int119911(119909)

119911(119886)

119908(119911minus1 (120577)) 119891 (119911minus1 (120577))[119911 (119909119896) minus 120577]1minus120572 119889120577

(43)

Let 119892(120577) = 119908(119911minus1(120577))119891(119911minus1(120577)) then the generalized frac-tional integral of 119891(119909) is converted to classical fractionalintegral of 119892(120577) in the following form

(119868120572119886+[119911119908]119891) (119909) = [119908 (119909)]minus1 1205770119868120572120577119892 (119911 (119909)) (44)

Lemma is proved

Lemma 22 Let 119892(120577) = 119908(119911minus1(120577))119891(119911minus1(120577)) 0 lt 120572 lt 1 thenthe generalized fractional derivative of order 120572 is equivalent tothe following classical fractional derivative

(119863120572119886+[1199111199082]119891) (119909) = [119908 (119909)]minus1 119888

1205770119863120572

120577119892 (119911 (119909)) (45)

Here 1205770 = 119911(119886)Proof Similar to the proof of Lemma 21 we have

(119863120572119886+[1199111199082]119891) (119909) = [119908 (119909)]minus1Γ (1 minus 120572)sdot int119911(119909)

119911(119886)

(120597120597120577) (119908 (119911minus1 (120577)) 119891 (119911minus1 (120577))) (120597120577120597119905)[119911 (119909) minus 120577]120572 119889119911minus1 (120577)

= [119908 (119909)]minus1Γ (1 minus 120572) int119911(119909)

119911(119886)

(120597120597120577) (119908 (119911minus1 (120577)) 119891 (119911minus1 (120577)))[119911 (119909) minus 120577]120572 119889120577

(46)

Let 119892(120577) = 119908(119911minus1(120577))119891(119911minus1(120577)) then the generalized frac-tional derivative of 119891(119909) is expressed through the classicalfractional derivative of 119892(120577)

(119863120572119886+[1199111199082]119891) (119909) = [119908 (119909)]minus1 119888

1205770119863120572

120577119892 (119911 (119909)) (47)

Lemma is proved

Remark 23 These two lemmas establish important relationbetween classical and generalized fractional operators Withthese lemmas generalized fractional differential equationscan be solved via classical fractional differential equationsand vice versa Which kind of transform should be takendepends on characters of the problem to be solved

In order to design a high order numerical approximationof the generalized fractional operator we define a scaled spaceP119899

[119908119911] such that

P119899[119908119911] = V (119909) = [119908 (119909)]minus1 119892 (119911 (119909)) 119892 (119909) isin P

119899 119909isin Ω = [119886 119887] (48)

whereP119899 is a polynomial space of up to order 119899Define a inner product and norm for space P119899

[119908119911] suchthat

(119906 (119909) V (119909))120596 = int119887

119886119906 (119909) V (119909) 120596 (119909) 119889119909

and V120596 = radic(V V)120596(49)

International Journal of Differential Equations 7

Define projection 119876119873 into space P119873119908119911 such that for any

function 119906(119909)(119906 minus 119876119873119906 V)120596 = 0 forallV isin P

119873119908119911 (50)

SupposeΦ119895(119909) 119895 = 0 119873 are a set of orthogonal basisfunctions in spaceP119873

119911119908 satisfying

(Φ119894 (119909) Φ119895 (119909))120596 = 1 119894 = 1198950 119894 = 119895 (51)

Let119873 997888rarr infin then Φ119895(119909) 119895 = 0 1 119873 form a 1198712120596(Ω)space and for any 119906(119909) isin 1198712120596(Ω) the projection 119876119873119906(119909) canbe written as

119876119873119906 (119909) = 119873sum119895=0

119895Φ119895 (119909) (52)

Here 119895 (119895 = 0 119873) are expansion coefficients such that

119895 = (119906 (119909) Φ119895 (119909))12059610038171003817100381710038171003817Φ119895 (119909)100381710038171003817100381710038172120596 (53)

The weight function 120596(119909) plays an important role in thecomputational process and analysis of the method Herewe choose a proper weight function to use properties oforthogonal polynomials and make the computation moreefficient Let 119901119895(119911(119909)) = 119908(119909)Φ119895(119909) and note that

(119876119873119906 (119909) Φ119894 (119909))120596 = int119887

119886( 119873sum

119895=0

119895119908minus1 (119909) 119901119895 (119911 (119909)))sdot 119908minus1 (119909) 119901119894 (119911 (119909)) 120596 (119909) 119889119909

(54)

Take 120596(119909) = 1199082(119909)1199111015840(119909) then(119876119873119906 (119909) Φ119894 (119909))120596 = int119911(119887)

119911(119886)( 119873sum

119895=0

119895119901119895 (119911))119901119894 (119911) 119889119911 (55)

Since 119901119895(119911) is polynomial of order 119895 the computation can becarried out easily through properties of orthogonal polyno-mials

Suppose 119875119895lowast(119909) is shifted Legendre polynomial definedon [119911(119886) 119911(119887)] Then given any function 119906(119909) isin 1198712120596 withweight 120596(119909) = 1199082(119909)1199111015840(119909) we have

119876119873119906 (119909) = [119908 (119909)]minus1 119873sum119895=0

119895119875119895lowast (119911 (119909)) (56)

Recalling Lemmas 21 and 22 the generalized fractionalintegral and derivative can be obtained in the form

(119868120572119886+[119911119908]119876119873119906) (119909)= [119908 (119909)]minus1 119873sum

119895=0

119895 ( 1205770119868120572120577119875119895lowast (120577)1003816100381610038161003816120577=119911(119909)) (57)

(119863120572119886+[1199111199082]119876119873119906) (119909)= [119908 (119909)]minus1 119873sum

119895=0

119895 ( 1198881205770119863120572

120577119875119895lowast (120577)1003816100381610038161003816120577=119911(119909)) (58)

From Lemma 19 the following corollary can be obtainedimmediately

Corollary 24 120577(119909) = 119911(119909) is a monotone increasing function119908(119909) gt 0 0 lt 120572 lt 1 119876119873 is a projection into space P119873119908119911119906(119911minus1(120577)) isin 119862119898(Ω) 119898 isin N then there exists a constant 119862120572

such that 10038171003817100381710038171003817119863120572119886+[1199111199082] (119906 (119909) minus 119876119873119906 (119909))100381710038171003817100381710038171205962le 1198621205721198731minus119898 10038171003817100381710038171003817 1205770119863119898+120572

119911119906 (120577minus1 (119911))100381710038171003817100381710038171205961

(59)

Here the weight function 1205961(120577) = (120577 minus 119886)minus120572(119887 minus 120577)120572 1205962(119909) =1205961(119911(119909))1199082(119909)1199111015840(119909)Remark 25 Unlike the convergence theory in classical poly-nomial space the convergence order in space P119873

119908119911 dependson regularity of the function 119906(119909) with respect to 119911(119909) Wewill illustrate this through numerical examples

Most of the time it is more convenient to consider theproblems in nodal form We assume the given interpolationpoints are 120577119895 = 119911(119909119895) (119895 = 0 1 119873) then the Lagrangebasis functions L119895(119909) (119895 = 0 1 119873) can be defined asfollows

L119894 (120577) = prod119895=0119873119895 =119894

(120577 minus 120577119895)(120577119894 minus 120577119895) (60)

The function 119906(119909) can be expressed using both Jacobipolynomials and Lagrange polynomials The following rela-tion is derived

119906 (119909) = [119908 (119909)]minus1 119873sum119895=0

119892119895119875119895lowast (119911 (119909))

= [119908 (119909)]minus1 119873sum119895=0

119908119895119906119895L119895 (119911 (119909)) (61)

Considering the equivalence between Legendre basis andLagrange basis the following equality holds

L119895 (119911) = 119873sum119896=0

119897119896119895119875119896lowast (119911) 119896 = 0 1 119873 (62)

where 119897119894119895 = (119871119895(119911) 119875119894lowast(119911))119875119894lowast(119911)21198712 Then the nodal form expansion of 119906(119909) is obtained

119906 (119909) = [119908 (119909)]minus1 119873sum119895=0

119873sum119896=0

119908119895119906119895119897119896119895119875119896lowast (119911 (119909)) (63)

8 International Journal of Differential Equations

From (57) and (58) the corresponding nodal form of gener-alized fractional integral and derivative are obtained

(119868120572119886+[119911119908]119906) (119909)= [119908 (119909)]minus1 119873sum

119895=0

119873sum119896=0

119908119895119906119895119897119896119895 ( 1205770119868120572120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119909)) (64)

(119863120572119886+[1199111199082]119906) (119909)= [119908 (119909)]minus1 119873sum

119895=0

119873sum119896=0

119908119895119906119895119897119896119895 ( 1198881205770119863120572

120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119909)) (65)

Example 26 Now we give an example to show the effec-tiveness and accuracy of the method Assuming 119911(119909) =radic1199093 119908(119909) = 1 we consider generalized fractional derivativeof 119910(119909) on the interval [0 1] with the following form

119910 (119909) = 5sum119895=1

(minus1)119895+1 1199093119895(2119895) (66)

The exact generalized fractional derivative of 119910(119909) is119863120572

0+[1199111199082]119910 (119909) = 5sum119895=1

(minus1)119895+1 1199093119895minus31205722Γ (2119895 + 1 minus 120572) (67)

Numerical approximation to generalized fractionalderivative of 119910(119909) can be obtained using (58) We considerthe maximum absolute error

119890 = max119909isin[01]

100381610038161003816100381610038161198631205720+[1199111199082]119910 (119909) minus 119863120572

0+[1199111199082]119876119873119910 (119909)10038161003816100381610038161003816 (68)

of the numerical derivative Results for 120572 = 02 05 08 areshown in Figure 1

For this example it is easy to check that 119910 isin P10119908119911 From

theory of spectral approximation the error would decreaseexponentially when 119873 lt 10 and the numerical fractionalderivative would be exact when119873 ge 10 Our numerical resultcoincides with the theory exactly

Example 27 In this example we test the spectral approxi-mation of Hadamard integral Considering Hadamard-typefractional integral of 119910(119909) = sin(120587119909) on the interval [1 2]when 119908(119909) = 1 and 120572 = 1 the Hadamard integral of 119910(119909) isSine integral function Si(120587119909) minus Si(120587) for more general 119908(119909)and 120572 the exact Hadamard-type integral of 119910(119909) is unknownHere we consider several pairs of 119908(119909) and 120572 NumericalHadamard-type integral of 119910(119909) would be computed using(57) with 119911(119909) = log(119909) To evaluate the approximationaccuracy for the case 119908(119909) = 1 120572 = 1 the exact Hada-mard fractional integral is computed usingMATLAB built-infunction 119904119894119899119894119899119905 for other cases ldquoexactrdquo Hadamard fractionalintegral is computed using (57) with large 119873 (eg 119873 =50) which is treated as reference solution The results fornumerical Hadamard integral and approximation error areshown in Figures 2 and 3

For this example 119906 is not in the spaceP119899119908119911 for any 119899 Max-

imum error of approximated fractional integral convergesexponentially until reaching machine accuracy

10minus16

10minus14

10minus12

10minus10

10minus8

10minus6

10minus4

10minus2

max

imum

erro

r

5 6 7 8 94 11 1210

N

alpha=02alpha=05alpha=08

Figure 1 Error of numerical approximation to generalized frac-tional derivative of 119910(119909)

11 12 13 14 15 16 17 18 19 21x

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Had

amar

d fra

ctio

nal i

nteg

ral

w(x)=1=1w(x)=1=06

w(x)=x=1w(x)=x=06

Figure 2 Hadamard fractional integral for different weight 119908(119909)and 120572

33 Fractional IntegralDifferential Matrices Suppose 119906(119909) isinP119873

119911119908 119909119894 (119894 = 0 1 119873) are the interpolation points and119910119895 (119895 = 0 1 119873) are the collocation points we define thefollowing symbols

U = [119906 (1199090) 1199061199091 119906 (119909119873)]119879 (69)

U(120572) = [119863120572

119886+[119911119908]119906 (1199100) 119863120572119886+[119911119908]119906 (1199101)

119863120572119886+[119911119908]119906 (119910119873)]119879 120572 gt 0 (70)

International Journal of Differential Equations 9

10minus16

10minus14

10minus12

10minus10

10minus8

10minus6

10minus4

max

imum

erro

r

w(x)=1=1w(x)=1=06

w(x)=x=1w(x)=x=06

10 12 14 16 18 20 228x

Figure 3 Approximation error of Hadamard fractional integral fordifferent weight 119908 and 120572

U(120572) = [119868minus120572119886+[119911119908]119906 (1199100) 119868minus120572119886+[119911119908]119906 (1199101) 119868minus120572119886+[119911119908]119906 (119910119873)]119879 120572 lt 0 (71)

We define a generalized fractional differentialintegralmatrixM120572 such that

M120572119894119895 = 119873sum

119896=0

119908(119909119895) 119897119896119895119908 (119910119894) ( 1198881205770119863120572

120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119910119894))

for 120572 gt 0(72)

M120572119894119895 = 119873sum

119896=0

119908(119909119895) 119897119896119895119908 (119910119894) ( 1205770119868120572120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119910119894))

for 120572 lt 0(73)

In order to compute the fractional matrix M120572 moreefficiently we define a few more matrices L and L120572 are(119873 + 1) times (119873 + 1) matrices such thatL119894119895 = L119895(119911(119910119894))L120572

119894119895 =1205770119863120572

120577L119895(119911)|119911=119911(119910119894) 119894 119895 = 0 1 119873 V is defined based on

values of Legendre polynomials at interpolation points 119911(119909119894)such that

V119894119895 = 119875119895lowast (119911 (119909119894)) 119894 119895 = 0 1 119873 (74)

D120572 is defined as fractional derivativeintegral of Legendrepolynomials at collocation points 120577119894 = 119911(119910119894)

D120572119894119895 = 119888

1205770119863120572

120577119875119895lowast (120577119894) 120572 gt 0 (75)

D120572119894119895 = 1205770

119868120572120577119875119895lowast (120577119894) 120572 lt 0 (76)

W119897 andW119903 are the weight matrices defined as follows

W119903119894119895 =

119908(119909119895) 119894 = 1198950 otherwise (77)

W119897119894119895 =

1119908(119910119895) 119894 = 119895 gt 10 otherwise (78)

Theorem 28 For 119906(119909) isin P119873119911119908 vectors defined in (69)ndash(71)

and matrices defined in (72)ndash(78) the following relation holds

U(120572) = M

120572U = (W119897D120572V

minus1W

119903)U (79)

Proof From the definition (69)ndash(71) (72)ndash(78) and (63)-(64) it is easy to obtain

U(120572) = M

120572U (80)

Next we prove thatM120572 = W119897D120572Vminus1W119903From the definition of the interpolation function L119895(119911)

119875119895lowast (119911) = 119873sum119894=0

119875119895lowast (119911 (119909119894)) L119894 (119911) (81)

Suppose 997888rarr119875(119911) = (1198750lowast(119911) 1198751lowast(119911) 119875119873lowast(119911)) 997888rarrL (119911) =(L0(119911) L1(119911) L119873(119911)) for 120572 gt 0 we have997888rarr119875 (119911) = 997888rarrL (119911)V

1205770119863120572

120577

997888rarr119875 (119911) = ( 1205770119863120572

120577

997888rarrL (119911))V (82)

Evaluating the matrices multiplication for each elementthe following relation is derived

(W119897D120572V

minus1W119903)119894119895 = (W119897L

120572W119903)119894119895

= 119908 (119909119895)119908 (119910119894) L(120572)119895 (119911 (119910119894))(83)

= 119873sum119896=0

119908(119909119895) 119897119896119895119908 (119910119894) ( 1205770119863120572

120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119910119894)) = M

120572119894119895 (84)

The case 120572 lt 0 can be proved similarly

Remark 29 Collocation points 119910119895 and the interpolationpoints 119909119894 are not necessary the same To obtain a goodapproximation interpolation points ofGauss-type are usuallyused At the same time collocation points should be chosenproperly to guarantee stability properties of the methodIn the following numerical examples for computation andstability aim both interpolation and collocation points arechosen based on Gauss-type points with respect to 119911(119909)

10 International Journal of Differential Equations

4 Collocation Methods for FractionalDifferential and Integral Equations

41 Fractional Ordinary Differential Equations In this sub-section we consider collocation method for the generalizedfractional ordinary differential equation of the form

1198631205720+[1199111199082]119906 (119909) = 120582 (119909) 119906 (119909) + 119891 (119909)

119909 isin Ω = (0 119887] (85)

119906 (0) = 1199060 (86)

Here 0 lt 120572 lt 1 119908(119909) gt 0 119911(119909) gt 0 and 119911(119909) is a monotonefunction inΩ

We assume 119906119873(119909) isin P119873119911119908 is a numerical solution of the

equation 119909119894(119894 = 0 1 119873) are the chosen interpolationpoints 119906119895119873 = 119906119873(119909119895) The following discretized equation isobtained

[119908 (119909)]minus1 119873sum119895=0

119873sum119896=0

119908(119909119895) 119906119895119873119897119896119895 ( 1198881205770119863120572

120577119875119896lowast (119911 (119909)))

= 120582 (119909) 119906119873 (119909) + 119891 (119909) (87)

Let (87) hold on collocation points 119910119895 (119895 = 1 119873) thematrix form is obtained

(M120572 minus Λ)U = 119865 (88)

Here Λ is a diagonal matrix with Λ 119894119894 = 120582(119910119894) 119865 = (1199060 119891(1199101) 119891(119910119873))119879Considering the initial condition we set S = M120572 minus Λ

with the first row replaced by (1 0 0) 1198651015840 = 119865with its firstelement replaced by 1199060 Then the solution U is obtained bysolving the matrix equation SU = 1198651015840

Example 30 Consider the following example

1198631205720+[1199111199082]119906 (119909) = (1 + 119909) 119906 (119909)

+ Γ (15119903 + 25) 119909119903minus(23)120572Γ (15119903 + 25 minus 120572)minus 119909119903 (1 + 119909) 119909 isin (0 1]

(89)

119906 (0) = 0 (90)

Here 119911(119909) = 11990923119908(119909) = 119909 119903 is an arbitrary positive numberThe exact solution of the ordinary differential equation is119906(119909) = 119909119903 Maximum absolute errors of numerical solutions

for 119903 = 6 7 and 120572 = 03 06 09 are shown in Figures 4 and 5When 119911(119909) = 11990923 119908(119909) = 119909 the scaled polynomial

spaceP119873119911119908 becomes

P119873119911119908 = 119904119901119886119899 11990921198993minus1 119899 = 1 2 119873 (91)

For 119903 = 7 the error converges exponentially and reachesmachine accuracy at 119873 = 12 It is faster than any finitedifference method while for 119903 = 6 solution convergesalgebraically as119873 increases however it still reaches machineaccuracy at 119873 = 23 The major reason for this is that 119906(119909) =1199097 isin P12

119911119908 and 119906(119909) = 1199096 notin P119873119911119908 for any119873 isin N

6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

N

10minus15

10minus10

10minus5

100

max

erro

r

=03=06=09

Figure 4 Log-log plot of the maximum error for 119903 = 6

=03=06=09

10minus15

10minus10

10minus5

100

max

erro

r

15 20 25 30 3510N

Figure 5 Semilog plot of the maximum error for 119903 = 7

Remark 31 Since space P119873119908119911 is transformed from classical

polynomial space with respect to 119911(119909) the convergenceof spectral collocation method for ODEs with generalizedfractional operators depends not only on the smoothness ofthe solution itself but also on the scale function 119911(119905)42 Hadamard-Type Integral Equations We consider thefollowing Hadamard-type boundary value problem

1Γ (120572) int119909

119886( 119904119909)

120583 (log 119909119904 )120572minus1 119891 (119904) 119889119904119904 = 119892 (119909)

119909 isin Ω = (119886 119887] (92)

International Journal of Differential Equations 11

In [42] Kilbas discussed the existence of the solution of(92) Explicit formulas for the solution 119891(119905) were establishedin the following theorem

Theorem 32 (see [42]) If 119909120583119892(119909) isin 119860119862[119886 119887] then theHadamard-type integral equation (92) with 0 lt 120572 lt 1 issolvable in 119883120583(119886 119887) and its solution may be represented in theform

119891 (119909) = 119909minus120583Γ (1 minus 120572) [119886120583119892 (119886) (log 119909119886)minus120572

+ int119909

119886(log 119909119904 )

minus120572 (119904120583119892 (119904))1015840 119889119904] (93)

Here 119860119862[119886 119887] is the set of absolutely continuous functions on[119886 119887] and119883120583(119886 119887) is space of those Lebesgue measurable func-tions 119891 on [119886 119887] for which 119909120583minus1119891(119909) is absolutely integrable

Solution of the Hadamard-type integral equation isexactly the same as generalized fractional derivative ofRiemann-Liouville type with 119911(119904) = log(119904) 119908(119904) = 119904120583 Therelationship between (93) and Caputo type generalized frac-tional derivative is

119891 (119909) = 119886120583119892 (119886) 119909minus120583Γ (1 minus 120572) (log 119909119886)minus120572 + 119863120572

119886+[1199111199082]119892 (119909) (94)

Suppose 119909119895 119895 = 0 1 119873 are interpolation points thediscretized solution of equation (92) is

119891119873 (119909) = 119886120583119892 (119886) 119909minus120583Γ (1 minus 120572) (log 119909119886)minus120572

+ 119909minus120583 119873sum119895=0

119873sum119896=0

119909120583119895119892 (119909119895) 119897119896119895( 119888120577119886119863120572

120577119875119896lowast (log (119909))

(95)

Rewriting (95) in matrix form we have

119865 = 119866119886 +M120572119866 (96)

where M120572 is generalized fractional differential matrix119866 = (119892(1199090) 119892(1199091) 119892(119909119873))119879 119865 = (119891119873(1199090) 119891119873(1199091) 119891119873(119909119873))119879 119866119886 is a vector about the initial condition of theintegral equation defined by

119866119886 = 119886120583119892 (119886)Γ (1 minus 120572) (119909minus1205830 (log 1199090119886 )minus120572 119909minus1205831 (log 1199091119886 )

minus120572 119909minus120583119873 (log 119909119873119886 )minus120572)119879

(97)

Example 33 Assume 119892(119909) = sin(119909 minus 1) 120583 = 13 Ω = [110] Solutions of (92) for 120572 = 03 06 09 are shown inFigure 6

2 4 6 8 10 120X

minus8

minus6

minus4

minus2

0

2

4

6

8

f(X)

=03=06=09

Figure 6 Solution of (92)

43 Erdelyi-Kober Fractional Diffusion Equation In this sub-section we consider the following Erdelyi-Kober fractionaldiffusion equation [12]

119906 (119909 119905) = 1199060 (119909) + 1Γ (120573)sdot int119905

0

120572120573119904120572120573minus1 (119905120572120573 minus 119904120572120573)120573minus1 12059721205971199092 119906 (119909 119904) d119904(98)

Erdelyi-Kober fractional diffusion equation which is alsocalled stretched time-fractional diffusion equation is themaster equation of a kind of generalized grey Brownianmotion (ggBm) The ggBm is a parametric class of stochasticprocesses that provides models for both fast and slow anoma-lous diffusion This class is made up of self-similar processes119861120572120573(119905)with stationary increments and it depends on two realparameters 0 lt 120572 le 2 and 0 lt 120573 le 1 It includes thefractional Brownian motion when 0 lt 120572 le 2 and 120573 = 1 thetime-fractional diffusion stochastic processes when 0 lt 120572 =120573 lt 1 and the standard Brownian motion when 120572 = 120573 = 1About the relationship between stochastic process119861120572120573(119905) andstretched time-fractional diffusion equation the followingproposition is presented in [12]

Proposition 34 The marginal probability density function119891120572120573(119909 119905) of the process 119861120572120573(119905) 119905 ge 0 is the fundamentalsolution of the stretched time-fractional diffusion equation

119906 (119909 119905) = 1199060 (119909) + 1Γ (120573)sdot int119905

0

120572120573119904120572120573minus1 (119905120572120573 minus 119904120572120573)120573minus1 12059721205971199092 119906 (119909 119904) d119904(99)

12 International Journal of Differential Equations

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 7 Standard Brownian motion 120572 = 1 120573 = 1

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 8 Time-fractional diffusion with 120572 = 06 120573 = 06

Recalling the definition of generalized fractional integraland setting 119911(119905) = 119905120572120573 119908(119905) = 1 the equation can be rewrit-ten as

119906 (119909 119905) = 1199060 (119909) + 1198681205730+[119911119908]119906119909119909 (119909 119905) (100)

We use collocation method for both space and timediscretization We choose Legendre-Gauss-Lobatto (L-G-L)points 119909119894 (119894 = 0 1 119872) as the space collocation points andchoose 119905119895 (119895 = 0 1 119873) such that 119911(119905119895) are L-G-L pointsas the time collocation points

Define space collocation matrix M2 such that M2119894119895 =(11988921198891199092)L119895(119909119894) and generalized fractional integral matrix

M120573 Matrix M120573 is computed through Theorem 28 and thespace-time collocation matrices are obtained using Kro-necker product 1015840otimes1015840 Suppose A and B are space-timecollocation matrices with dimension (119872 + 1)(119873 + 1) times (119872 +1)(119873 + 1) for the second order derivative and fractionalintegral of order 120573 separately Then

A = M2 otimesI119873+1

B = I119872+1 otimesM120573 (101)

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 9 Fractional Brownian motion with 120572 = 15 120573 = 1

Suppose 119906119873(119909 119905) is the numerical solution of (98) defin-ing 119906119873119894119895 = 119906119873(119909119894 119905119895) solution vectorU and initial vector 1198800such that

U = [11990611987300 11990611987301 1199061198730119873 11990611987310 1199061198731119873 1199061198731198720 119906119873119872119873]119879

1198800 = [1199060 (1199090) 1199060 (1199090) 1199060 (1199091) 1199060 (1199091) 1199060 (119909119872) 1199060 (119909119872)]119879

(102)

where in the definition of 1198800 each 1199060(119909119895) is repeated 119873 + 1times

The matrix form discretized equation of (98) is obtainedas

U = 1198800 +BAU (103)

In the discretized equation initial condition is explicitlyinvolved After boundary condition added properly numeri-cal solution can be obtained by solving the matrix equation

Example 35 Assume 119909 isin Ω = (minus1 1) 119905 isin (0 05] 1199060(119909) =119890minus101199092 minus 119890minus10 119906(sdot 119905)|120597Ω = 0 Numerical solutions with 119872 =119873 = 50 are shown in Figures 7ndash10

Erdelyi-Kober diffusion equation characterizes the mar-ginal density function of the process 119861120572120573(119905) 119905 ge 0 When120572 = 120573 = 1 we recover the standard diffusion equationWhen0 lt 120572 = 120573 lt 1 we get the time-fractional diffusion equationof order 120573 When 120573 = 1 and 0 lt 120572 lt 2 we have the equationof the fractional Brownian motion marginal density

As shown in Figures 7 and 8 when 1 lt 120572 lt 2 the diffu-sion is fast and the increments exhibit long-range depen-dence when 0 lt 120572 lt 1 the diffusion is slow and theincrements form a stationary process which does not exhibitlong-range dependenceThe results coincide with theoreticalanalysis in [12 14]

International Journal of Differential Equations 13

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 10 Fractional Brownian motion with 120572 = 05 120573 = 1

5 Conclusion

In this paper we propose a spectral collocation method fordifferential and integral equations with generalized fractionaloperators To deal with the difficulty in designing spectralapproximation scheme due to complexity of integral kerneland weight a variable transform technique is applied to thegeneralized fractional operator and a spectral approximationmethod is proposed for the generalized fractional operatorOperational matrices for generalized fractional operatorsare derived Spectral collocation methods are designed forfractional ordinary differential equations Hadamard-typeintegral equations and Erdelyi-Kober diffusion equationsseparately Numerical experiments are carried out to verifythe accuracy and efficiency of the method and characteristicsof the Erdelyi-Kober diffusion equation are analyzed basedon numerical results

Data Availability

(i) The programs used to support the findings of this studyhave been deposited in the GitHub repository (httpsgithubcomqinwuxuSpectralGFPDE ) (ii) No data were used tosupport this study

Disclosure

An earlier version of this workwas presented at the ldquo8th Inter-national Congress on Industrial and Applied Mathematics(ICIAM 2015)rdquo

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The first author is supported by the National Key RampD Pro-gram of China (No 2017YFC0209804) the National ScienceFoundation for Young Scientists of China (No 11701273) andYouth Foundation of Jiangsu Province (No BK20170628)

The second author is supported by the National Key RampDProgram of China (No 2017YFB0305601)

References

[1] B BaeumerDA BensonMMMeerschaert and SWWheat-craft ldquoSubordinated advection-dispersion equation for contam-inant transportrdquo Water Resources Research vol 37 no 6 pp1543ndash1550 2001

[2] E Barkai RMetzler and J Klafter ldquoFrom continuous time ran-dom walks to the fractional Fokker-Planck equationrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 61no 1 pp 132ndash138 2000

[3] A Blumen G Zumofen and J Klafter ldquoTransport aspects inanomalous diffusion Levy walksrdquo Physical Review A AtomicMolecular and Optical Physics vol 40 no 7 pp 3964ndash39731989

[4] J P Bouchaud and A Georges ldquoAnomalous diffusion in dis-ordered media statistical mechanisms models and physicalapplicationsrdquoPhysics Reports vol 195 no 4-5 pp 127ndash293 1990

[5] M Raberto E Scalas and F Mainardi ldquoWaiting-times andreturns in high-frequency financial data an empirical studyrdquoPhysica A Statistical Mechanics and its Applications vol 314 no1ndash4 pp 749ndash755 2002

[6] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations New York NYUSA Elsevier 2006

[7] S G Samko A A Kilbas and O I Marichev Fractional Inte-grals and Derivatives Theory and Applications Gordon andBreach Yverdon Switzerland 1993

[8] V Kiryakova ldquoA brief story about the operators of the general-ized fractional calculusrdquo Fractional CalculusampAppliedAnalysisAn International Journal forTheory and Applications vol 11 no2 pp 203ndash220 2008

[9] S L Kalla ldquoOn operators of fractional integration Irdquo Mathe-maticae Notae vol 22 pp 89ndash93 197071

[10] S L Kalla ldquoOn operators of fractional integration IIrdquo Mathe-maticae Notae vol 25 pp 29ndash35 1976

[11] V S Kiryakova Generalized Fractional Calculus and Applica-tions Long-man amp J Wiley Harlow New York NY USA 1994

[12] A Mura and F Mainardi ldquoA class of self-similar stochasticprocesses with stationary increments to model anomalousdiffusion in physicsrdquo Integral Transforms and Special Functionsvol 20 no 3-4 pp 185ndash198 2009

[13] O P Agrawal ldquoSome generalized fractional calculus operatorsand their applications in integral equationsrdquo Fractional Calculusand Applied Analysis An International Journal for Theory andApplications vol 15 no 4 pp 700ndash711 2012

[14] G Pagnini ldquoErdelyi-Kober fractional diffusionrdquo FractionalCalculus and Applied Analysis An International Journal forTheory and Applications vol 15 no 1 pp 117ndash127 2012

[15] A Mura and G Pagnini ldquoCharacterizations and simulations ofa class of stochastic processes to model anomalous diffusionrdquoJournal of Physics A Mathematical and General vol 41 no 28285003 22 pages 2008

[16] E K Lenzi L R Evangelista M K Lenzi H V Ribeiro and EC de Oliveira ldquoSolutions for a non-Markovian diffusion equa-tionrdquo Physics Letters A vol 374 no 41 pp 4193ndash4198 2010

[17] B Al-Saqabi and V S Kiryakova ldquoExplicit solutions of frac-tional integral and differential equations involving Erderyi-Kober operatorsrdquo Applied Mathematics and Computation vol95 no 1 pp 1ndash13 1998

14 International Journal of Differential Equations

[18] L A Hanna and Y F Luchko ldquoOperational calculus for theCaputo-type fractional Erdelyi-Kober derivative and its appli-cationsrdquo Integral Transforms and Special Functions vol 25 no5 pp 359ndash373 2014

[19] Y Xu Z He and Q Xu ldquoNumerical solutions of fractionaladvection-diffusion equations with a kind of new generalizedfractional derivativerdquo International Journal of Computer Math-ematics vol 91 no 3 pp 588ndash600 2014

[20] Y Xu Z He and O P Agrawal ldquoNumerical and analytical solu-tions of new generalized fractional diffusion equationrdquo Com-puters amp Mathematics with Applications vol 66 no 10 pp2019ndash2029 2013

[21] Y Xu and O P Agrawal ldquoNumerical solutions and analysisof diffusion for new generalized fractional Burgers equationrdquoFractional Calculus and Applied Analysis An International Jour-nal forTheory and Applications vol 16 no 3 pp 709ndash736 2013

[22] X Li and C Xu ldquoA space-time spectral method for the timefractional diffusion equationrdquo SIAM Journal on NumericalAnalysis vol 47 no 3 pp 2108ndash2131 2009

[23] Q Xu and J S Hesthaven ldquoStable multi-domain spectralpenalty methods for fractional partial differential equationsrdquoJournal of Computational Physics vol 257 pp 241ndash258 2014

[24] RMittal and S Pandit ldquoQuasilinearized Scale-3 Haar wavelets-based algorithm for numerical simulation of fractional dynam-ical systemsrdquo Engineering Computations vol 35 no 5 pp 1907ndash1931 2018

[25] C Li F Zeng and F Liu ldquoSpectral approximations to the frac-tional integral and derivativerdquo Fractional Calculus and AppliedAnalysis vol 15 no 3 pp 383ndash406 2012

[26] M Zheng F Liu I Turner and V Anh ldquoA novel high orderspace-time spectral method for the time fractional Fokker-Planck equationrdquo SIAM Journal on Scientific Computing vol 37no 2 pp A701ndashA724 2015

[27] F Zeng F Liu C Li K Burrage I Turner and V Anh ldquoACrank-Nicolson ADI spectral method for a two-dimensionalRiesz space fractional nonlinear reaction-diffusion equationrdquoSIAM Journal on Numerical Analysis vol 52 no 6 pp 2599ndash2622 2014

[28] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical ModellingSimulation andComputation for Engineering and EnvironmentalSystems vol 35 no 12 pp 5662ndash5672 2011

[29] M Zayernouri and G E Karniadakis ldquoExponentially accuratespectral and spectral element methods for fractional ODEsrdquoJournal of Computational Physics vol 257 pp 460ndash480 2014

[30] X Zhao and Z Zhang ldquoSuperconvergence points of fractionalspectral interpolationrdquo SIAM Journal on Scientific Computingvol 38 no 1 pp A598ndashA613 2016

[31] F Chen Q Xu and J S Hesthaven ldquoA multi-domain spectralmethod for time-fractional differential equationsrdquo Journal ofComputational Physics vol 293 pp 157ndash172 2015

[32] Z Mao and J Shen ldquoEfficient spectral-Galerkin methods forfractional partial differential equations with variable coeffi-cientsrdquo Journal of Computational Physics vol 307 pp 243ndash2612016

[33] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[34] J Hadamard ldquoEssai sur lrsquoetude des fonctions donnees par leurdeveloppement de Taylorrdquo Journal de Mathematiques Pures etAppliquees vol 4 pp 101ndash186 1892

[35] A Erdelyi andHKober ldquoSome remarks onHankel transformsrdquoQuarterly Journal of Mathematics vol 11 pp 212ndash221 1940

[36] I Dimovski ldquoOperational calculus for a class of differentialoperatorsrdquo Comptes Rendus De L Academie Bulgare Des Sci-ences vol 19 pp 1111ndash1114 1966

[37] S B Yakubovich and Y F LuchkoThe hypergeometric approachto integral transforms and convolutions vol 287 ofMathematicsand its Applications Kluwer Academic Publishers Dordrecht-Boston-London 1994

[38] Y Luchko ldquoOperational rules for a mixed operator of theErdelyi-Kober typerdquo Fractional Calculus and Applied Analysisvol 7 no 3 pp 339ndash364 2004

[39] G Szego Orthogonal polynomials American MathematicalSociety Providence 1992

[40] M Zayernouri and G E Karniadakis ldquoFractional Sturm-Liou-ville eigen-problems theory and numerical approximationrdquoJournal of Computational Physics vol 252 pp 495ndash517 2013

[41] S Chen J Shen and L-L Wang ldquoGeneralized Jacobi functionsand their applications to fractional differential equationsrdquoMathematics of Computation vol 85 no 300 pp 1603ndash16382016

[42] A A Kilbas ldquoHadamard-type integral equations and fractionalcalculus operatorsrdquo in Singular integral operators factorizationand applications vol 142 ofOperTheory Adv Appl pp 175ndash188Birkhauser Basel 2003

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4 International Journal of Differential Equations

approximation for generalized fractional operators basedon Agrawalrsquos definitions Fractional derivativesintegrals ofothers type can be obtained as special cases

31 Fractional Derivative of Orthogonal Polynomials Denoteby 119869120573120574119895 (119909) the 119895-th order Jacobi polynomial with index (120573 120574)defined on [minus1 1]

As a set of orthogonal polynomials 119869120573120574119895 (119909)119873119895=0 satisfiesthe following three-term-recurrence relation [39]

1198691205731205740 (119909) = 11198691205731205741 (119909) = (120573 + 120574 + 2) 119909 + (120573 minus 120574) 119869120573120574119895+1 (119909) = (119860120573120574

119895 119909 minus 119861120573120574119895 ) 119869120573120574119895 (119909) minus 119862120573120574

119895 119869120573120574119895minus1 (119909) 1 ⩽ 119895 ⩽ 119873 minus 1

(16)

where the recursive coefficients are defined as

119860120573120574119895 = (2119895 + 120573 + 120574 + 1) (2119895 + 120573 + 120574 + 2)2 (119895 + 1) (119895 + 120573 + 120574 + 1)

119861120573120574119895 = (1205742 minus 1205732) (2119895 + 120573 + 120574 + 1)

2 (119895 + 1) (119895 + 120573 + 120574 + 1) (2119895 + 120573 + 120574) 119862120573120574119895 = (119895 + 120573) (119895 + 120574) (2119895 + 120573 + 120574 + 2)(119895 + 1) (119895 + 120573 + 120574 + 1) (2119895 + 120573 + 120574)

(17)

In order to derive fractional derivative of Jacobi polyno-mials we introduce some useful lemmas first

Lemma 14 For any 119899 isin N+ 120573 120574 isin R 120572 gt 0 the followingrelation holds

119869(minus1minus120572120574+120572+1)119899 (119909)= minus(119899 + 120574 + 1) (1 minus 119909)2 (119899 minus 120572) 119869(1minus120572120574+120572+1)119899minus1 (119909)minus 120572119869(minus120572120574+120572)119899 (119909)119899 minus 120572

(18)

119869(120573+120572+1minus1minus120572)119899 (119909)= (119899 + 120573 + 1) (1 + 119909)2 (119899 minus 120572) 119869(120573+120572+11minus120572)119899minus1 (119909)minus 120572119869(120573+120572minus120572)119899 (119909)119899 minus 120572

(19)

Proof According to [39 421] Jacobi polynomials withparameters 120573 120574 isin R are defined by

119869(120573120574)119899 (119909)= 119899sum

119895=0

1119895 (119899 minus 119895) Γ (119899 + 119895 + 120573 + 120574 + 1)Γ (119899 + 120573 + 120574 + 1) Γ (119899 + 120573 + 1)Γ (119895 + 120573 + 1) (119909 minus 12 )119895 (20)

Considering property of Jacobi polynomials

119869(120573120574)119899 (119909) = (minus1)119899 119869(120574120573)119899 (minus119909) (21)

Jacobi polynomials can be rewritten in the following form

119869(120573120574)119899 (119909)= 119899sum

119895=0

(minus1)119895+119899119895 (119899 minus 119895) Γ (119899 + 119895 + 120573 + 120574 + 1)Γ (119899 + 120573 + 120574 + 1) Γ (119899 + 120574 + 1)Γ (119895 + 120574 + 1) (119909 + 12 )119895 (22)

Define the following symbols

119875119860 = (119899 + 120573 + 1) (1 + 119909) 119869(120573+120572+11minus120572)119899minus1 (119909) 119875119861 = 2120572119869(120573+120572minus120572)119899 (119909) 119875119862 = 2 (119899 minus 120572) 119869(120573+120572+1minus1minus120572)119899 (119909)

(23)

Through a series of calculation we have

119875119861 + 119875119862= 119899sum

119895=0

2120572 (minus1)119895+119899119895 (119899 minus 119895) Γ (119899 + 119895 + 120573 + 1)Γ (119899 + 120573 + 1) Γ (119899 minus 120572 + 1)Γ (119895 minus 120572 + 1) (119909 + 12 )119895

+ 119899sum119895=0

2 (119899 minus 120572) (minus1)119895+119899119895 (119899 minus 119895) Γ (119899 + 119895 + 120573 + 1)Γ (119899 + 120573 + 1) Γ (119899 minus 120572)Γ (119895 minus 120572) (119909 + 12 )119895

= 119899sum119895=0

(minus1)119895+119899119895 (119899 minus 119895) Γ (119899 + 119895 + 120573 + 1)Γ (119899 + 120573 + 1) Γ (119899 minus 120572 + 1)Γ (119895 minus 120572 + 1) (119909 + 12 )119895 (2119895) 119875119860= 119899sum

119895=1

2 (minus1)119895+119899(119895 minus 1) (119899 minus 119895) Γ (119899 + 119895 + 120573 + 1)Γ (119899 + 120573 + 1) Γ (119899 minus 120572 + 1)Γ (119895 minus 120572 + 1) (119909 + 12 )119895

= 119875119861 + 119875119862

(24)

The equality (19) is proved Equality (18) can be provedsimilarly

Lemma 15 (see [39 P96]) For120572 gt 0 minus1 lt 119909 lt 1 120573 gt minus1 120574 isinR

(1 minus 119909)120573+120572 119869(120573+120572120574minus120572)119899 (119909)119869(120573+120572120574minus120572)119899 (1)

= Γ (120573 + 120572 + 1)Γ (120573 + 1) Γ (120572) int1

119909

(1 minus 119910)120573(119910 minus 119909)1minus120572

119869(120573120574)119899 (119910)119869(120573120574)119899 (1) 119889119910

(25)

For 120572 gt 0 minus1 lt 119909 lt 1 120574 gt minus1 120573 isin R

(1 + 119909)120574+120572 119869(120573minus120572120574+120572)119899 (119909)119869(120573minus120572120574+120572)119899 (1)

= Γ (120574 + 120572 + 1)Γ (120574 + 1) Γ (120572) int119909

minus1

(1 + 119910)120574(119909 minus 119910)1minus120572

119869(120573120574)119899 (119910)119869(120573120574)119899 (1) 119889119910

(26)

Lemma 16 (see [29 40]) For 120572 gt 0 minus1 lt 119909 lt 1 120573 120574 isin R

1199091198681205721119869(0120574)119899 (119909) = 1Γ (120572) int1

119909

119869(0120574)119899 (119910)(119910 minus 119909)1minus120572 119889119910

= 119899 (1 minus 119909)120572Γ (119899 + 120572 + 1)119869(120572120574minus120572)119899 (119909) (27)

International Journal of Differential Equations 5

minus1119868120572119909119869(1205730)119899 (119909) = 1Γ (120572) int119909

minus1

119869(1205730)119899 (119910)(119909 minus 119910)1minus120572 119889119910

= 119899 (1 + 119909)120572Γ (119899 + 120572 + 1)119869(120573minus120572120572)119899 (119909) (28)

Lemma 17 forall120572 gt 0 120572 notin N minus1 lt 119909 lt 1 120573 120574 isin R

1199091198631205721119869(0120574)119899 (119909) = 119899 (1 minus 119909)minus120572Γ (119899 + 1 minus 120572)119869(minus120572120574+120572)119899 (119909) (29)

minus1119863120572119909119869(1205730)119899 (119909) = 119899 (1 + 119909)minus120572Γ (119899 + 1 minus 120572)119869(120573+120572minus120572)119899 (119909) (30)

Proof First for the case 0 lt 120572 lt 1 formulas (29) and (30) canbe obtained by setting 120573 = minus120572 in formula (25) setting 120574 = minus120572in formula (26) and applying Riemann-Liouville fractionalderivative operator to both sides of themWe refer to [22] fordetailed discussion

For the case 120572 gt 1 120572 notin N the formulas cannot beobtained from Lemma 15 due to the constrains 120573 gt minus1 and120574 gt minus1 in Lemma 15 Here we prove the lemma inductively

For left Riemann-Liouville derivative from case 1 equal-ity (30) holds for lfloor120572rfloor = 0 Assume the equality holds forlfloor120572rfloor = 119896 minus 1 119896 isin N+ when lfloor120572rfloor = 119896 and let = 120572 minus 1from the assumption it holds that

minus1119863119909119869(1205730)119899 (119909) = 119899 (1 + 119909)minusΓ (119899 + 1 minus )119869(120573+minus)119899 (119909) (31)

We apply 119889119889119909 to both sides of (31) and then

minus1119863120572119909119869(1205730)119899 (119909) = minus1119863+1

119909 119869(1205730)119899 (119909)= 119899Γ (119899 + 1 minus ) (minus (1 + 119909)minusminus1 119869(120573+minus)119899 (119909)+ (1 + 119909)minus 119889119889119909119869(120573+minus)119899 (119909))= 119899 (1 + 119909)minusminus1Γ (119899 + 1 minus ) (minus119869(120573+minus)119899 (119909)+ (119899 + 120573 + 1) (1 + 119909)2 119869(120573++11minus)119899minus1 (119909))

(32)

Applying Lemma 14 we have

minus1119863120572119909119869(1205730)119899 (119909)= 119899 (1 + 119909)minusminus1Γ (119899 + 1 minus ) (minus (119899 minus 120572) 119869(120573++1minusminus1)119899 (119909))= 119899 (1 + 119909)minusminus1Γ (119899 + 1 minus ) ((119899 minus ) 119869(120573++1minusminus1)119899 (119909))= 119899 (1 + 119909)minus120572Γ (119899 + 1 minus 120572)119869(120573+120572minus120572)119899 (119909)

(33)

Equality (30) is proved Equality (29) can be proved similarly

Through the relationship between Caputo derivative andRiemann-Liouville derivative we immediately obtain thefractional derivative for Caputo derivative

Lemma 18 For 120572 gt 0 minus1 lt 119909 lt 1119888119909119863120572

1119869(0120574)119899 (119909) = 119899 (1 minus 119909)minus120572Γ (119899 + 1 minus 120572)119869(minus120572120574+120572)119899 (119909)minus 119898minus1sum

119895=0

120597119895119909119869(0120574)119899 (1) (1 minus 119909)119895minus120572(minus1)119895 Γ (1 + 119895 minus 120572)

119888minus1119863120572

119909119869(1205730)119899 (119909) = 119899 (1 + 119909)minus120572Γ (119899 + 1 minus 120572)119869(120573+120572minus120572)119899 (119909)minus 119898minus1sum

119895=0

120597119895119909119869(1205730)119899 (minus1) (1 + 119909)119895minus120572Γ (1 + 119895 minus 120572)

(34)

where119898 = lceil120572rceilGiven a function 119906(119909) isin 119867119898

120596 [minus1 1] and polynomialsspace P119873 the projection of 119906(119909) in space P119873 120587119873119906(119909)satisfies the following relation

(119906 minus 120587119873119906 V)120596 = 0 forallV isin P119873 (35)

According properties of space P119873 and Jacobi polynomials119906119873 can be expressed as

120587119873119906 (119909) = 119873sum119895=0

119895119869(120573120574)119895 (119909) (36)

where 119895 = (119906(119909) 119869(120573120574)(119909))120596119869(120573120574)(119909)2120596 120596 = 120596(119909) is theweight function

Then fractional derivative of 119906(119909) can be approximatedas

minus1119863120572119909119906 (119909) asymp 119873sum

119895=0

119895 ( minus1119863120572119909119869(120573120574)119895 (119909)) (37)

And we have the following lemma

Lemma 19 (see [41]) forall119906 isin 119867119898+1120596 [minus1 1] 0 le 119897 le 119898 119897 119898 isin

N 0 lt 120572 lt 1 119906119873 is a spectral approximation to 119906 in polynomialspace such that (119906 minus 120587119873119906 V)120596 = 0 forallV isin P119873 and then thereexists a constant 119862 such that10038171003817100381710038171003817 minus1119863119897+120572

119909 (119906 (119909) minus 120587119873119906 (119909))10038171003817100381710038171003817120596(120573+120572+119897120572+119897)le 119862119873119897minus119898 1003817100381710038171003817 minus1119863119898+120572

119909 1199061003817100381710038171003817120596(120573+120572+119898120572+119898) (38)

where 120596(120573120574) = (1 minus 119909)120573(1 + 119909)120574Combining Lemmas 18 and 19 and (37) approximation

method can be obtained immediately for Caputo derivativeand we have the following corollary

Corollary 20 forall119906 isin 119867119898+1120596 [minus1 1] 0 le 119897 le 119898 119897 119898 isin N 0 lt120572 lt 1 119906119873 is a spectral approximation to 119906 in polynomial space

6 International Journal of Differential Equations

such that (119906 minus 120587119873119906 V)120596 = 0 forallV isin P119873 and then there exists aconstant 119862 such that100381710038171003817100381710038171003817 119888

minus1119863119897+120572

119909 (119906 (119909) minus 120587119873119906 (119909))100381710038171003817100381710038171003817120596 le 119862119873119897minus119898 1003817100381710038171003817 minus1119863119898+120572119909 1199061003817100381710038171003817120596 (39)

Based on above lemmas fractional integral and derivativeof Jacobi polynomials in the standard interval [minus1 1] can beexpressed explicitly Fractional integrals and derivatives inshifted interval [119886 119887] can be obtained through proper variabletransforms

For any 119910 isin [119886 119887] we assume 119869120573120574119899lowast(119910) is defined in [119886 119887]Let ℎ = 119887 minus 119886 119909 = minus1 + 2((119910 minus 119886)ℎ) then 119909 isin [minus1 1]Substituting 119909 = minus1 + 2((119910 minus 119886)(119887 minus 119886)) into equation (27)-(28) and (29)-(30) we obtained

119910119868120572119887119869(0120574)119899lowast (119910) = (ℎ2)120572

1199091198681205721119869(0120574)119899 (119909) 119886119868120572119910119869(1205730)119899lowast (119910) = (ℎ2)

120572

minus1119868120572119909119869(1205730)119899 (119909) 119888119910119863120572

119887119869(0120574)119899lowast (119910) = (2ℎ)

120572119888119909119863120572

1119869(0120574)119899 (119909) 119888119886119863120572

119910119869(1205730)119899lowast (119910) = (2ℎ)120572

119888minus1119863120572

119909119869(1205730)119899 (119909)

(40)

32 Spectral Approximation to Generalized Fractional Opera-tors We assume 119911(119909) and 119908(119909) are positive monotone func-tions and 119911 119908 isin 119862[119886 119887] Obviously 119911(119909) and119908(119909) are invert-ible The following lemmas can be derived for generalizedfractional operators

Lemma 21 Let 119892(120577) = 119908(119911minus1(120577))119891(119911minus1(120577)) 0 lt 120572 lt 1 thenthe generalized fractional integral operator is equivalent to thefollowing classical fractional integral

(119868120572119886+[119911119908]119891) (119909) = [119908 (119909)]minus1 1205770119868120572120577119892 (119911 (119909)) (41)

Here 1205770 = 119911(119886)Proof From the definition of generalized fractional integralwe have

(119868120572119886+[119911119908]119891) (119909) = [119908 (119909)]minus1Γ (120572) int119909

119886

119908 (119905) 1199111015840 (119905) 119891 (119905)[119911 (119909) minus 119911 (119905)]1minus120572 119889119905 (42)

Since 120577 = 119911(119905) is positive monotone function then 120577 = 119911(119905) isinvertible and we have 119905 = 119911minus1(120577)(119868120572119886+[119911119908]119891) (119909) = [119908 (119909)]minus1Γ (120572)sdot int119911(119909)

119911(119886)

119908(119911minus1 (120577)) 1199111015840 (119911minus1 (120577)) 119891 (119911minus1 (120577))[119911 (119909) minus 120577]1minus120572 119889119911minus1 (120577)

= [119908 (119909)]minus1Γ (120572) int119911(119909)

119911(119886)

119908(119911minus1 (120577)) 119891 (119911minus1 (120577))[119911 (119909119896) minus 120577]1minus120572 119889120577

(43)

Let 119892(120577) = 119908(119911minus1(120577))119891(119911minus1(120577)) then the generalized frac-tional integral of 119891(119909) is converted to classical fractionalintegral of 119892(120577) in the following form

(119868120572119886+[119911119908]119891) (119909) = [119908 (119909)]minus1 1205770119868120572120577119892 (119911 (119909)) (44)

Lemma is proved

Lemma 22 Let 119892(120577) = 119908(119911minus1(120577))119891(119911minus1(120577)) 0 lt 120572 lt 1 thenthe generalized fractional derivative of order 120572 is equivalent tothe following classical fractional derivative

(119863120572119886+[1199111199082]119891) (119909) = [119908 (119909)]minus1 119888

1205770119863120572

120577119892 (119911 (119909)) (45)

Here 1205770 = 119911(119886)Proof Similar to the proof of Lemma 21 we have

(119863120572119886+[1199111199082]119891) (119909) = [119908 (119909)]minus1Γ (1 minus 120572)sdot int119911(119909)

119911(119886)

(120597120597120577) (119908 (119911minus1 (120577)) 119891 (119911minus1 (120577))) (120597120577120597119905)[119911 (119909) minus 120577]120572 119889119911minus1 (120577)

= [119908 (119909)]minus1Γ (1 minus 120572) int119911(119909)

119911(119886)

(120597120597120577) (119908 (119911minus1 (120577)) 119891 (119911minus1 (120577)))[119911 (119909) minus 120577]120572 119889120577

(46)

Let 119892(120577) = 119908(119911minus1(120577))119891(119911minus1(120577)) then the generalized frac-tional derivative of 119891(119909) is expressed through the classicalfractional derivative of 119892(120577)

(119863120572119886+[1199111199082]119891) (119909) = [119908 (119909)]minus1 119888

1205770119863120572

120577119892 (119911 (119909)) (47)

Lemma is proved

Remark 23 These two lemmas establish important relationbetween classical and generalized fractional operators Withthese lemmas generalized fractional differential equationscan be solved via classical fractional differential equationsand vice versa Which kind of transform should be takendepends on characters of the problem to be solved

In order to design a high order numerical approximationof the generalized fractional operator we define a scaled spaceP119899

[119908119911] such that

P119899[119908119911] = V (119909) = [119908 (119909)]minus1 119892 (119911 (119909)) 119892 (119909) isin P

119899 119909isin Ω = [119886 119887] (48)

whereP119899 is a polynomial space of up to order 119899Define a inner product and norm for space P119899

[119908119911] suchthat

(119906 (119909) V (119909))120596 = int119887

119886119906 (119909) V (119909) 120596 (119909) 119889119909

and V120596 = radic(V V)120596(49)

International Journal of Differential Equations 7

Define projection 119876119873 into space P119873119908119911 such that for any

function 119906(119909)(119906 minus 119876119873119906 V)120596 = 0 forallV isin P

119873119908119911 (50)

SupposeΦ119895(119909) 119895 = 0 119873 are a set of orthogonal basisfunctions in spaceP119873

119911119908 satisfying

(Φ119894 (119909) Φ119895 (119909))120596 = 1 119894 = 1198950 119894 = 119895 (51)

Let119873 997888rarr infin then Φ119895(119909) 119895 = 0 1 119873 form a 1198712120596(Ω)space and for any 119906(119909) isin 1198712120596(Ω) the projection 119876119873119906(119909) canbe written as

119876119873119906 (119909) = 119873sum119895=0

119895Φ119895 (119909) (52)

Here 119895 (119895 = 0 119873) are expansion coefficients such that

119895 = (119906 (119909) Φ119895 (119909))12059610038171003817100381710038171003817Φ119895 (119909)100381710038171003817100381710038172120596 (53)

The weight function 120596(119909) plays an important role in thecomputational process and analysis of the method Herewe choose a proper weight function to use properties oforthogonal polynomials and make the computation moreefficient Let 119901119895(119911(119909)) = 119908(119909)Φ119895(119909) and note that

(119876119873119906 (119909) Φ119894 (119909))120596 = int119887

119886( 119873sum

119895=0

119895119908minus1 (119909) 119901119895 (119911 (119909)))sdot 119908minus1 (119909) 119901119894 (119911 (119909)) 120596 (119909) 119889119909

(54)

Take 120596(119909) = 1199082(119909)1199111015840(119909) then(119876119873119906 (119909) Φ119894 (119909))120596 = int119911(119887)

119911(119886)( 119873sum

119895=0

119895119901119895 (119911))119901119894 (119911) 119889119911 (55)

Since 119901119895(119911) is polynomial of order 119895 the computation can becarried out easily through properties of orthogonal polyno-mials

Suppose 119875119895lowast(119909) is shifted Legendre polynomial definedon [119911(119886) 119911(119887)] Then given any function 119906(119909) isin 1198712120596 withweight 120596(119909) = 1199082(119909)1199111015840(119909) we have

119876119873119906 (119909) = [119908 (119909)]minus1 119873sum119895=0

119895119875119895lowast (119911 (119909)) (56)

Recalling Lemmas 21 and 22 the generalized fractionalintegral and derivative can be obtained in the form

(119868120572119886+[119911119908]119876119873119906) (119909)= [119908 (119909)]minus1 119873sum

119895=0

119895 ( 1205770119868120572120577119875119895lowast (120577)1003816100381610038161003816120577=119911(119909)) (57)

(119863120572119886+[1199111199082]119876119873119906) (119909)= [119908 (119909)]minus1 119873sum

119895=0

119895 ( 1198881205770119863120572

120577119875119895lowast (120577)1003816100381610038161003816120577=119911(119909)) (58)

From Lemma 19 the following corollary can be obtainedimmediately

Corollary 24 120577(119909) = 119911(119909) is a monotone increasing function119908(119909) gt 0 0 lt 120572 lt 1 119876119873 is a projection into space P119873119908119911119906(119911minus1(120577)) isin 119862119898(Ω) 119898 isin N then there exists a constant 119862120572

such that 10038171003817100381710038171003817119863120572119886+[1199111199082] (119906 (119909) minus 119876119873119906 (119909))100381710038171003817100381710038171205962le 1198621205721198731minus119898 10038171003817100381710038171003817 1205770119863119898+120572

119911119906 (120577minus1 (119911))100381710038171003817100381710038171205961

(59)

Here the weight function 1205961(120577) = (120577 minus 119886)minus120572(119887 minus 120577)120572 1205962(119909) =1205961(119911(119909))1199082(119909)1199111015840(119909)Remark 25 Unlike the convergence theory in classical poly-nomial space the convergence order in space P119873

119908119911 dependson regularity of the function 119906(119909) with respect to 119911(119909) Wewill illustrate this through numerical examples

Most of the time it is more convenient to consider theproblems in nodal form We assume the given interpolationpoints are 120577119895 = 119911(119909119895) (119895 = 0 1 119873) then the Lagrangebasis functions L119895(119909) (119895 = 0 1 119873) can be defined asfollows

L119894 (120577) = prod119895=0119873119895 =119894

(120577 minus 120577119895)(120577119894 minus 120577119895) (60)

The function 119906(119909) can be expressed using both Jacobipolynomials and Lagrange polynomials The following rela-tion is derived

119906 (119909) = [119908 (119909)]minus1 119873sum119895=0

119892119895119875119895lowast (119911 (119909))

= [119908 (119909)]minus1 119873sum119895=0

119908119895119906119895L119895 (119911 (119909)) (61)

Considering the equivalence between Legendre basis andLagrange basis the following equality holds

L119895 (119911) = 119873sum119896=0

119897119896119895119875119896lowast (119911) 119896 = 0 1 119873 (62)

where 119897119894119895 = (119871119895(119911) 119875119894lowast(119911))119875119894lowast(119911)21198712 Then the nodal form expansion of 119906(119909) is obtained

119906 (119909) = [119908 (119909)]minus1 119873sum119895=0

119873sum119896=0

119908119895119906119895119897119896119895119875119896lowast (119911 (119909)) (63)

8 International Journal of Differential Equations

From (57) and (58) the corresponding nodal form of gener-alized fractional integral and derivative are obtained

(119868120572119886+[119911119908]119906) (119909)= [119908 (119909)]minus1 119873sum

119895=0

119873sum119896=0

119908119895119906119895119897119896119895 ( 1205770119868120572120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119909)) (64)

(119863120572119886+[1199111199082]119906) (119909)= [119908 (119909)]minus1 119873sum

119895=0

119873sum119896=0

119908119895119906119895119897119896119895 ( 1198881205770119863120572

120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119909)) (65)

Example 26 Now we give an example to show the effec-tiveness and accuracy of the method Assuming 119911(119909) =radic1199093 119908(119909) = 1 we consider generalized fractional derivativeof 119910(119909) on the interval [0 1] with the following form

119910 (119909) = 5sum119895=1

(minus1)119895+1 1199093119895(2119895) (66)

The exact generalized fractional derivative of 119910(119909) is119863120572

0+[1199111199082]119910 (119909) = 5sum119895=1

(minus1)119895+1 1199093119895minus31205722Γ (2119895 + 1 minus 120572) (67)

Numerical approximation to generalized fractionalderivative of 119910(119909) can be obtained using (58) We considerthe maximum absolute error

119890 = max119909isin[01]

100381610038161003816100381610038161198631205720+[1199111199082]119910 (119909) minus 119863120572

0+[1199111199082]119876119873119910 (119909)10038161003816100381610038161003816 (68)

of the numerical derivative Results for 120572 = 02 05 08 areshown in Figure 1

For this example it is easy to check that 119910 isin P10119908119911 From

theory of spectral approximation the error would decreaseexponentially when 119873 lt 10 and the numerical fractionalderivative would be exact when119873 ge 10 Our numerical resultcoincides with the theory exactly

Example 27 In this example we test the spectral approxi-mation of Hadamard integral Considering Hadamard-typefractional integral of 119910(119909) = sin(120587119909) on the interval [1 2]when 119908(119909) = 1 and 120572 = 1 the Hadamard integral of 119910(119909) isSine integral function Si(120587119909) minus Si(120587) for more general 119908(119909)and 120572 the exact Hadamard-type integral of 119910(119909) is unknownHere we consider several pairs of 119908(119909) and 120572 NumericalHadamard-type integral of 119910(119909) would be computed using(57) with 119911(119909) = log(119909) To evaluate the approximationaccuracy for the case 119908(119909) = 1 120572 = 1 the exact Hada-mard fractional integral is computed usingMATLAB built-infunction 119904119894119899119894119899119905 for other cases ldquoexactrdquo Hadamard fractionalintegral is computed using (57) with large 119873 (eg 119873 =50) which is treated as reference solution The results fornumerical Hadamard integral and approximation error areshown in Figures 2 and 3

For this example 119906 is not in the spaceP119899119908119911 for any 119899 Max-

imum error of approximated fractional integral convergesexponentially until reaching machine accuracy

10minus16

10minus14

10minus12

10minus10

10minus8

10minus6

10minus4

10minus2

max

imum

erro

r

5 6 7 8 94 11 1210

N

alpha=02alpha=05alpha=08

Figure 1 Error of numerical approximation to generalized frac-tional derivative of 119910(119909)

11 12 13 14 15 16 17 18 19 21x

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Had

amar

d fra

ctio

nal i

nteg

ral

w(x)=1=1w(x)=1=06

w(x)=x=1w(x)=x=06

Figure 2 Hadamard fractional integral for different weight 119908(119909)and 120572

33 Fractional IntegralDifferential Matrices Suppose 119906(119909) isinP119873

119911119908 119909119894 (119894 = 0 1 119873) are the interpolation points and119910119895 (119895 = 0 1 119873) are the collocation points we define thefollowing symbols

U = [119906 (1199090) 1199061199091 119906 (119909119873)]119879 (69)

U(120572) = [119863120572

119886+[119911119908]119906 (1199100) 119863120572119886+[119911119908]119906 (1199101)

119863120572119886+[119911119908]119906 (119910119873)]119879 120572 gt 0 (70)

International Journal of Differential Equations 9

10minus16

10minus14

10minus12

10minus10

10minus8

10minus6

10minus4

max

imum

erro

r

w(x)=1=1w(x)=1=06

w(x)=x=1w(x)=x=06

10 12 14 16 18 20 228x

Figure 3 Approximation error of Hadamard fractional integral fordifferent weight 119908 and 120572

U(120572) = [119868minus120572119886+[119911119908]119906 (1199100) 119868minus120572119886+[119911119908]119906 (1199101) 119868minus120572119886+[119911119908]119906 (119910119873)]119879 120572 lt 0 (71)

We define a generalized fractional differentialintegralmatrixM120572 such that

M120572119894119895 = 119873sum

119896=0

119908(119909119895) 119897119896119895119908 (119910119894) ( 1198881205770119863120572

120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119910119894))

for 120572 gt 0(72)

M120572119894119895 = 119873sum

119896=0

119908(119909119895) 119897119896119895119908 (119910119894) ( 1205770119868120572120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119910119894))

for 120572 lt 0(73)

In order to compute the fractional matrix M120572 moreefficiently we define a few more matrices L and L120572 are(119873 + 1) times (119873 + 1) matrices such thatL119894119895 = L119895(119911(119910119894))L120572

119894119895 =1205770119863120572

120577L119895(119911)|119911=119911(119910119894) 119894 119895 = 0 1 119873 V is defined based on

values of Legendre polynomials at interpolation points 119911(119909119894)such that

V119894119895 = 119875119895lowast (119911 (119909119894)) 119894 119895 = 0 1 119873 (74)

D120572 is defined as fractional derivativeintegral of Legendrepolynomials at collocation points 120577119894 = 119911(119910119894)

D120572119894119895 = 119888

1205770119863120572

120577119875119895lowast (120577119894) 120572 gt 0 (75)

D120572119894119895 = 1205770

119868120572120577119875119895lowast (120577119894) 120572 lt 0 (76)

W119897 andW119903 are the weight matrices defined as follows

W119903119894119895 =

119908(119909119895) 119894 = 1198950 otherwise (77)

W119897119894119895 =

1119908(119910119895) 119894 = 119895 gt 10 otherwise (78)

Theorem 28 For 119906(119909) isin P119873119911119908 vectors defined in (69)ndash(71)

and matrices defined in (72)ndash(78) the following relation holds

U(120572) = M

120572U = (W119897D120572V

minus1W

119903)U (79)

Proof From the definition (69)ndash(71) (72)ndash(78) and (63)-(64) it is easy to obtain

U(120572) = M

120572U (80)

Next we prove thatM120572 = W119897D120572Vminus1W119903From the definition of the interpolation function L119895(119911)

119875119895lowast (119911) = 119873sum119894=0

119875119895lowast (119911 (119909119894)) L119894 (119911) (81)

Suppose 997888rarr119875(119911) = (1198750lowast(119911) 1198751lowast(119911) 119875119873lowast(119911)) 997888rarrL (119911) =(L0(119911) L1(119911) L119873(119911)) for 120572 gt 0 we have997888rarr119875 (119911) = 997888rarrL (119911)V

1205770119863120572

120577

997888rarr119875 (119911) = ( 1205770119863120572

120577

997888rarrL (119911))V (82)

Evaluating the matrices multiplication for each elementthe following relation is derived

(W119897D120572V

minus1W119903)119894119895 = (W119897L

120572W119903)119894119895

= 119908 (119909119895)119908 (119910119894) L(120572)119895 (119911 (119910119894))(83)

= 119873sum119896=0

119908(119909119895) 119897119896119895119908 (119910119894) ( 1205770119863120572

120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119910119894)) = M

120572119894119895 (84)

The case 120572 lt 0 can be proved similarly

Remark 29 Collocation points 119910119895 and the interpolationpoints 119909119894 are not necessary the same To obtain a goodapproximation interpolation points ofGauss-type are usuallyused At the same time collocation points should be chosenproperly to guarantee stability properties of the methodIn the following numerical examples for computation andstability aim both interpolation and collocation points arechosen based on Gauss-type points with respect to 119911(119909)

10 International Journal of Differential Equations

4 Collocation Methods for FractionalDifferential and Integral Equations

41 Fractional Ordinary Differential Equations In this sub-section we consider collocation method for the generalizedfractional ordinary differential equation of the form

1198631205720+[1199111199082]119906 (119909) = 120582 (119909) 119906 (119909) + 119891 (119909)

119909 isin Ω = (0 119887] (85)

119906 (0) = 1199060 (86)

Here 0 lt 120572 lt 1 119908(119909) gt 0 119911(119909) gt 0 and 119911(119909) is a monotonefunction inΩ

We assume 119906119873(119909) isin P119873119911119908 is a numerical solution of the

equation 119909119894(119894 = 0 1 119873) are the chosen interpolationpoints 119906119895119873 = 119906119873(119909119895) The following discretized equation isobtained

[119908 (119909)]minus1 119873sum119895=0

119873sum119896=0

119908(119909119895) 119906119895119873119897119896119895 ( 1198881205770119863120572

120577119875119896lowast (119911 (119909)))

= 120582 (119909) 119906119873 (119909) + 119891 (119909) (87)

Let (87) hold on collocation points 119910119895 (119895 = 1 119873) thematrix form is obtained

(M120572 minus Λ)U = 119865 (88)

Here Λ is a diagonal matrix with Λ 119894119894 = 120582(119910119894) 119865 = (1199060 119891(1199101) 119891(119910119873))119879Considering the initial condition we set S = M120572 minus Λ

with the first row replaced by (1 0 0) 1198651015840 = 119865with its firstelement replaced by 1199060 Then the solution U is obtained bysolving the matrix equation SU = 1198651015840

Example 30 Consider the following example

1198631205720+[1199111199082]119906 (119909) = (1 + 119909) 119906 (119909)

+ Γ (15119903 + 25) 119909119903minus(23)120572Γ (15119903 + 25 minus 120572)minus 119909119903 (1 + 119909) 119909 isin (0 1]

(89)

119906 (0) = 0 (90)

Here 119911(119909) = 11990923119908(119909) = 119909 119903 is an arbitrary positive numberThe exact solution of the ordinary differential equation is119906(119909) = 119909119903 Maximum absolute errors of numerical solutions

for 119903 = 6 7 and 120572 = 03 06 09 are shown in Figures 4 and 5When 119911(119909) = 11990923 119908(119909) = 119909 the scaled polynomial

spaceP119873119911119908 becomes

P119873119911119908 = 119904119901119886119899 11990921198993minus1 119899 = 1 2 119873 (91)

For 119903 = 7 the error converges exponentially and reachesmachine accuracy at 119873 = 12 It is faster than any finitedifference method while for 119903 = 6 solution convergesalgebraically as119873 increases however it still reaches machineaccuracy at 119873 = 23 The major reason for this is that 119906(119909) =1199097 isin P12

119911119908 and 119906(119909) = 1199096 notin P119873119911119908 for any119873 isin N

6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

N

10minus15

10minus10

10minus5

100

max

erro

r

=03=06=09

Figure 4 Log-log plot of the maximum error for 119903 = 6

=03=06=09

10minus15

10minus10

10minus5

100

max

erro

r

15 20 25 30 3510N

Figure 5 Semilog plot of the maximum error for 119903 = 7

Remark 31 Since space P119873119908119911 is transformed from classical

polynomial space with respect to 119911(119909) the convergenceof spectral collocation method for ODEs with generalizedfractional operators depends not only on the smoothness ofthe solution itself but also on the scale function 119911(119905)42 Hadamard-Type Integral Equations We consider thefollowing Hadamard-type boundary value problem

1Γ (120572) int119909

119886( 119904119909)

120583 (log 119909119904 )120572minus1 119891 (119904) 119889119904119904 = 119892 (119909)

119909 isin Ω = (119886 119887] (92)

International Journal of Differential Equations 11

In [42] Kilbas discussed the existence of the solution of(92) Explicit formulas for the solution 119891(119905) were establishedin the following theorem

Theorem 32 (see [42]) If 119909120583119892(119909) isin 119860119862[119886 119887] then theHadamard-type integral equation (92) with 0 lt 120572 lt 1 issolvable in 119883120583(119886 119887) and its solution may be represented in theform

119891 (119909) = 119909minus120583Γ (1 minus 120572) [119886120583119892 (119886) (log 119909119886)minus120572

+ int119909

119886(log 119909119904 )

minus120572 (119904120583119892 (119904))1015840 119889119904] (93)

Here 119860119862[119886 119887] is the set of absolutely continuous functions on[119886 119887] and119883120583(119886 119887) is space of those Lebesgue measurable func-tions 119891 on [119886 119887] for which 119909120583minus1119891(119909) is absolutely integrable

Solution of the Hadamard-type integral equation isexactly the same as generalized fractional derivative ofRiemann-Liouville type with 119911(119904) = log(119904) 119908(119904) = 119904120583 Therelationship between (93) and Caputo type generalized frac-tional derivative is

119891 (119909) = 119886120583119892 (119886) 119909minus120583Γ (1 minus 120572) (log 119909119886)minus120572 + 119863120572

119886+[1199111199082]119892 (119909) (94)

Suppose 119909119895 119895 = 0 1 119873 are interpolation points thediscretized solution of equation (92) is

119891119873 (119909) = 119886120583119892 (119886) 119909minus120583Γ (1 minus 120572) (log 119909119886)minus120572

+ 119909minus120583 119873sum119895=0

119873sum119896=0

119909120583119895119892 (119909119895) 119897119896119895( 119888120577119886119863120572

120577119875119896lowast (log (119909))

(95)

Rewriting (95) in matrix form we have

119865 = 119866119886 +M120572119866 (96)

where M120572 is generalized fractional differential matrix119866 = (119892(1199090) 119892(1199091) 119892(119909119873))119879 119865 = (119891119873(1199090) 119891119873(1199091) 119891119873(119909119873))119879 119866119886 is a vector about the initial condition of theintegral equation defined by

119866119886 = 119886120583119892 (119886)Γ (1 minus 120572) (119909minus1205830 (log 1199090119886 )minus120572 119909minus1205831 (log 1199091119886 )

minus120572 119909minus120583119873 (log 119909119873119886 )minus120572)119879

(97)

Example 33 Assume 119892(119909) = sin(119909 minus 1) 120583 = 13 Ω = [110] Solutions of (92) for 120572 = 03 06 09 are shown inFigure 6

2 4 6 8 10 120X

minus8

minus6

minus4

minus2

0

2

4

6

8

f(X)

=03=06=09

Figure 6 Solution of (92)

43 Erdelyi-Kober Fractional Diffusion Equation In this sub-section we consider the following Erdelyi-Kober fractionaldiffusion equation [12]

119906 (119909 119905) = 1199060 (119909) + 1Γ (120573)sdot int119905

0

120572120573119904120572120573minus1 (119905120572120573 minus 119904120572120573)120573minus1 12059721205971199092 119906 (119909 119904) d119904(98)

Erdelyi-Kober fractional diffusion equation which is alsocalled stretched time-fractional diffusion equation is themaster equation of a kind of generalized grey Brownianmotion (ggBm) The ggBm is a parametric class of stochasticprocesses that provides models for both fast and slow anoma-lous diffusion This class is made up of self-similar processes119861120572120573(119905)with stationary increments and it depends on two realparameters 0 lt 120572 le 2 and 0 lt 120573 le 1 It includes thefractional Brownian motion when 0 lt 120572 le 2 and 120573 = 1 thetime-fractional diffusion stochastic processes when 0 lt 120572 =120573 lt 1 and the standard Brownian motion when 120572 = 120573 = 1About the relationship between stochastic process119861120572120573(119905) andstretched time-fractional diffusion equation the followingproposition is presented in [12]

Proposition 34 The marginal probability density function119891120572120573(119909 119905) of the process 119861120572120573(119905) 119905 ge 0 is the fundamentalsolution of the stretched time-fractional diffusion equation

119906 (119909 119905) = 1199060 (119909) + 1Γ (120573)sdot int119905

0

120572120573119904120572120573minus1 (119905120572120573 minus 119904120572120573)120573minus1 12059721205971199092 119906 (119909 119904) d119904(99)

12 International Journal of Differential Equations

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 7 Standard Brownian motion 120572 = 1 120573 = 1

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 8 Time-fractional diffusion with 120572 = 06 120573 = 06

Recalling the definition of generalized fractional integraland setting 119911(119905) = 119905120572120573 119908(119905) = 1 the equation can be rewrit-ten as

119906 (119909 119905) = 1199060 (119909) + 1198681205730+[119911119908]119906119909119909 (119909 119905) (100)

We use collocation method for both space and timediscretization We choose Legendre-Gauss-Lobatto (L-G-L)points 119909119894 (119894 = 0 1 119872) as the space collocation points andchoose 119905119895 (119895 = 0 1 119873) such that 119911(119905119895) are L-G-L pointsas the time collocation points

Define space collocation matrix M2 such that M2119894119895 =(11988921198891199092)L119895(119909119894) and generalized fractional integral matrix

M120573 Matrix M120573 is computed through Theorem 28 and thespace-time collocation matrices are obtained using Kro-necker product 1015840otimes1015840 Suppose A and B are space-timecollocation matrices with dimension (119872 + 1)(119873 + 1) times (119872 +1)(119873 + 1) for the second order derivative and fractionalintegral of order 120573 separately Then

A = M2 otimesI119873+1

B = I119872+1 otimesM120573 (101)

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 9 Fractional Brownian motion with 120572 = 15 120573 = 1

Suppose 119906119873(119909 119905) is the numerical solution of (98) defin-ing 119906119873119894119895 = 119906119873(119909119894 119905119895) solution vectorU and initial vector 1198800such that

U = [11990611987300 11990611987301 1199061198730119873 11990611987310 1199061198731119873 1199061198731198720 119906119873119872119873]119879

1198800 = [1199060 (1199090) 1199060 (1199090) 1199060 (1199091) 1199060 (1199091) 1199060 (119909119872) 1199060 (119909119872)]119879

(102)

where in the definition of 1198800 each 1199060(119909119895) is repeated 119873 + 1times

The matrix form discretized equation of (98) is obtainedas

U = 1198800 +BAU (103)

In the discretized equation initial condition is explicitlyinvolved After boundary condition added properly numeri-cal solution can be obtained by solving the matrix equation

Example 35 Assume 119909 isin Ω = (minus1 1) 119905 isin (0 05] 1199060(119909) =119890minus101199092 minus 119890minus10 119906(sdot 119905)|120597Ω = 0 Numerical solutions with 119872 =119873 = 50 are shown in Figures 7ndash10

Erdelyi-Kober diffusion equation characterizes the mar-ginal density function of the process 119861120572120573(119905) 119905 ge 0 When120572 = 120573 = 1 we recover the standard diffusion equationWhen0 lt 120572 = 120573 lt 1 we get the time-fractional diffusion equationof order 120573 When 120573 = 1 and 0 lt 120572 lt 2 we have the equationof the fractional Brownian motion marginal density

As shown in Figures 7 and 8 when 1 lt 120572 lt 2 the diffu-sion is fast and the increments exhibit long-range depen-dence when 0 lt 120572 lt 1 the diffusion is slow and theincrements form a stationary process which does not exhibitlong-range dependenceThe results coincide with theoreticalanalysis in [12 14]

International Journal of Differential Equations 13

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 10 Fractional Brownian motion with 120572 = 05 120573 = 1

5 Conclusion

In this paper we propose a spectral collocation method fordifferential and integral equations with generalized fractionaloperators To deal with the difficulty in designing spectralapproximation scheme due to complexity of integral kerneland weight a variable transform technique is applied to thegeneralized fractional operator and a spectral approximationmethod is proposed for the generalized fractional operatorOperational matrices for generalized fractional operatorsare derived Spectral collocation methods are designed forfractional ordinary differential equations Hadamard-typeintegral equations and Erdelyi-Kober diffusion equationsseparately Numerical experiments are carried out to verifythe accuracy and efficiency of the method and characteristicsof the Erdelyi-Kober diffusion equation are analyzed basedon numerical results

Data Availability

(i) The programs used to support the findings of this studyhave been deposited in the GitHub repository (httpsgithubcomqinwuxuSpectralGFPDE ) (ii) No data were used tosupport this study

Disclosure

An earlier version of this workwas presented at the ldquo8th Inter-national Congress on Industrial and Applied Mathematics(ICIAM 2015)rdquo

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The first author is supported by the National Key RampD Pro-gram of China (No 2017YFC0209804) the National ScienceFoundation for Young Scientists of China (No 11701273) andYouth Foundation of Jiangsu Province (No BK20170628)

The second author is supported by the National Key RampDProgram of China (No 2017YFB0305601)

References

[1] B BaeumerDA BensonMMMeerschaert and SWWheat-craft ldquoSubordinated advection-dispersion equation for contam-inant transportrdquo Water Resources Research vol 37 no 6 pp1543ndash1550 2001

[2] E Barkai RMetzler and J Klafter ldquoFrom continuous time ran-dom walks to the fractional Fokker-Planck equationrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 61no 1 pp 132ndash138 2000

[3] A Blumen G Zumofen and J Klafter ldquoTransport aspects inanomalous diffusion Levy walksrdquo Physical Review A AtomicMolecular and Optical Physics vol 40 no 7 pp 3964ndash39731989

[4] J P Bouchaud and A Georges ldquoAnomalous diffusion in dis-ordered media statistical mechanisms models and physicalapplicationsrdquoPhysics Reports vol 195 no 4-5 pp 127ndash293 1990

[5] M Raberto E Scalas and F Mainardi ldquoWaiting-times andreturns in high-frequency financial data an empirical studyrdquoPhysica A Statistical Mechanics and its Applications vol 314 no1ndash4 pp 749ndash755 2002

[6] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations New York NYUSA Elsevier 2006

[7] S G Samko A A Kilbas and O I Marichev Fractional Inte-grals and Derivatives Theory and Applications Gordon andBreach Yverdon Switzerland 1993

[8] V Kiryakova ldquoA brief story about the operators of the general-ized fractional calculusrdquo Fractional CalculusampAppliedAnalysisAn International Journal forTheory and Applications vol 11 no2 pp 203ndash220 2008

[9] S L Kalla ldquoOn operators of fractional integration Irdquo Mathe-maticae Notae vol 22 pp 89ndash93 197071

[10] S L Kalla ldquoOn operators of fractional integration IIrdquo Mathe-maticae Notae vol 25 pp 29ndash35 1976

[11] V S Kiryakova Generalized Fractional Calculus and Applica-tions Long-man amp J Wiley Harlow New York NY USA 1994

[12] A Mura and F Mainardi ldquoA class of self-similar stochasticprocesses with stationary increments to model anomalousdiffusion in physicsrdquo Integral Transforms and Special Functionsvol 20 no 3-4 pp 185ndash198 2009

[13] O P Agrawal ldquoSome generalized fractional calculus operatorsand their applications in integral equationsrdquo Fractional Calculusand Applied Analysis An International Journal for Theory andApplications vol 15 no 4 pp 700ndash711 2012

[14] G Pagnini ldquoErdelyi-Kober fractional diffusionrdquo FractionalCalculus and Applied Analysis An International Journal forTheory and Applications vol 15 no 1 pp 117ndash127 2012

[15] A Mura and G Pagnini ldquoCharacterizations and simulations ofa class of stochastic processes to model anomalous diffusionrdquoJournal of Physics A Mathematical and General vol 41 no 28285003 22 pages 2008

[16] E K Lenzi L R Evangelista M K Lenzi H V Ribeiro and EC de Oliveira ldquoSolutions for a non-Markovian diffusion equa-tionrdquo Physics Letters A vol 374 no 41 pp 4193ndash4198 2010

[17] B Al-Saqabi and V S Kiryakova ldquoExplicit solutions of frac-tional integral and differential equations involving Erderyi-Kober operatorsrdquo Applied Mathematics and Computation vol95 no 1 pp 1ndash13 1998

14 International Journal of Differential Equations

[18] L A Hanna and Y F Luchko ldquoOperational calculus for theCaputo-type fractional Erdelyi-Kober derivative and its appli-cationsrdquo Integral Transforms and Special Functions vol 25 no5 pp 359ndash373 2014

[19] Y Xu Z He and Q Xu ldquoNumerical solutions of fractionaladvection-diffusion equations with a kind of new generalizedfractional derivativerdquo International Journal of Computer Math-ematics vol 91 no 3 pp 588ndash600 2014

[20] Y Xu Z He and O P Agrawal ldquoNumerical and analytical solu-tions of new generalized fractional diffusion equationrdquo Com-puters amp Mathematics with Applications vol 66 no 10 pp2019ndash2029 2013

[21] Y Xu and O P Agrawal ldquoNumerical solutions and analysisof diffusion for new generalized fractional Burgers equationrdquoFractional Calculus and Applied Analysis An International Jour-nal forTheory and Applications vol 16 no 3 pp 709ndash736 2013

[22] X Li and C Xu ldquoA space-time spectral method for the timefractional diffusion equationrdquo SIAM Journal on NumericalAnalysis vol 47 no 3 pp 2108ndash2131 2009

[23] Q Xu and J S Hesthaven ldquoStable multi-domain spectralpenalty methods for fractional partial differential equationsrdquoJournal of Computational Physics vol 257 pp 241ndash258 2014

[24] RMittal and S Pandit ldquoQuasilinearized Scale-3 Haar wavelets-based algorithm for numerical simulation of fractional dynam-ical systemsrdquo Engineering Computations vol 35 no 5 pp 1907ndash1931 2018

[25] C Li F Zeng and F Liu ldquoSpectral approximations to the frac-tional integral and derivativerdquo Fractional Calculus and AppliedAnalysis vol 15 no 3 pp 383ndash406 2012

[26] M Zheng F Liu I Turner and V Anh ldquoA novel high orderspace-time spectral method for the time fractional Fokker-Planck equationrdquo SIAM Journal on Scientific Computing vol 37no 2 pp A701ndashA724 2015

[27] F Zeng F Liu C Li K Burrage I Turner and V Anh ldquoACrank-Nicolson ADI spectral method for a two-dimensionalRiesz space fractional nonlinear reaction-diffusion equationrdquoSIAM Journal on Numerical Analysis vol 52 no 6 pp 2599ndash2622 2014

[28] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical ModellingSimulation andComputation for Engineering and EnvironmentalSystems vol 35 no 12 pp 5662ndash5672 2011

[29] M Zayernouri and G E Karniadakis ldquoExponentially accuratespectral and spectral element methods for fractional ODEsrdquoJournal of Computational Physics vol 257 pp 460ndash480 2014

[30] X Zhao and Z Zhang ldquoSuperconvergence points of fractionalspectral interpolationrdquo SIAM Journal on Scientific Computingvol 38 no 1 pp A598ndashA613 2016

[31] F Chen Q Xu and J S Hesthaven ldquoA multi-domain spectralmethod for time-fractional differential equationsrdquo Journal ofComputational Physics vol 293 pp 157ndash172 2015

[32] Z Mao and J Shen ldquoEfficient spectral-Galerkin methods forfractional partial differential equations with variable coeffi-cientsrdquo Journal of Computational Physics vol 307 pp 243ndash2612016

[33] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[34] J Hadamard ldquoEssai sur lrsquoetude des fonctions donnees par leurdeveloppement de Taylorrdquo Journal de Mathematiques Pures etAppliquees vol 4 pp 101ndash186 1892

[35] A Erdelyi andHKober ldquoSome remarks onHankel transformsrdquoQuarterly Journal of Mathematics vol 11 pp 212ndash221 1940

[36] I Dimovski ldquoOperational calculus for a class of differentialoperatorsrdquo Comptes Rendus De L Academie Bulgare Des Sci-ences vol 19 pp 1111ndash1114 1966

[37] S B Yakubovich and Y F LuchkoThe hypergeometric approachto integral transforms and convolutions vol 287 ofMathematicsand its Applications Kluwer Academic Publishers Dordrecht-Boston-London 1994

[38] Y Luchko ldquoOperational rules for a mixed operator of theErdelyi-Kober typerdquo Fractional Calculus and Applied Analysisvol 7 no 3 pp 339ndash364 2004

[39] G Szego Orthogonal polynomials American MathematicalSociety Providence 1992

[40] M Zayernouri and G E Karniadakis ldquoFractional Sturm-Liou-ville eigen-problems theory and numerical approximationrdquoJournal of Computational Physics vol 252 pp 495ndash517 2013

[41] S Chen J Shen and L-L Wang ldquoGeneralized Jacobi functionsand their applications to fractional differential equationsrdquoMathematics of Computation vol 85 no 300 pp 1603ndash16382016

[42] A A Kilbas ldquoHadamard-type integral equations and fractionalcalculus operatorsrdquo in Singular integral operators factorizationand applications vol 142 ofOperTheory Adv Appl pp 175ndash188Birkhauser Basel 2003

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International Journal of Differential Equations 5

minus1119868120572119909119869(1205730)119899 (119909) = 1Γ (120572) int119909

minus1

119869(1205730)119899 (119910)(119909 minus 119910)1minus120572 119889119910

= 119899 (1 + 119909)120572Γ (119899 + 120572 + 1)119869(120573minus120572120572)119899 (119909) (28)

Lemma 17 forall120572 gt 0 120572 notin N minus1 lt 119909 lt 1 120573 120574 isin R

1199091198631205721119869(0120574)119899 (119909) = 119899 (1 minus 119909)minus120572Γ (119899 + 1 minus 120572)119869(minus120572120574+120572)119899 (119909) (29)

minus1119863120572119909119869(1205730)119899 (119909) = 119899 (1 + 119909)minus120572Γ (119899 + 1 minus 120572)119869(120573+120572minus120572)119899 (119909) (30)

Proof First for the case 0 lt 120572 lt 1 formulas (29) and (30) canbe obtained by setting 120573 = minus120572 in formula (25) setting 120574 = minus120572in formula (26) and applying Riemann-Liouville fractionalderivative operator to both sides of themWe refer to [22] fordetailed discussion

For the case 120572 gt 1 120572 notin N the formulas cannot beobtained from Lemma 15 due to the constrains 120573 gt minus1 and120574 gt minus1 in Lemma 15 Here we prove the lemma inductively

For left Riemann-Liouville derivative from case 1 equal-ity (30) holds for lfloor120572rfloor = 0 Assume the equality holds forlfloor120572rfloor = 119896 minus 1 119896 isin N+ when lfloor120572rfloor = 119896 and let = 120572 minus 1from the assumption it holds that

minus1119863119909119869(1205730)119899 (119909) = 119899 (1 + 119909)minusΓ (119899 + 1 minus )119869(120573+minus)119899 (119909) (31)

We apply 119889119889119909 to both sides of (31) and then

minus1119863120572119909119869(1205730)119899 (119909) = minus1119863+1

119909 119869(1205730)119899 (119909)= 119899Γ (119899 + 1 minus ) (minus (1 + 119909)minusminus1 119869(120573+minus)119899 (119909)+ (1 + 119909)minus 119889119889119909119869(120573+minus)119899 (119909))= 119899 (1 + 119909)minusminus1Γ (119899 + 1 minus ) (minus119869(120573+minus)119899 (119909)+ (119899 + 120573 + 1) (1 + 119909)2 119869(120573++11minus)119899minus1 (119909))

(32)

Applying Lemma 14 we have

minus1119863120572119909119869(1205730)119899 (119909)= 119899 (1 + 119909)minusminus1Γ (119899 + 1 minus ) (minus (119899 minus 120572) 119869(120573++1minusminus1)119899 (119909))= 119899 (1 + 119909)minusminus1Γ (119899 + 1 minus ) ((119899 minus ) 119869(120573++1minusminus1)119899 (119909))= 119899 (1 + 119909)minus120572Γ (119899 + 1 minus 120572)119869(120573+120572minus120572)119899 (119909)

(33)

Equality (30) is proved Equality (29) can be proved similarly

Through the relationship between Caputo derivative andRiemann-Liouville derivative we immediately obtain thefractional derivative for Caputo derivative

Lemma 18 For 120572 gt 0 minus1 lt 119909 lt 1119888119909119863120572

1119869(0120574)119899 (119909) = 119899 (1 minus 119909)minus120572Γ (119899 + 1 minus 120572)119869(minus120572120574+120572)119899 (119909)minus 119898minus1sum

119895=0

120597119895119909119869(0120574)119899 (1) (1 minus 119909)119895minus120572(minus1)119895 Γ (1 + 119895 minus 120572)

119888minus1119863120572

119909119869(1205730)119899 (119909) = 119899 (1 + 119909)minus120572Γ (119899 + 1 minus 120572)119869(120573+120572minus120572)119899 (119909)minus 119898minus1sum

119895=0

120597119895119909119869(1205730)119899 (minus1) (1 + 119909)119895minus120572Γ (1 + 119895 minus 120572)

(34)

where119898 = lceil120572rceilGiven a function 119906(119909) isin 119867119898

120596 [minus1 1] and polynomialsspace P119873 the projection of 119906(119909) in space P119873 120587119873119906(119909)satisfies the following relation

(119906 minus 120587119873119906 V)120596 = 0 forallV isin P119873 (35)

According properties of space P119873 and Jacobi polynomials119906119873 can be expressed as

120587119873119906 (119909) = 119873sum119895=0

119895119869(120573120574)119895 (119909) (36)

where 119895 = (119906(119909) 119869(120573120574)(119909))120596119869(120573120574)(119909)2120596 120596 = 120596(119909) is theweight function

Then fractional derivative of 119906(119909) can be approximatedas

minus1119863120572119909119906 (119909) asymp 119873sum

119895=0

119895 ( minus1119863120572119909119869(120573120574)119895 (119909)) (37)

And we have the following lemma

Lemma 19 (see [41]) forall119906 isin 119867119898+1120596 [minus1 1] 0 le 119897 le 119898 119897 119898 isin

N 0 lt 120572 lt 1 119906119873 is a spectral approximation to 119906 in polynomialspace such that (119906 minus 120587119873119906 V)120596 = 0 forallV isin P119873 and then thereexists a constant 119862 such that10038171003817100381710038171003817 minus1119863119897+120572

119909 (119906 (119909) minus 120587119873119906 (119909))10038171003817100381710038171003817120596(120573+120572+119897120572+119897)le 119862119873119897minus119898 1003817100381710038171003817 minus1119863119898+120572

119909 1199061003817100381710038171003817120596(120573+120572+119898120572+119898) (38)

where 120596(120573120574) = (1 minus 119909)120573(1 + 119909)120574Combining Lemmas 18 and 19 and (37) approximation

method can be obtained immediately for Caputo derivativeand we have the following corollary

Corollary 20 forall119906 isin 119867119898+1120596 [minus1 1] 0 le 119897 le 119898 119897 119898 isin N 0 lt120572 lt 1 119906119873 is a spectral approximation to 119906 in polynomial space

6 International Journal of Differential Equations

such that (119906 minus 120587119873119906 V)120596 = 0 forallV isin P119873 and then there exists aconstant 119862 such that100381710038171003817100381710038171003817 119888

minus1119863119897+120572

119909 (119906 (119909) minus 120587119873119906 (119909))100381710038171003817100381710038171003817120596 le 119862119873119897minus119898 1003817100381710038171003817 minus1119863119898+120572119909 1199061003817100381710038171003817120596 (39)

Based on above lemmas fractional integral and derivativeof Jacobi polynomials in the standard interval [minus1 1] can beexpressed explicitly Fractional integrals and derivatives inshifted interval [119886 119887] can be obtained through proper variabletransforms

For any 119910 isin [119886 119887] we assume 119869120573120574119899lowast(119910) is defined in [119886 119887]Let ℎ = 119887 minus 119886 119909 = minus1 + 2((119910 minus 119886)ℎ) then 119909 isin [minus1 1]Substituting 119909 = minus1 + 2((119910 minus 119886)(119887 minus 119886)) into equation (27)-(28) and (29)-(30) we obtained

119910119868120572119887119869(0120574)119899lowast (119910) = (ℎ2)120572

1199091198681205721119869(0120574)119899 (119909) 119886119868120572119910119869(1205730)119899lowast (119910) = (ℎ2)

120572

minus1119868120572119909119869(1205730)119899 (119909) 119888119910119863120572

119887119869(0120574)119899lowast (119910) = (2ℎ)

120572119888119909119863120572

1119869(0120574)119899 (119909) 119888119886119863120572

119910119869(1205730)119899lowast (119910) = (2ℎ)120572

119888minus1119863120572

119909119869(1205730)119899 (119909)

(40)

32 Spectral Approximation to Generalized Fractional Opera-tors We assume 119911(119909) and 119908(119909) are positive monotone func-tions and 119911 119908 isin 119862[119886 119887] Obviously 119911(119909) and119908(119909) are invert-ible The following lemmas can be derived for generalizedfractional operators

Lemma 21 Let 119892(120577) = 119908(119911minus1(120577))119891(119911minus1(120577)) 0 lt 120572 lt 1 thenthe generalized fractional integral operator is equivalent to thefollowing classical fractional integral

(119868120572119886+[119911119908]119891) (119909) = [119908 (119909)]minus1 1205770119868120572120577119892 (119911 (119909)) (41)

Here 1205770 = 119911(119886)Proof From the definition of generalized fractional integralwe have

(119868120572119886+[119911119908]119891) (119909) = [119908 (119909)]minus1Γ (120572) int119909

119886

119908 (119905) 1199111015840 (119905) 119891 (119905)[119911 (119909) minus 119911 (119905)]1minus120572 119889119905 (42)

Since 120577 = 119911(119905) is positive monotone function then 120577 = 119911(119905) isinvertible and we have 119905 = 119911minus1(120577)(119868120572119886+[119911119908]119891) (119909) = [119908 (119909)]minus1Γ (120572)sdot int119911(119909)

119911(119886)

119908(119911minus1 (120577)) 1199111015840 (119911minus1 (120577)) 119891 (119911minus1 (120577))[119911 (119909) minus 120577]1minus120572 119889119911minus1 (120577)

= [119908 (119909)]minus1Γ (120572) int119911(119909)

119911(119886)

119908(119911minus1 (120577)) 119891 (119911minus1 (120577))[119911 (119909119896) minus 120577]1minus120572 119889120577

(43)

Let 119892(120577) = 119908(119911minus1(120577))119891(119911minus1(120577)) then the generalized frac-tional integral of 119891(119909) is converted to classical fractionalintegral of 119892(120577) in the following form

(119868120572119886+[119911119908]119891) (119909) = [119908 (119909)]minus1 1205770119868120572120577119892 (119911 (119909)) (44)

Lemma is proved

Lemma 22 Let 119892(120577) = 119908(119911minus1(120577))119891(119911minus1(120577)) 0 lt 120572 lt 1 thenthe generalized fractional derivative of order 120572 is equivalent tothe following classical fractional derivative

(119863120572119886+[1199111199082]119891) (119909) = [119908 (119909)]minus1 119888

1205770119863120572

120577119892 (119911 (119909)) (45)

Here 1205770 = 119911(119886)Proof Similar to the proof of Lemma 21 we have

(119863120572119886+[1199111199082]119891) (119909) = [119908 (119909)]minus1Γ (1 minus 120572)sdot int119911(119909)

119911(119886)

(120597120597120577) (119908 (119911minus1 (120577)) 119891 (119911minus1 (120577))) (120597120577120597119905)[119911 (119909) minus 120577]120572 119889119911minus1 (120577)

= [119908 (119909)]minus1Γ (1 minus 120572) int119911(119909)

119911(119886)

(120597120597120577) (119908 (119911minus1 (120577)) 119891 (119911minus1 (120577)))[119911 (119909) minus 120577]120572 119889120577

(46)

Let 119892(120577) = 119908(119911minus1(120577))119891(119911minus1(120577)) then the generalized frac-tional derivative of 119891(119909) is expressed through the classicalfractional derivative of 119892(120577)

(119863120572119886+[1199111199082]119891) (119909) = [119908 (119909)]minus1 119888

1205770119863120572

120577119892 (119911 (119909)) (47)

Lemma is proved

Remark 23 These two lemmas establish important relationbetween classical and generalized fractional operators Withthese lemmas generalized fractional differential equationscan be solved via classical fractional differential equationsand vice versa Which kind of transform should be takendepends on characters of the problem to be solved

In order to design a high order numerical approximationof the generalized fractional operator we define a scaled spaceP119899

[119908119911] such that

P119899[119908119911] = V (119909) = [119908 (119909)]minus1 119892 (119911 (119909)) 119892 (119909) isin P

119899 119909isin Ω = [119886 119887] (48)

whereP119899 is a polynomial space of up to order 119899Define a inner product and norm for space P119899

[119908119911] suchthat

(119906 (119909) V (119909))120596 = int119887

119886119906 (119909) V (119909) 120596 (119909) 119889119909

and V120596 = radic(V V)120596(49)

International Journal of Differential Equations 7

Define projection 119876119873 into space P119873119908119911 such that for any

function 119906(119909)(119906 minus 119876119873119906 V)120596 = 0 forallV isin P

119873119908119911 (50)

SupposeΦ119895(119909) 119895 = 0 119873 are a set of orthogonal basisfunctions in spaceP119873

119911119908 satisfying

(Φ119894 (119909) Φ119895 (119909))120596 = 1 119894 = 1198950 119894 = 119895 (51)

Let119873 997888rarr infin then Φ119895(119909) 119895 = 0 1 119873 form a 1198712120596(Ω)space and for any 119906(119909) isin 1198712120596(Ω) the projection 119876119873119906(119909) canbe written as

119876119873119906 (119909) = 119873sum119895=0

119895Φ119895 (119909) (52)

Here 119895 (119895 = 0 119873) are expansion coefficients such that

119895 = (119906 (119909) Φ119895 (119909))12059610038171003817100381710038171003817Φ119895 (119909)100381710038171003817100381710038172120596 (53)

The weight function 120596(119909) plays an important role in thecomputational process and analysis of the method Herewe choose a proper weight function to use properties oforthogonal polynomials and make the computation moreefficient Let 119901119895(119911(119909)) = 119908(119909)Φ119895(119909) and note that

(119876119873119906 (119909) Φ119894 (119909))120596 = int119887

119886( 119873sum

119895=0

119895119908minus1 (119909) 119901119895 (119911 (119909)))sdot 119908minus1 (119909) 119901119894 (119911 (119909)) 120596 (119909) 119889119909

(54)

Take 120596(119909) = 1199082(119909)1199111015840(119909) then(119876119873119906 (119909) Φ119894 (119909))120596 = int119911(119887)

119911(119886)( 119873sum

119895=0

119895119901119895 (119911))119901119894 (119911) 119889119911 (55)

Since 119901119895(119911) is polynomial of order 119895 the computation can becarried out easily through properties of orthogonal polyno-mials

Suppose 119875119895lowast(119909) is shifted Legendre polynomial definedon [119911(119886) 119911(119887)] Then given any function 119906(119909) isin 1198712120596 withweight 120596(119909) = 1199082(119909)1199111015840(119909) we have

119876119873119906 (119909) = [119908 (119909)]minus1 119873sum119895=0

119895119875119895lowast (119911 (119909)) (56)

Recalling Lemmas 21 and 22 the generalized fractionalintegral and derivative can be obtained in the form

(119868120572119886+[119911119908]119876119873119906) (119909)= [119908 (119909)]minus1 119873sum

119895=0

119895 ( 1205770119868120572120577119875119895lowast (120577)1003816100381610038161003816120577=119911(119909)) (57)

(119863120572119886+[1199111199082]119876119873119906) (119909)= [119908 (119909)]minus1 119873sum

119895=0

119895 ( 1198881205770119863120572

120577119875119895lowast (120577)1003816100381610038161003816120577=119911(119909)) (58)

From Lemma 19 the following corollary can be obtainedimmediately

Corollary 24 120577(119909) = 119911(119909) is a monotone increasing function119908(119909) gt 0 0 lt 120572 lt 1 119876119873 is a projection into space P119873119908119911119906(119911minus1(120577)) isin 119862119898(Ω) 119898 isin N then there exists a constant 119862120572

such that 10038171003817100381710038171003817119863120572119886+[1199111199082] (119906 (119909) minus 119876119873119906 (119909))100381710038171003817100381710038171205962le 1198621205721198731minus119898 10038171003817100381710038171003817 1205770119863119898+120572

119911119906 (120577minus1 (119911))100381710038171003817100381710038171205961

(59)

Here the weight function 1205961(120577) = (120577 minus 119886)minus120572(119887 minus 120577)120572 1205962(119909) =1205961(119911(119909))1199082(119909)1199111015840(119909)Remark 25 Unlike the convergence theory in classical poly-nomial space the convergence order in space P119873

119908119911 dependson regularity of the function 119906(119909) with respect to 119911(119909) Wewill illustrate this through numerical examples

Most of the time it is more convenient to consider theproblems in nodal form We assume the given interpolationpoints are 120577119895 = 119911(119909119895) (119895 = 0 1 119873) then the Lagrangebasis functions L119895(119909) (119895 = 0 1 119873) can be defined asfollows

L119894 (120577) = prod119895=0119873119895 =119894

(120577 minus 120577119895)(120577119894 minus 120577119895) (60)

The function 119906(119909) can be expressed using both Jacobipolynomials and Lagrange polynomials The following rela-tion is derived

119906 (119909) = [119908 (119909)]minus1 119873sum119895=0

119892119895119875119895lowast (119911 (119909))

= [119908 (119909)]minus1 119873sum119895=0

119908119895119906119895L119895 (119911 (119909)) (61)

Considering the equivalence between Legendre basis andLagrange basis the following equality holds

L119895 (119911) = 119873sum119896=0

119897119896119895119875119896lowast (119911) 119896 = 0 1 119873 (62)

where 119897119894119895 = (119871119895(119911) 119875119894lowast(119911))119875119894lowast(119911)21198712 Then the nodal form expansion of 119906(119909) is obtained

119906 (119909) = [119908 (119909)]minus1 119873sum119895=0

119873sum119896=0

119908119895119906119895119897119896119895119875119896lowast (119911 (119909)) (63)

8 International Journal of Differential Equations

From (57) and (58) the corresponding nodal form of gener-alized fractional integral and derivative are obtained

(119868120572119886+[119911119908]119906) (119909)= [119908 (119909)]minus1 119873sum

119895=0

119873sum119896=0

119908119895119906119895119897119896119895 ( 1205770119868120572120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119909)) (64)

(119863120572119886+[1199111199082]119906) (119909)= [119908 (119909)]minus1 119873sum

119895=0

119873sum119896=0

119908119895119906119895119897119896119895 ( 1198881205770119863120572

120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119909)) (65)

Example 26 Now we give an example to show the effec-tiveness and accuracy of the method Assuming 119911(119909) =radic1199093 119908(119909) = 1 we consider generalized fractional derivativeof 119910(119909) on the interval [0 1] with the following form

119910 (119909) = 5sum119895=1

(minus1)119895+1 1199093119895(2119895) (66)

The exact generalized fractional derivative of 119910(119909) is119863120572

0+[1199111199082]119910 (119909) = 5sum119895=1

(minus1)119895+1 1199093119895minus31205722Γ (2119895 + 1 minus 120572) (67)

Numerical approximation to generalized fractionalderivative of 119910(119909) can be obtained using (58) We considerthe maximum absolute error

119890 = max119909isin[01]

100381610038161003816100381610038161198631205720+[1199111199082]119910 (119909) minus 119863120572

0+[1199111199082]119876119873119910 (119909)10038161003816100381610038161003816 (68)

of the numerical derivative Results for 120572 = 02 05 08 areshown in Figure 1

For this example it is easy to check that 119910 isin P10119908119911 From

theory of spectral approximation the error would decreaseexponentially when 119873 lt 10 and the numerical fractionalderivative would be exact when119873 ge 10 Our numerical resultcoincides with the theory exactly

Example 27 In this example we test the spectral approxi-mation of Hadamard integral Considering Hadamard-typefractional integral of 119910(119909) = sin(120587119909) on the interval [1 2]when 119908(119909) = 1 and 120572 = 1 the Hadamard integral of 119910(119909) isSine integral function Si(120587119909) minus Si(120587) for more general 119908(119909)and 120572 the exact Hadamard-type integral of 119910(119909) is unknownHere we consider several pairs of 119908(119909) and 120572 NumericalHadamard-type integral of 119910(119909) would be computed using(57) with 119911(119909) = log(119909) To evaluate the approximationaccuracy for the case 119908(119909) = 1 120572 = 1 the exact Hada-mard fractional integral is computed usingMATLAB built-infunction 119904119894119899119894119899119905 for other cases ldquoexactrdquo Hadamard fractionalintegral is computed using (57) with large 119873 (eg 119873 =50) which is treated as reference solution The results fornumerical Hadamard integral and approximation error areshown in Figures 2 and 3

For this example 119906 is not in the spaceP119899119908119911 for any 119899 Max-

imum error of approximated fractional integral convergesexponentially until reaching machine accuracy

10minus16

10minus14

10minus12

10minus10

10minus8

10minus6

10minus4

10minus2

max

imum

erro

r

5 6 7 8 94 11 1210

N

alpha=02alpha=05alpha=08

Figure 1 Error of numerical approximation to generalized frac-tional derivative of 119910(119909)

11 12 13 14 15 16 17 18 19 21x

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Had

amar

d fra

ctio

nal i

nteg

ral

w(x)=1=1w(x)=1=06

w(x)=x=1w(x)=x=06

Figure 2 Hadamard fractional integral for different weight 119908(119909)and 120572

33 Fractional IntegralDifferential Matrices Suppose 119906(119909) isinP119873

119911119908 119909119894 (119894 = 0 1 119873) are the interpolation points and119910119895 (119895 = 0 1 119873) are the collocation points we define thefollowing symbols

U = [119906 (1199090) 1199061199091 119906 (119909119873)]119879 (69)

U(120572) = [119863120572

119886+[119911119908]119906 (1199100) 119863120572119886+[119911119908]119906 (1199101)

119863120572119886+[119911119908]119906 (119910119873)]119879 120572 gt 0 (70)

International Journal of Differential Equations 9

10minus16

10minus14

10minus12

10minus10

10minus8

10minus6

10minus4

max

imum

erro

r

w(x)=1=1w(x)=1=06

w(x)=x=1w(x)=x=06

10 12 14 16 18 20 228x

Figure 3 Approximation error of Hadamard fractional integral fordifferent weight 119908 and 120572

U(120572) = [119868minus120572119886+[119911119908]119906 (1199100) 119868minus120572119886+[119911119908]119906 (1199101) 119868minus120572119886+[119911119908]119906 (119910119873)]119879 120572 lt 0 (71)

We define a generalized fractional differentialintegralmatrixM120572 such that

M120572119894119895 = 119873sum

119896=0

119908(119909119895) 119897119896119895119908 (119910119894) ( 1198881205770119863120572

120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119910119894))

for 120572 gt 0(72)

M120572119894119895 = 119873sum

119896=0

119908(119909119895) 119897119896119895119908 (119910119894) ( 1205770119868120572120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119910119894))

for 120572 lt 0(73)

In order to compute the fractional matrix M120572 moreefficiently we define a few more matrices L and L120572 are(119873 + 1) times (119873 + 1) matrices such thatL119894119895 = L119895(119911(119910119894))L120572

119894119895 =1205770119863120572

120577L119895(119911)|119911=119911(119910119894) 119894 119895 = 0 1 119873 V is defined based on

values of Legendre polynomials at interpolation points 119911(119909119894)such that

V119894119895 = 119875119895lowast (119911 (119909119894)) 119894 119895 = 0 1 119873 (74)

D120572 is defined as fractional derivativeintegral of Legendrepolynomials at collocation points 120577119894 = 119911(119910119894)

D120572119894119895 = 119888

1205770119863120572

120577119875119895lowast (120577119894) 120572 gt 0 (75)

D120572119894119895 = 1205770

119868120572120577119875119895lowast (120577119894) 120572 lt 0 (76)

W119897 andW119903 are the weight matrices defined as follows

W119903119894119895 =

119908(119909119895) 119894 = 1198950 otherwise (77)

W119897119894119895 =

1119908(119910119895) 119894 = 119895 gt 10 otherwise (78)

Theorem 28 For 119906(119909) isin P119873119911119908 vectors defined in (69)ndash(71)

and matrices defined in (72)ndash(78) the following relation holds

U(120572) = M

120572U = (W119897D120572V

minus1W

119903)U (79)

Proof From the definition (69)ndash(71) (72)ndash(78) and (63)-(64) it is easy to obtain

U(120572) = M

120572U (80)

Next we prove thatM120572 = W119897D120572Vminus1W119903From the definition of the interpolation function L119895(119911)

119875119895lowast (119911) = 119873sum119894=0

119875119895lowast (119911 (119909119894)) L119894 (119911) (81)

Suppose 997888rarr119875(119911) = (1198750lowast(119911) 1198751lowast(119911) 119875119873lowast(119911)) 997888rarrL (119911) =(L0(119911) L1(119911) L119873(119911)) for 120572 gt 0 we have997888rarr119875 (119911) = 997888rarrL (119911)V

1205770119863120572

120577

997888rarr119875 (119911) = ( 1205770119863120572

120577

997888rarrL (119911))V (82)

Evaluating the matrices multiplication for each elementthe following relation is derived

(W119897D120572V

minus1W119903)119894119895 = (W119897L

120572W119903)119894119895

= 119908 (119909119895)119908 (119910119894) L(120572)119895 (119911 (119910119894))(83)

= 119873sum119896=0

119908(119909119895) 119897119896119895119908 (119910119894) ( 1205770119863120572

120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119910119894)) = M

120572119894119895 (84)

The case 120572 lt 0 can be proved similarly

Remark 29 Collocation points 119910119895 and the interpolationpoints 119909119894 are not necessary the same To obtain a goodapproximation interpolation points ofGauss-type are usuallyused At the same time collocation points should be chosenproperly to guarantee stability properties of the methodIn the following numerical examples for computation andstability aim both interpolation and collocation points arechosen based on Gauss-type points with respect to 119911(119909)

10 International Journal of Differential Equations

4 Collocation Methods for FractionalDifferential and Integral Equations

41 Fractional Ordinary Differential Equations In this sub-section we consider collocation method for the generalizedfractional ordinary differential equation of the form

1198631205720+[1199111199082]119906 (119909) = 120582 (119909) 119906 (119909) + 119891 (119909)

119909 isin Ω = (0 119887] (85)

119906 (0) = 1199060 (86)

Here 0 lt 120572 lt 1 119908(119909) gt 0 119911(119909) gt 0 and 119911(119909) is a monotonefunction inΩ

We assume 119906119873(119909) isin P119873119911119908 is a numerical solution of the

equation 119909119894(119894 = 0 1 119873) are the chosen interpolationpoints 119906119895119873 = 119906119873(119909119895) The following discretized equation isobtained

[119908 (119909)]minus1 119873sum119895=0

119873sum119896=0

119908(119909119895) 119906119895119873119897119896119895 ( 1198881205770119863120572

120577119875119896lowast (119911 (119909)))

= 120582 (119909) 119906119873 (119909) + 119891 (119909) (87)

Let (87) hold on collocation points 119910119895 (119895 = 1 119873) thematrix form is obtained

(M120572 minus Λ)U = 119865 (88)

Here Λ is a diagonal matrix with Λ 119894119894 = 120582(119910119894) 119865 = (1199060 119891(1199101) 119891(119910119873))119879Considering the initial condition we set S = M120572 minus Λ

with the first row replaced by (1 0 0) 1198651015840 = 119865with its firstelement replaced by 1199060 Then the solution U is obtained bysolving the matrix equation SU = 1198651015840

Example 30 Consider the following example

1198631205720+[1199111199082]119906 (119909) = (1 + 119909) 119906 (119909)

+ Γ (15119903 + 25) 119909119903minus(23)120572Γ (15119903 + 25 minus 120572)minus 119909119903 (1 + 119909) 119909 isin (0 1]

(89)

119906 (0) = 0 (90)

Here 119911(119909) = 11990923119908(119909) = 119909 119903 is an arbitrary positive numberThe exact solution of the ordinary differential equation is119906(119909) = 119909119903 Maximum absolute errors of numerical solutions

for 119903 = 6 7 and 120572 = 03 06 09 are shown in Figures 4 and 5When 119911(119909) = 11990923 119908(119909) = 119909 the scaled polynomial

spaceP119873119911119908 becomes

P119873119911119908 = 119904119901119886119899 11990921198993minus1 119899 = 1 2 119873 (91)

For 119903 = 7 the error converges exponentially and reachesmachine accuracy at 119873 = 12 It is faster than any finitedifference method while for 119903 = 6 solution convergesalgebraically as119873 increases however it still reaches machineaccuracy at 119873 = 23 The major reason for this is that 119906(119909) =1199097 isin P12

119911119908 and 119906(119909) = 1199096 notin P119873119911119908 for any119873 isin N

6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

N

10minus15

10minus10

10minus5

100

max

erro

r

=03=06=09

Figure 4 Log-log plot of the maximum error for 119903 = 6

=03=06=09

10minus15

10minus10

10minus5

100

max

erro

r

15 20 25 30 3510N

Figure 5 Semilog plot of the maximum error for 119903 = 7

Remark 31 Since space P119873119908119911 is transformed from classical

polynomial space with respect to 119911(119909) the convergenceof spectral collocation method for ODEs with generalizedfractional operators depends not only on the smoothness ofthe solution itself but also on the scale function 119911(119905)42 Hadamard-Type Integral Equations We consider thefollowing Hadamard-type boundary value problem

1Γ (120572) int119909

119886( 119904119909)

120583 (log 119909119904 )120572minus1 119891 (119904) 119889119904119904 = 119892 (119909)

119909 isin Ω = (119886 119887] (92)

International Journal of Differential Equations 11

In [42] Kilbas discussed the existence of the solution of(92) Explicit formulas for the solution 119891(119905) were establishedin the following theorem

Theorem 32 (see [42]) If 119909120583119892(119909) isin 119860119862[119886 119887] then theHadamard-type integral equation (92) with 0 lt 120572 lt 1 issolvable in 119883120583(119886 119887) and its solution may be represented in theform

119891 (119909) = 119909minus120583Γ (1 minus 120572) [119886120583119892 (119886) (log 119909119886)minus120572

+ int119909

119886(log 119909119904 )

minus120572 (119904120583119892 (119904))1015840 119889119904] (93)

Here 119860119862[119886 119887] is the set of absolutely continuous functions on[119886 119887] and119883120583(119886 119887) is space of those Lebesgue measurable func-tions 119891 on [119886 119887] for which 119909120583minus1119891(119909) is absolutely integrable

Solution of the Hadamard-type integral equation isexactly the same as generalized fractional derivative ofRiemann-Liouville type with 119911(119904) = log(119904) 119908(119904) = 119904120583 Therelationship between (93) and Caputo type generalized frac-tional derivative is

119891 (119909) = 119886120583119892 (119886) 119909minus120583Γ (1 minus 120572) (log 119909119886)minus120572 + 119863120572

119886+[1199111199082]119892 (119909) (94)

Suppose 119909119895 119895 = 0 1 119873 are interpolation points thediscretized solution of equation (92) is

119891119873 (119909) = 119886120583119892 (119886) 119909minus120583Γ (1 minus 120572) (log 119909119886)minus120572

+ 119909minus120583 119873sum119895=0

119873sum119896=0

119909120583119895119892 (119909119895) 119897119896119895( 119888120577119886119863120572

120577119875119896lowast (log (119909))

(95)

Rewriting (95) in matrix form we have

119865 = 119866119886 +M120572119866 (96)

where M120572 is generalized fractional differential matrix119866 = (119892(1199090) 119892(1199091) 119892(119909119873))119879 119865 = (119891119873(1199090) 119891119873(1199091) 119891119873(119909119873))119879 119866119886 is a vector about the initial condition of theintegral equation defined by

119866119886 = 119886120583119892 (119886)Γ (1 minus 120572) (119909minus1205830 (log 1199090119886 )minus120572 119909minus1205831 (log 1199091119886 )

minus120572 119909minus120583119873 (log 119909119873119886 )minus120572)119879

(97)

Example 33 Assume 119892(119909) = sin(119909 minus 1) 120583 = 13 Ω = [110] Solutions of (92) for 120572 = 03 06 09 are shown inFigure 6

2 4 6 8 10 120X

minus8

minus6

minus4

minus2

0

2

4

6

8

f(X)

=03=06=09

Figure 6 Solution of (92)

43 Erdelyi-Kober Fractional Diffusion Equation In this sub-section we consider the following Erdelyi-Kober fractionaldiffusion equation [12]

119906 (119909 119905) = 1199060 (119909) + 1Γ (120573)sdot int119905

0

120572120573119904120572120573minus1 (119905120572120573 minus 119904120572120573)120573minus1 12059721205971199092 119906 (119909 119904) d119904(98)

Erdelyi-Kober fractional diffusion equation which is alsocalled stretched time-fractional diffusion equation is themaster equation of a kind of generalized grey Brownianmotion (ggBm) The ggBm is a parametric class of stochasticprocesses that provides models for both fast and slow anoma-lous diffusion This class is made up of self-similar processes119861120572120573(119905)with stationary increments and it depends on two realparameters 0 lt 120572 le 2 and 0 lt 120573 le 1 It includes thefractional Brownian motion when 0 lt 120572 le 2 and 120573 = 1 thetime-fractional diffusion stochastic processes when 0 lt 120572 =120573 lt 1 and the standard Brownian motion when 120572 = 120573 = 1About the relationship between stochastic process119861120572120573(119905) andstretched time-fractional diffusion equation the followingproposition is presented in [12]

Proposition 34 The marginal probability density function119891120572120573(119909 119905) of the process 119861120572120573(119905) 119905 ge 0 is the fundamentalsolution of the stretched time-fractional diffusion equation

119906 (119909 119905) = 1199060 (119909) + 1Γ (120573)sdot int119905

0

120572120573119904120572120573minus1 (119905120572120573 minus 119904120572120573)120573minus1 12059721205971199092 119906 (119909 119904) d119904(99)

12 International Journal of Differential Equations

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 7 Standard Brownian motion 120572 = 1 120573 = 1

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 8 Time-fractional diffusion with 120572 = 06 120573 = 06

Recalling the definition of generalized fractional integraland setting 119911(119905) = 119905120572120573 119908(119905) = 1 the equation can be rewrit-ten as

119906 (119909 119905) = 1199060 (119909) + 1198681205730+[119911119908]119906119909119909 (119909 119905) (100)

We use collocation method for both space and timediscretization We choose Legendre-Gauss-Lobatto (L-G-L)points 119909119894 (119894 = 0 1 119872) as the space collocation points andchoose 119905119895 (119895 = 0 1 119873) such that 119911(119905119895) are L-G-L pointsas the time collocation points

Define space collocation matrix M2 such that M2119894119895 =(11988921198891199092)L119895(119909119894) and generalized fractional integral matrix

M120573 Matrix M120573 is computed through Theorem 28 and thespace-time collocation matrices are obtained using Kro-necker product 1015840otimes1015840 Suppose A and B are space-timecollocation matrices with dimension (119872 + 1)(119873 + 1) times (119872 +1)(119873 + 1) for the second order derivative and fractionalintegral of order 120573 separately Then

A = M2 otimesI119873+1

B = I119872+1 otimesM120573 (101)

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 9 Fractional Brownian motion with 120572 = 15 120573 = 1

Suppose 119906119873(119909 119905) is the numerical solution of (98) defin-ing 119906119873119894119895 = 119906119873(119909119894 119905119895) solution vectorU and initial vector 1198800such that

U = [11990611987300 11990611987301 1199061198730119873 11990611987310 1199061198731119873 1199061198731198720 119906119873119872119873]119879

1198800 = [1199060 (1199090) 1199060 (1199090) 1199060 (1199091) 1199060 (1199091) 1199060 (119909119872) 1199060 (119909119872)]119879

(102)

where in the definition of 1198800 each 1199060(119909119895) is repeated 119873 + 1times

The matrix form discretized equation of (98) is obtainedas

U = 1198800 +BAU (103)

In the discretized equation initial condition is explicitlyinvolved After boundary condition added properly numeri-cal solution can be obtained by solving the matrix equation

Example 35 Assume 119909 isin Ω = (minus1 1) 119905 isin (0 05] 1199060(119909) =119890minus101199092 minus 119890minus10 119906(sdot 119905)|120597Ω = 0 Numerical solutions with 119872 =119873 = 50 are shown in Figures 7ndash10

Erdelyi-Kober diffusion equation characterizes the mar-ginal density function of the process 119861120572120573(119905) 119905 ge 0 When120572 = 120573 = 1 we recover the standard diffusion equationWhen0 lt 120572 = 120573 lt 1 we get the time-fractional diffusion equationof order 120573 When 120573 = 1 and 0 lt 120572 lt 2 we have the equationof the fractional Brownian motion marginal density

As shown in Figures 7 and 8 when 1 lt 120572 lt 2 the diffu-sion is fast and the increments exhibit long-range depen-dence when 0 lt 120572 lt 1 the diffusion is slow and theincrements form a stationary process which does not exhibitlong-range dependenceThe results coincide with theoreticalanalysis in [12 14]

International Journal of Differential Equations 13

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 10 Fractional Brownian motion with 120572 = 05 120573 = 1

5 Conclusion

In this paper we propose a spectral collocation method fordifferential and integral equations with generalized fractionaloperators To deal with the difficulty in designing spectralapproximation scheme due to complexity of integral kerneland weight a variable transform technique is applied to thegeneralized fractional operator and a spectral approximationmethod is proposed for the generalized fractional operatorOperational matrices for generalized fractional operatorsare derived Spectral collocation methods are designed forfractional ordinary differential equations Hadamard-typeintegral equations and Erdelyi-Kober diffusion equationsseparately Numerical experiments are carried out to verifythe accuracy and efficiency of the method and characteristicsof the Erdelyi-Kober diffusion equation are analyzed basedon numerical results

Data Availability

(i) The programs used to support the findings of this studyhave been deposited in the GitHub repository (httpsgithubcomqinwuxuSpectralGFPDE ) (ii) No data were used tosupport this study

Disclosure

An earlier version of this workwas presented at the ldquo8th Inter-national Congress on Industrial and Applied Mathematics(ICIAM 2015)rdquo

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The first author is supported by the National Key RampD Pro-gram of China (No 2017YFC0209804) the National ScienceFoundation for Young Scientists of China (No 11701273) andYouth Foundation of Jiangsu Province (No BK20170628)

The second author is supported by the National Key RampDProgram of China (No 2017YFB0305601)

References

[1] B BaeumerDA BensonMMMeerschaert and SWWheat-craft ldquoSubordinated advection-dispersion equation for contam-inant transportrdquo Water Resources Research vol 37 no 6 pp1543ndash1550 2001

[2] E Barkai RMetzler and J Klafter ldquoFrom continuous time ran-dom walks to the fractional Fokker-Planck equationrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 61no 1 pp 132ndash138 2000

[3] A Blumen G Zumofen and J Klafter ldquoTransport aspects inanomalous diffusion Levy walksrdquo Physical Review A AtomicMolecular and Optical Physics vol 40 no 7 pp 3964ndash39731989

[4] J P Bouchaud and A Georges ldquoAnomalous diffusion in dis-ordered media statistical mechanisms models and physicalapplicationsrdquoPhysics Reports vol 195 no 4-5 pp 127ndash293 1990

[5] M Raberto E Scalas and F Mainardi ldquoWaiting-times andreturns in high-frequency financial data an empirical studyrdquoPhysica A Statistical Mechanics and its Applications vol 314 no1ndash4 pp 749ndash755 2002

[6] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations New York NYUSA Elsevier 2006

[7] S G Samko A A Kilbas and O I Marichev Fractional Inte-grals and Derivatives Theory and Applications Gordon andBreach Yverdon Switzerland 1993

[8] V Kiryakova ldquoA brief story about the operators of the general-ized fractional calculusrdquo Fractional CalculusampAppliedAnalysisAn International Journal forTheory and Applications vol 11 no2 pp 203ndash220 2008

[9] S L Kalla ldquoOn operators of fractional integration Irdquo Mathe-maticae Notae vol 22 pp 89ndash93 197071

[10] S L Kalla ldquoOn operators of fractional integration IIrdquo Mathe-maticae Notae vol 25 pp 29ndash35 1976

[11] V S Kiryakova Generalized Fractional Calculus and Applica-tions Long-man amp J Wiley Harlow New York NY USA 1994

[12] A Mura and F Mainardi ldquoA class of self-similar stochasticprocesses with stationary increments to model anomalousdiffusion in physicsrdquo Integral Transforms and Special Functionsvol 20 no 3-4 pp 185ndash198 2009

[13] O P Agrawal ldquoSome generalized fractional calculus operatorsand their applications in integral equationsrdquo Fractional Calculusand Applied Analysis An International Journal for Theory andApplications vol 15 no 4 pp 700ndash711 2012

[14] G Pagnini ldquoErdelyi-Kober fractional diffusionrdquo FractionalCalculus and Applied Analysis An International Journal forTheory and Applications vol 15 no 1 pp 117ndash127 2012

[15] A Mura and G Pagnini ldquoCharacterizations and simulations ofa class of stochastic processes to model anomalous diffusionrdquoJournal of Physics A Mathematical and General vol 41 no 28285003 22 pages 2008

[16] E K Lenzi L R Evangelista M K Lenzi H V Ribeiro and EC de Oliveira ldquoSolutions for a non-Markovian diffusion equa-tionrdquo Physics Letters A vol 374 no 41 pp 4193ndash4198 2010

[17] B Al-Saqabi and V S Kiryakova ldquoExplicit solutions of frac-tional integral and differential equations involving Erderyi-Kober operatorsrdquo Applied Mathematics and Computation vol95 no 1 pp 1ndash13 1998

14 International Journal of Differential Equations

[18] L A Hanna and Y F Luchko ldquoOperational calculus for theCaputo-type fractional Erdelyi-Kober derivative and its appli-cationsrdquo Integral Transforms and Special Functions vol 25 no5 pp 359ndash373 2014

[19] Y Xu Z He and Q Xu ldquoNumerical solutions of fractionaladvection-diffusion equations with a kind of new generalizedfractional derivativerdquo International Journal of Computer Math-ematics vol 91 no 3 pp 588ndash600 2014

[20] Y Xu Z He and O P Agrawal ldquoNumerical and analytical solu-tions of new generalized fractional diffusion equationrdquo Com-puters amp Mathematics with Applications vol 66 no 10 pp2019ndash2029 2013

[21] Y Xu and O P Agrawal ldquoNumerical solutions and analysisof diffusion for new generalized fractional Burgers equationrdquoFractional Calculus and Applied Analysis An International Jour-nal forTheory and Applications vol 16 no 3 pp 709ndash736 2013

[22] X Li and C Xu ldquoA space-time spectral method for the timefractional diffusion equationrdquo SIAM Journal on NumericalAnalysis vol 47 no 3 pp 2108ndash2131 2009

[23] Q Xu and J S Hesthaven ldquoStable multi-domain spectralpenalty methods for fractional partial differential equationsrdquoJournal of Computational Physics vol 257 pp 241ndash258 2014

[24] RMittal and S Pandit ldquoQuasilinearized Scale-3 Haar wavelets-based algorithm for numerical simulation of fractional dynam-ical systemsrdquo Engineering Computations vol 35 no 5 pp 1907ndash1931 2018

[25] C Li F Zeng and F Liu ldquoSpectral approximations to the frac-tional integral and derivativerdquo Fractional Calculus and AppliedAnalysis vol 15 no 3 pp 383ndash406 2012

[26] M Zheng F Liu I Turner and V Anh ldquoA novel high orderspace-time spectral method for the time fractional Fokker-Planck equationrdquo SIAM Journal on Scientific Computing vol 37no 2 pp A701ndashA724 2015

[27] F Zeng F Liu C Li K Burrage I Turner and V Anh ldquoACrank-Nicolson ADI spectral method for a two-dimensionalRiesz space fractional nonlinear reaction-diffusion equationrdquoSIAM Journal on Numerical Analysis vol 52 no 6 pp 2599ndash2622 2014

[28] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical ModellingSimulation andComputation for Engineering and EnvironmentalSystems vol 35 no 12 pp 5662ndash5672 2011

[29] M Zayernouri and G E Karniadakis ldquoExponentially accuratespectral and spectral element methods for fractional ODEsrdquoJournal of Computational Physics vol 257 pp 460ndash480 2014

[30] X Zhao and Z Zhang ldquoSuperconvergence points of fractionalspectral interpolationrdquo SIAM Journal on Scientific Computingvol 38 no 1 pp A598ndashA613 2016

[31] F Chen Q Xu and J S Hesthaven ldquoA multi-domain spectralmethod for time-fractional differential equationsrdquo Journal ofComputational Physics vol 293 pp 157ndash172 2015

[32] Z Mao and J Shen ldquoEfficient spectral-Galerkin methods forfractional partial differential equations with variable coeffi-cientsrdquo Journal of Computational Physics vol 307 pp 243ndash2612016

[33] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[34] J Hadamard ldquoEssai sur lrsquoetude des fonctions donnees par leurdeveloppement de Taylorrdquo Journal de Mathematiques Pures etAppliquees vol 4 pp 101ndash186 1892

[35] A Erdelyi andHKober ldquoSome remarks onHankel transformsrdquoQuarterly Journal of Mathematics vol 11 pp 212ndash221 1940

[36] I Dimovski ldquoOperational calculus for a class of differentialoperatorsrdquo Comptes Rendus De L Academie Bulgare Des Sci-ences vol 19 pp 1111ndash1114 1966

[37] S B Yakubovich and Y F LuchkoThe hypergeometric approachto integral transforms and convolutions vol 287 ofMathematicsand its Applications Kluwer Academic Publishers Dordrecht-Boston-London 1994

[38] Y Luchko ldquoOperational rules for a mixed operator of theErdelyi-Kober typerdquo Fractional Calculus and Applied Analysisvol 7 no 3 pp 339ndash364 2004

[39] G Szego Orthogonal polynomials American MathematicalSociety Providence 1992

[40] M Zayernouri and G E Karniadakis ldquoFractional Sturm-Liou-ville eigen-problems theory and numerical approximationrdquoJournal of Computational Physics vol 252 pp 495ndash517 2013

[41] S Chen J Shen and L-L Wang ldquoGeneralized Jacobi functionsand their applications to fractional differential equationsrdquoMathematics of Computation vol 85 no 300 pp 1603ndash16382016

[42] A A Kilbas ldquoHadamard-type integral equations and fractionalcalculus operatorsrdquo in Singular integral operators factorizationand applications vol 142 ofOperTheory Adv Appl pp 175ndash188Birkhauser Basel 2003

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6 International Journal of Differential Equations

such that (119906 minus 120587119873119906 V)120596 = 0 forallV isin P119873 and then there exists aconstant 119862 such that100381710038171003817100381710038171003817 119888

minus1119863119897+120572

119909 (119906 (119909) minus 120587119873119906 (119909))100381710038171003817100381710038171003817120596 le 119862119873119897minus119898 1003817100381710038171003817 minus1119863119898+120572119909 1199061003817100381710038171003817120596 (39)

Based on above lemmas fractional integral and derivativeof Jacobi polynomials in the standard interval [minus1 1] can beexpressed explicitly Fractional integrals and derivatives inshifted interval [119886 119887] can be obtained through proper variabletransforms

For any 119910 isin [119886 119887] we assume 119869120573120574119899lowast(119910) is defined in [119886 119887]Let ℎ = 119887 minus 119886 119909 = minus1 + 2((119910 minus 119886)ℎ) then 119909 isin [minus1 1]Substituting 119909 = minus1 + 2((119910 minus 119886)(119887 minus 119886)) into equation (27)-(28) and (29)-(30) we obtained

119910119868120572119887119869(0120574)119899lowast (119910) = (ℎ2)120572

1199091198681205721119869(0120574)119899 (119909) 119886119868120572119910119869(1205730)119899lowast (119910) = (ℎ2)

120572

minus1119868120572119909119869(1205730)119899 (119909) 119888119910119863120572

119887119869(0120574)119899lowast (119910) = (2ℎ)

120572119888119909119863120572

1119869(0120574)119899 (119909) 119888119886119863120572

119910119869(1205730)119899lowast (119910) = (2ℎ)120572

119888minus1119863120572

119909119869(1205730)119899 (119909)

(40)

32 Spectral Approximation to Generalized Fractional Opera-tors We assume 119911(119909) and 119908(119909) are positive monotone func-tions and 119911 119908 isin 119862[119886 119887] Obviously 119911(119909) and119908(119909) are invert-ible The following lemmas can be derived for generalizedfractional operators

Lemma 21 Let 119892(120577) = 119908(119911minus1(120577))119891(119911minus1(120577)) 0 lt 120572 lt 1 thenthe generalized fractional integral operator is equivalent to thefollowing classical fractional integral

(119868120572119886+[119911119908]119891) (119909) = [119908 (119909)]minus1 1205770119868120572120577119892 (119911 (119909)) (41)

Here 1205770 = 119911(119886)Proof From the definition of generalized fractional integralwe have

(119868120572119886+[119911119908]119891) (119909) = [119908 (119909)]minus1Γ (120572) int119909

119886

119908 (119905) 1199111015840 (119905) 119891 (119905)[119911 (119909) minus 119911 (119905)]1minus120572 119889119905 (42)

Since 120577 = 119911(119905) is positive monotone function then 120577 = 119911(119905) isinvertible and we have 119905 = 119911minus1(120577)(119868120572119886+[119911119908]119891) (119909) = [119908 (119909)]minus1Γ (120572)sdot int119911(119909)

119911(119886)

119908(119911minus1 (120577)) 1199111015840 (119911minus1 (120577)) 119891 (119911minus1 (120577))[119911 (119909) minus 120577]1minus120572 119889119911minus1 (120577)

= [119908 (119909)]minus1Γ (120572) int119911(119909)

119911(119886)

119908(119911minus1 (120577)) 119891 (119911minus1 (120577))[119911 (119909119896) minus 120577]1minus120572 119889120577

(43)

Let 119892(120577) = 119908(119911minus1(120577))119891(119911minus1(120577)) then the generalized frac-tional integral of 119891(119909) is converted to classical fractionalintegral of 119892(120577) in the following form

(119868120572119886+[119911119908]119891) (119909) = [119908 (119909)]minus1 1205770119868120572120577119892 (119911 (119909)) (44)

Lemma is proved

Lemma 22 Let 119892(120577) = 119908(119911minus1(120577))119891(119911minus1(120577)) 0 lt 120572 lt 1 thenthe generalized fractional derivative of order 120572 is equivalent tothe following classical fractional derivative

(119863120572119886+[1199111199082]119891) (119909) = [119908 (119909)]minus1 119888

1205770119863120572

120577119892 (119911 (119909)) (45)

Here 1205770 = 119911(119886)Proof Similar to the proof of Lemma 21 we have

(119863120572119886+[1199111199082]119891) (119909) = [119908 (119909)]minus1Γ (1 minus 120572)sdot int119911(119909)

119911(119886)

(120597120597120577) (119908 (119911minus1 (120577)) 119891 (119911minus1 (120577))) (120597120577120597119905)[119911 (119909) minus 120577]120572 119889119911minus1 (120577)

= [119908 (119909)]minus1Γ (1 minus 120572) int119911(119909)

119911(119886)

(120597120597120577) (119908 (119911minus1 (120577)) 119891 (119911minus1 (120577)))[119911 (119909) minus 120577]120572 119889120577

(46)

Let 119892(120577) = 119908(119911minus1(120577))119891(119911minus1(120577)) then the generalized frac-tional derivative of 119891(119909) is expressed through the classicalfractional derivative of 119892(120577)

(119863120572119886+[1199111199082]119891) (119909) = [119908 (119909)]minus1 119888

1205770119863120572

120577119892 (119911 (119909)) (47)

Lemma is proved

Remark 23 These two lemmas establish important relationbetween classical and generalized fractional operators Withthese lemmas generalized fractional differential equationscan be solved via classical fractional differential equationsand vice versa Which kind of transform should be takendepends on characters of the problem to be solved

In order to design a high order numerical approximationof the generalized fractional operator we define a scaled spaceP119899

[119908119911] such that

P119899[119908119911] = V (119909) = [119908 (119909)]minus1 119892 (119911 (119909)) 119892 (119909) isin P

119899 119909isin Ω = [119886 119887] (48)

whereP119899 is a polynomial space of up to order 119899Define a inner product and norm for space P119899

[119908119911] suchthat

(119906 (119909) V (119909))120596 = int119887

119886119906 (119909) V (119909) 120596 (119909) 119889119909

and V120596 = radic(V V)120596(49)

International Journal of Differential Equations 7

Define projection 119876119873 into space P119873119908119911 such that for any

function 119906(119909)(119906 minus 119876119873119906 V)120596 = 0 forallV isin P

119873119908119911 (50)

SupposeΦ119895(119909) 119895 = 0 119873 are a set of orthogonal basisfunctions in spaceP119873

119911119908 satisfying

(Φ119894 (119909) Φ119895 (119909))120596 = 1 119894 = 1198950 119894 = 119895 (51)

Let119873 997888rarr infin then Φ119895(119909) 119895 = 0 1 119873 form a 1198712120596(Ω)space and for any 119906(119909) isin 1198712120596(Ω) the projection 119876119873119906(119909) canbe written as

119876119873119906 (119909) = 119873sum119895=0

119895Φ119895 (119909) (52)

Here 119895 (119895 = 0 119873) are expansion coefficients such that

119895 = (119906 (119909) Φ119895 (119909))12059610038171003817100381710038171003817Φ119895 (119909)100381710038171003817100381710038172120596 (53)

The weight function 120596(119909) plays an important role in thecomputational process and analysis of the method Herewe choose a proper weight function to use properties oforthogonal polynomials and make the computation moreefficient Let 119901119895(119911(119909)) = 119908(119909)Φ119895(119909) and note that

(119876119873119906 (119909) Φ119894 (119909))120596 = int119887

119886( 119873sum

119895=0

119895119908minus1 (119909) 119901119895 (119911 (119909)))sdot 119908minus1 (119909) 119901119894 (119911 (119909)) 120596 (119909) 119889119909

(54)

Take 120596(119909) = 1199082(119909)1199111015840(119909) then(119876119873119906 (119909) Φ119894 (119909))120596 = int119911(119887)

119911(119886)( 119873sum

119895=0

119895119901119895 (119911))119901119894 (119911) 119889119911 (55)

Since 119901119895(119911) is polynomial of order 119895 the computation can becarried out easily through properties of orthogonal polyno-mials

Suppose 119875119895lowast(119909) is shifted Legendre polynomial definedon [119911(119886) 119911(119887)] Then given any function 119906(119909) isin 1198712120596 withweight 120596(119909) = 1199082(119909)1199111015840(119909) we have

119876119873119906 (119909) = [119908 (119909)]minus1 119873sum119895=0

119895119875119895lowast (119911 (119909)) (56)

Recalling Lemmas 21 and 22 the generalized fractionalintegral and derivative can be obtained in the form

(119868120572119886+[119911119908]119876119873119906) (119909)= [119908 (119909)]minus1 119873sum

119895=0

119895 ( 1205770119868120572120577119875119895lowast (120577)1003816100381610038161003816120577=119911(119909)) (57)

(119863120572119886+[1199111199082]119876119873119906) (119909)= [119908 (119909)]minus1 119873sum

119895=0

119895 ( 1198881205770119863120572

120577119875119895lowast (120577)1003816100381610038161003816120577=119911(119909)) (58)

From Lemma 19 the following corollary can be obtainedimmediately

Corollary 24 120577(119909) = 119911(119909) is a monotone increasing function119908(119909) gt 0 0 lt 120572 lt 1 119876119873 is a projection into space P119873119908119911119906(119911minus1(120577)) isin 119862119898(Ω) 119898 isin N then there exists a constant 119862120572

such that 10038171003817100381710038171003817119863120572119886+[1199111199082] (119906 (119909) minus 119876119873119906 (119909))100381710038171003817100381710038171205962le 1198621205721198731minus119898 10038171003817100381710038171003817 1205770119863119898+120572

119911119906 (120577minus1 (119911))100381710038171003817100381710038171205961

(59)

Here the weight function 1205961(120577) = (120577 minus 119886)minus120572(119887 minus 120577)120572 1205962(119909) =1205961(119911(119909))1199082(119909)1199111015840(119909)Remark 25 Unlike the convergence theory in classical poly-nomial space the convergence order in space P119873

119908119911 dependson regularity of the function 119906(119909) with respect to 119911(119909) Wewill illustrate this through numerical examples

Most of the time it is more convenient to consider theproblems in nodal form We assume the given interpolationpoints are 120577119895 = 119911(119909119895) (119895 = 0 1 119873) then the Lagrangebasis functions L119895(119909) (119895 = 0 1 119873) can be defined asfollows

L119894 (120577) = prod119895=0119873119895 =119894

(120577 minus 120577119895)(120577119894 minus 120577119895) (60)

The function 119906(119909) can be expressed using both Jacobipolynomials and Lagrange polynomials The following rela-tion is derived

119906 (119909) = [119908 (119909)]minus1 119873sum119895=0

119892119895119875119895lowast (119911 (119909))

= [119908 (119909)]minus1 119873sum119895=0

119908119895119906119895L119895 (119911 (119909)) (61)

Considering the equivalence between Legendre basis andLagrange basis the following equality holds

L119895 (119911) = 119873sum119896=0

119897119896119895119875119896lowast (119911) 119896 = 0 1 119873 (62)

where 119897119894119895 = (119871119895(119911) 119875119894lowast(119911))119875119894lowast(119911)21198712 Then the nodal form expansion of 119906(119909) is obtained

119906 (119909) = [119908 (119909)]minus1 119873sum119895=0

119873sum119896=0

119908119895119906119895119897119896119895119875119896lowast (119911 (119909)) (63)

8 International Journal of Differential Equations

From (57) and (58) the corresponding nodal form of gener-alized fractional integral and derivative are obtained

(119868120572119886+[119911119908]119906) (119909)= [119908 (119909)]minus1 119873sum

119895=0

119873sum119896=0

119908119895119906119895119897119896119895 ( 1205770119868120572120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119909)) (64)

(119863120572119886+[1199111199082]119906) (119909)= [119908 (119909)]minus1 119873sum

119895=0

119873sum119896=0

119908119895119906119895119897119896119895 ( 1198881205770119863120572

120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119909)) (65)

Example 26 Now we give an example to show the effec-tiveness and accuracy of the method Assuming 119911(119909) =radic1199093 119908(119909) = 1 we consider generalized fractional derivativeof 119910(119909) on the interval [0 1] with the following form

119910 (119909) = 5sum119895=1

(minus1)119895+1 1199093119895(2119895) (66)

The exact generalized fractional derivative of 119910(119909) is119863120572

0+[1199111199082]119910 (119909) = 5sum119895=1

(minus1)119895+1 1199093119895minus31205722Γ (2119895 + 1 minus 120572) (67)

Numerical approximation to generalized fractionalderivative of 119910(119909) can be obtained using (58) We considerthe maximum absolute error

119890 = max119909isin[01]

100381610038161003816100381610038161198631205720+[1199111199082]119910 (119909) minus 119863120572

0+[1199111199082]119876119873119910 (119909)10038161003816100381610038161003816 (68)

of the numerical derivative Results for 120572 = 02 05 08 areshown in Figure 1

For this example it is easy to check that 119910 isin P10119908119911 From

theory of spectral approximation the error would decreaseexponentially when 119873 lt 10 and the numerical fractionalderivative would be exact when119873 ge 10 Our numerical resultcoincides with the theory exactly

Example 27 In this example we test the spectral approxi-mation of Hadamard integral Considering Hadamard-typefractional integral of 119910(119909) = sin(120587119909) on the interval [1 2]when 119908(119909) = 1 and 120572 = 1 the Hadamard integral of 119910(119909) isSine integral function Si(120587119909) minus Si(120587) for more general 119908(119909)and 120572 the exact Hadamard-type integral of 119910(119909) is unknownHere we consider several pairs of 119908(119909) and 120572 NumericalHadamard-type integral of 119910(119909) would be computed using(57) with 119911(119909) = log(119909) To evaluate the approximationaccuracy for the case 119908(119909) = 1 120572 = 1 the exact Hada-mard fractional integral is computed usingMATLAB built-infunction 119904119894119899119894119899119905 for other cases ldquoexactrdquo Hadamard fractionalintegral is computed using (57) with large 119873 (eg 119873 =50) which is treated as reference solution The results fornumerical Hadamard integral and approximation error areshown in Figures 2 and 3

For this example 119906 is not in the spaceP119899119908119911 for any 119899 Max-

imum error of approximated fractional integral convergesexponentially until reaching machine accuracy

10minus16

10minus14

10minus12

10minus10

10minus8

10minus6

10minus4

10minus2

max

imum

erro

r

5 6 7 8 94 11 1210

N

alpha=02alpha=05alpha=08

Figure 1 Error of numerical approximation to generalized frac-tional derivative of 119910(119909)

11 12 13 14 15 16 17 18 19 21x

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Had

amar

d fra

ctio

nal i

nteg

ral

w(x)=1=1w(x)=1=06

w(x)=x=1w(x)=x=06

Figure 2 Hadamard fractional integral for different weight 119908(119909)and 120572

33 Fractional IntegralDifferential Matrices Suppose 119906(119909) isinP119873

119911119908 119909119894 (119894 = 0 1 119873) are the interpolation points and119910119895 (119895 = 0 1 119873) are the collocation points we define thefollowing symbols

U = [119906 (1199090) 1199061199091 119906 (119909119873)]119879 (69)

U(120572) = [119863120572

119886+[119911119908]119906 (1199100) 119863120572119886+[119911119908]119906 (1199101)

119863120572119886+[119911119908]119906 (119910119873)]119879 120572 gt 0 (70)

International Journal of Differential Equations 9

10minus16

10minus14

10minus12

10minus10

10minus8

10minus6

10minus4

max

imum

erro

r

w(x)=1=1w(x)=1=06

w(x)=x=1w(x)=x=06

10 12 14 16 18 20 228x

Figure 3 Approximation error of Hadamard fractional integral fordifferent weight 119908 and 120572

U(120572) = [119868minus120572119886+[119911119908]119906 (1199100) 119868minus120572119886+[119911119908]119906 (1199101) 119868minus120572119886+[119911119908]119906 (119910119873)]119879 120572 lt 0 (71)

We define a generalized fractional differentialintegralmatrixM120572 such that

M120572119894119895 = 119873sum

119896=0

119908(119909119895) 119897119896119895119908 (119910119894) ( 1198881205770119863120572

120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119910119894))

for 120572 gt 0(72)

M120572119894119895 = 119873sum

119896=0

119908(119909119895) 119897119896119895119908 (119910119894) ( 1205770119868120572120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119910119894))

for 120572 lt 0(73)

In order to compute the fractional matrix M120572 moreefficiently we define a few more matrices L and L120572 are(119873 + 1) times (119873 + 1) matrices such thatL119894119895 = L119895(119911(119910119894))L120572

119894119895 =1205770119863120572

120577L119895(119911)|119911=119911(119910119894) 119894 119895 = 0 1 119873 V is defined based on

values of Legendre polynomials at interpolation points 119911(119909119894)such that

V119894119895 = 119875119895lowast (119911 (119909119894)) 119894 119895 = 0 1 119873 (74)

D120572 is defined as fractional derivativeintegral of Legendrepolynomials at collocation points 120577119894 = 119911(119910119894)

D120572119894119895 = 119888

1205770119863120572

120577119875119895lowast (120577119894) 120572 gt 0 (75)

D120572119894119895 = 1205770

119868120572120577119875119895lowast (120577119894) 120572 lt 0 (76)

W119897 andW119903 are the weight matrices defined as follows

W119903119894119895 =

119908(119909119895) 119894 = 1198950 otherwise (77)

W119897119894119895 =

1119908(119910119895) 119894 = 119895 gt 10 otherwise (78)

Theorem 28 For 119906(119909) isin P119873119911119908 vectors defined in (69)ndash(71)

and matrices defined in (72)ndash(78) the following relation holds

U(120572) = M

120572U = (W119897D120572V

minus1W

119903)U (79)

Proof From the definition (69)ndash(71) (72)ndash(78) and (63)-(64) it is easy to obtain

U(120572) = M

120572U (80)

Next we prove thatM120572 = W119897D120572Vminus1W119903From the definition of the interpolation function L119895(119911)

119875119895lowast (119911) = 119873sum119894=0

119875119895lowast (119911 (119909119894)) L119894 (119911) (81)

Suppose 997888rarr119875(119911) = (1198750lowast(119911) 1198751lowast(119911) 119875119873lowast(119911)) 997888rarrL (119911) =(L0(119911) L1(119911) L119873(119911)) for 120572 gt 0 we have997888rarr119875 (119911) = 997888rarrL (119911)V

1205770119863120572

120577

997888rarr119875 (119911) = ( 1205770119863120572

120577

997888rarrL (119911))V (82)

Evaluating the matrices multiplication for each elementthe following relation is derived

(W119897D120572V

minus1W119903)119894119895 = (W119897L

120572W119903)119894119895

= 119908 (119909119895)119908 (119910119894) L(120572)119895 (119911 (119910119894))(83)

= 119873sum119896=0

119908(119909119895) 119897119896119895119908 (119910119894) ( 1205770119863120572

120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119910119894)) = M

120572119894119895 (84)

The case 120572 lt 0 can be proved similarly

Remark 29 Collocation points 119910119895 and the interpolationpoints 119909119894 are not necessary the same To obtain a goodapproximation interpolation points ofGauss-type are usuallyused At the same time collocation points should be chosenproperly to guarantee stability properties of the methodIn the following numerical examples for computation andstability aim both interpolation and collocation points arechosen based on Gauss-type points with respect to 119911(119909)

10 International Journal of Differential Equations

4 Collocation Methods for FractionalDifferential and Integral Equations

41 Fractional Ordinary Differential Equations In this sub-section we consider collocation method for the generalizedfractional ordinary differential equation of the form

1198631205720+[1199111199082]119906 (119909) = 120582 (119909) 119906 (119909) + 119891 (119909)

119909 isin Ω = (0 119887] (85)

119906 (0) = 1199060 (86)

Here 0 lt 120572 lt 1 119908(119909) gt 0 119911(119909) gt 0 and 119911(119909) is a monotonefunction inΩ

We assume 119906119873(119909) isin P119873119911119908 is a numerical solution of the

equation 119909119894(119894 = 0 1 119873) are the chosen interpolationpoints 119906119895119873 = 119906119873(119909119895) The following discretized equation isobtained

[119908 (119909)]minus1 119873sum119895=0

119873sum119896=0

119908(119909119895) 119906119895119873119897119896119895 ( 1198881205770119863120572

120577119875119896lowast (119911 (119909)))

= 120582 (119909) 119906119873 (119909) + 119891 (119909) (87)

Let (87) hold on collocation points 119910119895 (119895 = 1 119873) thematrix form is obtained

(M120572 minus Λ)U = 119865 (88)

Here Λ is a diagonal matrix with Λ 119894119894 = 120582(119910119894) 119865 = (1199060 119891(1199101) 119891(119910119873))119879Considering the initial condition we set S = M120572 minus Λ

with the first row replaced by (1 0 0) 1198651015840 = 119865with its firstelement replaced by 1199060 Then the solution U is obtained bysolving the matrix equation SU = 1198651015840

Example 30 Consider the following example

1198631205720+[1199111199082]119906 (119909) = (1 + 119909) 119906 (119909)

+ Γ (15119903 + 25) 119909119903minus(23)120572Γ (15119903 + 25 minus 120572)minus 119909119903 (1 + 119909) 119909 isin (0 1]

(89)

119906 (0) = 0 (90)

Here 119911(119909) = 11990923119908(119909) = 119909 119903 is an arbitrary positive numberThe exact solution of the ordinary differential equation is119906(119909) = 119909119903 Maximum absolute errors of numerical solutions

for 119903 = 6 7 and 120572 = 03 06 09 are shown in Figures 4 and 5When 119911(119909) = 11990923 119908(119909) = 119909 the scaled polynomial

spaceP119873119911119908 becomes

P119873119911119908 = 119904119901119886119899 11990921198993minus1 119899 = 1 2 119873 (91)

For 119903 = 7 the error converges exponentially and reachesmachine accuracy at 119873 = 12 It is faster than any finitedifference method while for 119903 = 6 solution convergesalgebraically as119873 increases however it still reaches machineaccuracy at 119873 = 23 The major reason for this is that 119906(119909) =1199097 isin P12

119911119908 and 119906(119909) = 1199096 notin P119873119911119908 for any119873 isin N

6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

N

10minus15

10minus10

10minus5

100

max

erro

r

=03=06=09

Figure 4 Log-log plot of the maximum error for 119903 = 6

=03=06=09

10minus15

10minus10

10minus5

100

max

erro

r

15 20 25 30 3510N

Figure 5 Semilog plot of the maximum error for 119903 = 7

Remark 31 Since space P119873119908119911 is transformed from classical

polynomial space with respect to 119911(119909) the convergenceof spectral collocation method for ODEs with generalizedfractional operators depends not only on the smoothness ofthe solution itself but also on the scale function 119911(119905)42 Hadamard-Type Integral Equations We consider thefollowing Hadamard-type boundary value problem

1Γ (120572) int119909

119886( 119904119909)

120583 (log 119909119904 )120572minus1 119891 (119904) 119889119904119904 = 119892 (119909)

119909 isin Ω = (119886 119887] (92)

International Journal of Differential Equations 11

In [42] Kilbas discussed the existence of the solution of(92) Explicit formulas for the solution 119891(119905) were establishedin the following theorem

Theorem 32 (see [42]) If 119909120583119892(119909) isin 119860119862[119886 119887] then theHadamard-type integral equation (92) with 0 lt 120572 lt 1 issolvable in 119883120583(119886 119887) and its solution may be represented in theform

119891 (119909) = 119909minus120583Γ (1 minus 120572) [119886120583119892 (119886) (log 119909119886)minus120572

+ int119909

119886(log 119909119904 )

minus120572 (119904120583119892 (119904))1015840 119889119904] (93)

Here 119860119862[119886 119887] is the set of absolutely continuous functions on[119886 119887] and119883120583(119886 119887) is space of those Lebesgue measurable func-tions 119891 on [119886 119887] for which 119909120583minus1119891(119909) is absolutely integrable

Solution of the Hadamard-type integral equation isexactly the same as generalized fractional derivative ofRiemann-Liouville type with 119911(119904) = log(119904) 119908(119904) = 119904120583 Therelationship between (93) and Caputo type generalized frac-tional derivative is

119891 (119909) = 119886120583119892 (119886) 119909minus120583Γ (1 minus 120572) (log 119909119886)minus120572 + 119863120572

119886+[1199111199082]119892 (119909) (94)

Suppose 119909119895 119895 = 0 1 119873 are interpolation points thediscretized solution of equation (92) is

119891119873 (119909) = 119886120583119892 (119886) 119909minus120583Γ (1 minus 120572) (log 119909119886)minus120572

+ 119909minus120583 119873sum119895=0

119873sum119896=0

119909120583119895119892 (119909119895) 119897119896119895( 119888120577119886119863120572

120577119875119896lowast (log (119909))

(95)

Rewriting (95) in matrix form we have

119865 = 119866119886 +M120572119866 (96)

where M120572 is generalized fractional differential matrix119866 = (119892(1199090) 119892(1199091) 119892(119909119873))119879 119865 = (119891119873(1199090) 119891119873(1199091) 119891119873(119909119873))119879 119866119886 is a vector about the initial condition of theintegral equation defined by

119866119886 = 119886120583119892 (119886)Γ (1 minus 120572) (119909minus1205830 (log 1199090119886 )minus120572 119909minus1205831 (log 1199091119886 )

minus120572 119909minus120583119873 (log 119909119873119886 )minus120572)119879

(97)

Example 33 Assume 119892(119909) = sin(119909 minus 1) 120583 = 13 Ω = [110] Solutions of (92) for 120572 = 03 06 09 are shown inFigure 6

2 4 6 8 10 120X

minus8

minus6

minus4

minus2

0

2

4

6

8

f(X)

=03=06=09

Figure 6 Solution of (92)

43 Erdelyi-Kober Fractional Diffusion Equation In this sub-section we consider the following Erdelyi-Kober fractionaldiffusion equation [12]

119906 (119909 119905) = 1199060 (119909) + 1Γ (120573)sdot int119905

0

120572120573119904120572120573minus1 (119905120572120573 minus 119904120572120573)120573minus1 12059721205971199092 119906 (119909 119904) d119904(98)

Erdelyi-Kober fractional diffusion equation which is alsocalled stretched time-fractional diffusion equation is themaster equation of a kind of generalized grey Brownianmotion (ggBm) The ggBm is a parametric class of stochasticprocesses that provides models for both fast and slow anoma-lous diffusion This class is made up of self-similar processes119861120572120573(119905)with stationary increments and it depends on two realparameters 0 lt 120572 le 2 and 0 lt 120573 le 1 It includes thefractional Brownian motion when 0 lt 120572 le 2 and 120573 = 1 thetime-fractional diffusion stochastic processes when 0 lt 120572 =120573 lt 1 and the standard Brownian motion when 120572 = 120573 = 1About the relationship between stochastic process119861120572120573(119905) andstretched time-fractional diffusion equation the followingproposition is presented in [12]

Proposition 34 The marginal probability density function119891120572120573(119909 119905) of the process 119861120572120573(119905) 119905 ge 0 is the fundamentalsolution of the stretched time-fractional diffusion equation

119906 (119909 119905) = 1199060 (119909) + 1Γ (120573)sdot int119905

0

120572120573119904120572120573minus1 (119905120572120573 minus 119904120572120573)120573minus1 12059721205971199092 119906 (119909 119904) d119904(99)

12 International Journal of Differential Equations

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 7 Standard Brownian motion 120572 = 1 120573 = 1

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 8 Time-fractional diffusion with 120572 = 06 120573 = 06

Recalling the definition of generalized fractional integraland setting 119911(119905) = 119905120572120573 119908(119905) = 1 the equation can be rewrit-ten as

119906 (119909 119905) = 1199060 (119909) + 1198681205730+[119911119908]119906119909119909 (119909 119905) (100)

We use collocation method for both space and timediscretization We choose Legendre-Gauss-Lobatto (L-G-L)points 119909119894 (119894 = 0 1 119872) as the space collocation points andchoose 119905119895 (119895 = 0 1 119873) such that 119911(119905119895) are L-G-L pointsas the time collocation points

Define space collocation matrix M2 such that M2119894119895 =(11988921198891199092)L119895(119909119894) and generalized fractional integral matrix

M120573 Matrix M120573 is computed through Theorem 28 and thespace-time collocation matrices are obtained using Kro-necker product 1015840otimes1015840 Suppose A and B are space-timecollocation matrices with dimension (119872 + 1)(119873 + 1) times (119872 +1)(119873 + 1) for the second order derivative and fractionalintegral of order 120573 separately Then

A = M2 otimesI119873+1

B = I119872+1 otimesM120573 (101)

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 9 Fractional Brownian motion with 120572 = 15 120573 = 1

Suppose 119906119873(119909 119905) is the numerical solution of (98) defin-ing 119906119873119894119895 = 119906119873(119909119894 119905119895) solution vectorU and initial vector 1198800such that

U = [11990611987300 11990611987301 1199061198730119873 11990611987310 1199061198731119873 1199061198731198720 119906119873119872119873]119879

1198800 = [1199060 (1199090) 1199060 (1199090) 1199060 (1199091) 1199060 (1199091) 1199060 (119909119872) 1199060 (119909119872)]119879

(102)

where in the definition of 1198800 each 1199060(119909119895) is repeated 119873 + 1times

The matrix form discretized equation of (98) is obtainedas

U = 1198800 +BAU (103)

In the discretized equation initial condition is explicitlyinvolved After boundary condition added properly numeri-cal solution can be obtained by solving the matrix equation

Example 35 Assume 119909 isin Ω = (minus1 1) 119905 isin (0 05] 1199060(119909) =119890minus101199092 minus 119890minus10 119906(sdot 119905)|120597Ω = 0 Numerical solutions with 119872 =119873 = 50 are shown in Figures 7ndash10

Erdelyi-Kober diffusion equation characterizes the mar-ginal density function of the process 119861120572120573(119905) 119905 ge 0 When120572 = 120573 = 1 we recover the standard diffusion equationWhen0 lt 120572 = 120573 lt 1 we get the time-fractional diffusion equationof order 120573 When 120573 = 1 and 0 lt 120572 lt 2 we have the equationof the fractional Brownian motion marginal density

As shown in Figures 7 and 8 when 1 lt 120572 lt 2 the diffu-sion is fast and the increments exhibit long-range depen-dence when 0 lt 120572 lt 1 the diffusion is slow and theincrements form a stationary process which does not exhibitlong-range dependenceThe results coincide with theoreticalanalysis in [12 14]

International Journal of Differential Equations 13

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 10 Fractional Brownian motion with 120572 = 05 120573 = 1

5 Conclusion

In this paper we propose a spectral collocation method fordifferential and integral equations with generalized fractionaloperators To deal with the difficulty in designing spectralapproximation scheme due to complexity of integral kerneland weight a variable transform technique is applied to thegeneralized fractional operator and a spectral approximationmethod is proposed for the generalized fractional operatorOperational matrices for generalized fractional operatorsare derived Spectral collocation methods are designed forfractional ordinary differential equations Hadamard-typeintegral equations and Erdelyi-Kober diffusion equationsseparately Numerical experiments are carried out to verifythe accuracy and efficiency of the method and characteristicsof the Erdelyi-Kober diffusion equation are analyzed basedon numerical results

Data Availability

(i) The programs used to support the findings of this studyhave been deposited in the GitHub repository (httpsgithubcomqinwuxuSpectralGFPDE ) (ii) No data were used tosupport this study

Disclosure

An earlier version of this workwas presented at the ldquo8th Inter-national Congress on Industrial and Applied Mathematics(ICIAM 2015)rdquo

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The first author is supported by the National Key RampD Pro-gram of China (No 2017YFC0209804) the National ScienceFoundation for Young Scientists of China (No 11701273) andYouth Foundation of Jiangsu Province (No BK20170628)

The second author is supported by the National Key RampDProgram of China (No 2017YFB0305601)

References

[1] B BaeumerDA BensonMMMeerschaert and SWWheat-craft ldquoSubordinated advection-dispersion equation for contam-inant transportrdquo Water Resources Research vol 37 no 6 pp1543ndash1550 2001

[2] E Barkai RMetzler and J Klafter ldquoFrom continuous time ran-dom walks to the fractional Fokker-Planck equationrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 61no 1 pp 132ndash138 2000

[3] A Blumen G Zumofen and J Klafter ldquoTransport aspects inanomalous diffusion Levy walksrdquo Physical Review A AtomicMolecular and Optical Physics vol 40 no 7 pp 3964ndash39731989

[4] J P Bouchaud and A Georges ldquoAnomalous diffusion in dis-ordered media statistical mechanisms models and physicalapplicationsrdquoPhysics Reports vol 195 no 4-5 pp 127ndash293 1990

[5] M Raberto E Scalas and F Mainardi ldquoWaiting-times andreturns in high-frequency financial data an empirical studyrdquoPhysica A Statistical Mechanics and its Applications vol 314 no1ndash4 pp 749ndash755 2002

[6] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations New York NYUSA Elsevier 2006

[7] S G Samko A A Kilbas and O I Marichev Fractional Inte-grals and Derivatives Theory and Applications Gordon andBreach Yverdon Switzerland 1993

[8] V Kiryakova ldquoA brief story about the operators of the general-ized fractional calculusrdquo Fractional CalculusampAppliedAnalysisAn International Journal forTheory and Applications vol 11 no2 pp 203ndash220 2008

[9] S L Kalla ldquoOn operators of fractional integration Irdquo Mathe-maticae Notae vol 22 pp 89ndash93 197071

[10] S L Kalla ldquoOn operators of fractional integration IIrdquo Mathe-maticae Notae vol 25 pp 29ndash35 1976

[11] V S Kiryakova Generalized Fractional Calculus and Applica-tions Long-man amp J Wiley Harlow New York NY USA 1994

[12] A Mura and F Mainardi ldquoA class of self-similar stochasticprocesses with stationary increments to model anomalousdiffusion in physicsrdquo Integral Transforms and Special Functionsvol 20 no 3-4 pp 185ndash198 2009

[13] O P Agrawal ldquoSome generalized fractional calculus operatorsand their applications in integral equationsrdquo Fractional Calculusand Applied Analysis An International Journal for Theory andApplications vol 15 no 4 pp 700ndash711 2012

[14] G Pagnini ldquoErdelyi-Kober fractional diffusionrdquo FractionalCalculus and Applied Analysis An International Journal forTheory and Applications vol 15 no 1 pp 117ndash127 2012

[15] A Mura and G Pagnini ldquoCharacterizations and simulations ofa class of stochastic processes to model anomalous diffusionrdquoJournal of Physics A Mathematical and General vol 41 no 28285003 22 pages 2008

[16] E K Lenzi L R Evangelista M K Lenzi H V Ribeiro and EC de Oliveira ldquoSolutions for a non-Markovian diffusion equa-tionrdquo Physics Letters A vol 374 no 41 pp 4193ndash4198 2010

[17] B Al-Saqabi and V S Kiryakova ldquoExplicit solutions of frac-tional integral and differential equations involving Erderyi-Kober operatorsrdquo Applied Mathematics and Computation vol95 no 1 pp 1ndash13 1998

14 International Journal of Differential Equations

[18] L A Hanna and Y F Luchko ldquoOperational calculus for theCaputo-type fractional Erdelyi-Kober derivative and its appli-cationsrdquo Integral Transforms and Special Functions vol 25 no5 pp 359ndash373 2014

[19] Y Xu Z He and Q Xu ldquoNumerical solutions of fractionaladvection-diffusion equations with a kind of new generalizedfractional derivativerdquo International Journal of Computer Math-ematics vol 91 no 3 pp 588ndash600 2014

[20] Y Xu Z He and O P Agrawal ldquoNumerical and analytical solu-tions of new generalized fractional diffusion equationrdquo Com-puters amp Mathematics with Applications vol 66 no 10 pp2019ndash2029 2013

[21] Y Xu and O P Agrawal ldquoNumerical solutions and analysisof diffusion for new generalized fractional Burgers equationrdquoFractional Calculus and Applied Analysis An International Jour-nal forTheory and Applications vol 16 no 3 pp 709ndash736 2013

[22] X Li and C Xu ldquoA space-time spectral method for the timefractional diffusion equationrdquo SIAM Journal on NumericalAnalysis vol 47 no 3 pp 2108ndash2131 2009

[23] Q Xu and J S Hesthaven ldquoStable multi-domain spectralpenalty methods for fractional partial differential equationsrdquoJournal of Computational Physics vol 257 pp 241ndash258 2014

[24] RMittal and S Pandit ldquoQuasilinearized Scale-3 Haar wavelets-based algorithm for numerical simulation of fractional dynam-ical systemsrdquo Engineering Computations vol 35 no 5 pp 1907ndash1931 2018

[25] C Li F Zeng and F Liu ldquoSpectral approximations to the frac-tional integral and derivativerdquo Fractional Calculus and AppliedAnalysis vol 15 no 3 pp 383ndash406 2012

[26] M Zheng F Liu I Turner and V Anh ldquoA novel high orderspace-time spectral method for the time fractional Fokker-Planck equationrdquo SIAM Journal on Scientific Computing vol 37no 2 pp A701ndashA724 2015

[27] F Zeng F Liu C Li K Burrage I Turner and V Anh ldquoACrank-Nicolson ADI spectral method for a two-dimensionalRiesz space fractional nonlinear reaction-diffusion equationrdquoSIAM Journal on Numerical Analysis vol 52 no 6 pp 2599ndash2622 2014

[28] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical ModellingSimulation andComputation for Engineering and EnvironmentalSystems vol 35 no 12 pp 5662ndash5672 2011

[29] M Zayernouri and G E Karniadakis ldquoExponentially accuratespectral and spectral element methods for fractional ODEsrdquoJournal of Computational Physics vol 257 pp 460ndash480 2014

[30] X Zhao and Z Zhang ldquoSuperconvergence points of fractionalspectral interpolationrdquo SIAM Journal on Scientific Computingvol 38 no 1 pp A598ndashA613 2016

[31] F Chen Q Xu and J S Hesthaven ldquoA multi-domain spectralmethod for time-fractional differential equationsrdquo Journal ofComputational Physics vol 293 pp 157ndash172 2015

[32] Z Mao and J Shen ldquoEfficient spectral-Galerkin methods forfractional partial differential equations with variable coeffi-cientsrdquo Journal of Computational Physics vol 307 pp 243ndash2612016

[33] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[34] J Hadamard ldquoEssai sur lrsquoetude des fonctions donnees par leurdeveloppement de Taylorrdquo Journal de Mathematiques Pures etAppliquees vol 4 pp 101ndash186 1892

[35] A Erdelyi andHKober ldquoSome remarks onHankel transformsrdquoQuarterly Journal of Mathematics vol 11 pp 212ndash221 1940

[36] I Dimovski ldquoOperational calculus for a class of differentialoperatorsrdquo Comptes Rendus De L Academie Bulgare Des Sci-ences vol 19 pp 1111ndash1114 1966

[37] S B Yakubovich and Y F LuchkoThe hypergeometric approachto integral transforms and convolutions vol 287 ofMathematicsand its Applications Kluwer Academic Publishers Dordrecht-Boston-London 1994

[38] Y Luchko ldquoOperational rules for a mixed operator of theErdelyi-Kober typerdquo Fractional Calculus and Applied Analysisvol 7 no 3 pp 339ndash364 2004

[39] G Szego Orthogonal polynomials American MathematicalSociety Providence 1992

[40] M Zayernouri and G E Karniadakis ldquoFractional Sturm-Liou-ville eigen-problems theory and numerical approximationrdquoJournal of Computational Physics vol 252 pp 495ndash517 2013

[41] S Chen J Shen and L-L Wang ldquoGeneralized Jacobi functionsand their applications to fractional differential equationsrdquoMathematics of Computation vol 85 no 300 pp 1603ndash16382016

[42] A A Kilbas ldquoHadamard-type integral equations and fractionalcalculus operatorsrdquo in Singular integral operators factorizationand applications vol 142 ofOperTheory Adv Appl pp 175ndash188Birkhauser Basel 2003

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International Journal of Differential Equations 7

Define projection 119876119873 into space P119873119908119911 such that for any

function 119906(119909)(119906 minus 119876119873119906 V)120596 = 0 forallV isin P

119873119908119911 (50)

SupposeΦ119895(119909) 119895 = 0 119873 are a set of orthogonal basisfunctions in spaceP119873

119911119908 satisfying

(Φ119894 (119909) Φ119895 (119909))120596 = 1 119894 = 1198950 119894 = 119895 (51)

Let119873 997888rarr infin then Φ119895(119909) 119895 = 0 1 119873 form a 1198712120596(Ω)space and for any 119906(119909) isin 1198712120596(Ω) the projection 119876119873119906(119909) canbe written as

119876119873119906 (119909) = 119873sum119895=0

119895Φ119895 (119909) (52)

Here 119895 (119895 = 0 119873) are expansion coefficients such that

119895 = (119906 (119909) Φ119895 (119909))12059610038171003817100381710038171003817Φ119895 (119909)100381710038171003817100381710038172120596 (53)

The weight function 120596(119909) plays an important role in thecomputational process and analysis of the method Herewe choose a proper weight function to use properties oforthogonal polynomials and make the computation moreefficient Let 119901119895(119911(119909)) = 119908(119909)Φ119895(119909) and note that

(119876119873119906 (119909) Φ119894 (119909))120596 = int119887

119886( 119873sum

119895=0

119895119908minus1 (119909) 119901119895 (119911 (119909)))sdot 119908minus1 (119909) 119901119894 (119911 (119909)) 120596 (119909) 119889119909

(54)

Take 120596(119909) = 1199082(119909)1199111015840(119909) then(119876119873119906 (119909) Φ119894 (119909))120596 = int119911(119887)

119911(119886)( 119873sum

119895=0

119895119901119895 (119911))119901119894 (119911) 119889119911 (55)

Since 119901119895(119911) is polynomial of order 119895 the computation can becarried out easily through properties of orthogonal polyno-mials

Suppose 119875119895lowast(119909) is shifted Legendre polynomial definedon [119911(119886) 119911(119887)] Then given any function 119906(119909) isin 1198712120596 withweight 120596(119909) = 1199082(119909)1199111015840(119909) we have

119876119873119906 (119909) = [119908 (119909)]minus1 119873sum119895=0

119895119875119895lowast (119911 (119909)) (56)

Recalling Lemmas 21 and 22 the generalized fractionalintegral and derivative can be obtained in the form

(119868120572119886+[119911119908]119876119873119906) (119909)= [119908 (119909)]minus1 119873sum

119895=0

119895 ( 1205770119868120572120577119875119895lowast (120577)1003816100381610038161003816120577=119911(119909)) (57)

(119863120572119886+[1199111199082]119876119873119906) (119909)= [119908 (119909)]minus1 119873sum

119895=0

119895 ( 1198881205770119863120572

120577119875119895lowast (120577)1003816100381610038161003816120577=119911(119909)) (58)

From Lemma 19 the following corollary can be obtainedimmediately

Corollary 24 120577(119909) = 119911(119909) is a monotone increasing function119908(119909) gt 0 0 lt 120572 lt 1 119876119873 is a projection into space P119873119908119911119906(119911minus1(120577)) isin 119862119898(Ω) 119898 isin N then there exists a constant 119862120572

such that 10038171003817100381710038171003817119863120572119886+[1199111199082] (119906 (119909) minus 119876119873119906 (119909))100381710038171003817100381710038171205962le 1198621205721198731minus119898 10038171003817100381710038171003817 1205770119863119898+120572

119911119906 (120577minus1 (119911))100381710038171003817100381710038171205961

(59)

Here the weight function 1205961(120577) = (120577 minus 119886)minus120572(119887 minus 120577)120572 1205962(119909) =1205961(119911(119909))1199082(119909)1199111015840(119909)Remark 25 Unlike the convergence theory in classical poly-nomial space the convergence order in space P119873

119908119911 dependson regularity of the function 119906(119909) with respect to 119911(119909) Wewill illustrate this through numerical examples

Most of the time it is more convenient to consider theproblems in nodal form We assume the given interpolationpoints are 120577119895 = 119911(119909119895) (119895 = 0 1 119873) then the Lagrangebasis functions L119895(119909) (119895 = 0 1 119873) can be defined asfollows

L119894 (120577) = prod119895=0119873119895 =119894

(120577 minus 120577119895)(120577119894 minus 120577119895) (60)

The function 119906(119909) can be expressed using both Jacobipolynomials and Lagrange polynomials The following rela-tion is derived

119906 (119909) = [119908 (119909)]minus1 119873sum119895=0

119892119895119875119895lowast (119911 (119909))

= [119908 (119909)]minus1 119873sum119895=0

119908119895119906119895L119895 (119911 (119909)) (61)

Considering the equivalence between Legendre basis andLagrange basis the following equality holds

L119895 (119911) = 119873sum119896=0

119897119896119895119875119896lowast (119911) 119896 = 0 1 119873 (62)

where 119897119894119895 = (119871119895(119911) 119875119894lowast(119911))119875119894lowast(119911)21198712 Then the nodal form expansion of 119906(119909) is obtained

119906 (119909) = [119908 (119909)]minus1 119873sum119895=0

119873sum119896=0

119908119895119906119895119897119896119895119875119896lowast (119911 (119909)) (63)

8 International Journal of Differential Equations

From (57) and (58) the corresponding nodal form of gener-alized fractional integral and derivative are obtained

(119868120572119886+[119911119908]119906) (119909)= [119908 (119909)]minus1 119873sum

119895=0

119873sum119896=0

119908119895119906119895119897119896119895 ( 1205770119868120572120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119909)) (64)

(119863120572119886+[1199111199082]119906) (119909)= [119908 (119909)]minus1 119873sum

119895=0

119873sum119896=0

119908119895119906119895119897119896119895 ( 1198881205770119863120572

120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119909)) (65)

Example 26 Now we give an example to show the effec-tiveness and accuracy of the method Assuming 119911(119909) =radic1199093 119908(119909) = 1 we consider generalized fractional derivativeof 119910(119909) on the interval [0 1] with the following form

119910 (119909) = 5sum119895=1

(minus1)119895+1 1199093119895(2119895) (66)

The exact generalized fractional derivative of 119910(119909) is119863120572

0+[1199111199082]119910 (119909) = 5sum119895=1

(minus1)119895+1 1199093119895minus31205722Γ (2119895 + 1 minus 120572) (67)

Numerical approximation to generalized fractionalderivative of 119910(119909) can be obtained using (58) We considerthe maximum absolute error

119890 = max119909isin[01]

100381610038161003816100381610038161198631205720+[1199111199082]119910 (119909) minus 119863120572

0+[1199111199082]119876119873119910 (119909)10038161003816100381610038161003816 (68)

of the numerical derivative Results for 120572 = 02 05 08 areshown in Figure 1

For this example it is easy to check that 119910 isin P10119908119911 From

theory of spectral approximation the error would decreaseexponentially when 119873 lt 10 and the numerical fractionalderivative would be exact when119873 ge 10 Our numerical resultcoincides with the theory exactly

Example 27 In this example we test the spectral approxi-mation of Hadamard integral Considering Hadamard-typefractional integral of 119910(119909) = sin(120587119909) on the interval [1 2]when 119908(119909) = 1 and 120572 = 1 the Hadamard integral of 119910(119909) isSine integral function Si(120587119909) minus Si(120587) for more general 119908(119909)and 120572 the exact Hadamard-type integral of 119910(119909) is unknownHere we consider several pairs of 119908(119909) and 120572 NumericalHadamard-type integral of 119910(119909) would be computed using(57) with 119911(119909) = log(119909) To evaluate the approximationaccuracy for the case 119908(119909) = 1 120572 = 1 the exact Hada-mard fractional integral is computed usingMATLAB built-infunction 119904119894119899119894119899119905 for other cases ldquoexactrdquo Hadamard fractionalintegral is computed using (57) with large 119873 (eg 119873 =50) which is treated as reference solution The results fornumerical Hadamard integral and approximation error areshown in Figures 2 and 3

For this example 119906 is not in the spaceP119899119908119911 for any 119899 Max-

imum error of approximated fractional integral convergesexponentially until reaching machine accuracy

10minus16

10minus14

10minus12

10minus10

10minus8

10minus6

10minus4

10minus2

max

imum

erro

r

5 6 7 8 94 11 1210

N

alpha=02alpha=05alpha=08

Figure 1 Error of numerical approximation to generalized frac-tional derivative of 119910(119909)

11 12 13 14 15 16 17 18 19 21x

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Had

amar

d fra

ctio

nal i

nteg

ral

w(x)=1=1w(x)=1=06

w(x)=x=1w(x)=x=06

Figure 2 Hadamard fractional integral for different weight 119908(119909)and 120572

33 Fractional IntegralDifferential Matrices Suppose 119906(119909) isinP119873

119911119908 119909119894 (119894 = 0 1 119873) are the interpolation points and119910119895 (119895 = 0 1 119873) are the collocation points we define thefollowing symbols

U = [119906 (1199090) 1199061199091 119906 (119909119873)]119879 (69)

U(120572) = [119863120572

119886+[119911119908]119906 (1199100) 119863120572119886+[119911119908]119906 (1199101)

119863120572119886+[119911119908]119906 (119910119873)]119879 120572 gt 0 (70)

International Journal of Differential Equations 9

10minus16

10minus14

10minus12

10minus10

10minus8

10minus6

10minus4

max

imum

erro

r

w(x)=1=1w(x)=1=06

w(x)=x=1w(x)=x=06

10 12 14 16 18 20 228x

Figure 3 Approximation error of Hadamard fractional integral fordifferent weight 119908 and 120572

U(120572) = [119868minus120572119886+[119911119908]119906 (1199100) 119868minus120572119886+[119911119908]119906 (1199101) 119868minus120572119886+[119911119908]119906 (119910119873)]119879 120572 lt 0 (71)

We define a generalized fractional differentialintegralmatrixM120572 such that

M120572119894119895 = 119873sum

119896=0

119908(119909119895) 119897119896119895119908 (119910119894) ( 1198881205770119863120572

120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119910119894))

for 120572 gt 0(72)

M120572119894119895 = 119873sum

119896=0

119908(119909119895) 119897119896119895119908 (119910119894) ( 1205770119868120572120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119910119894))

for 120572 lt 0(73)

In order to compute the fractional matrix M120572 moreefficiently we define a few more matrices L and L120572 are(119873 + 1) times (119873 + 1) matrices such thatL119894119895 = L119895(119911(119910119894))L120572

119894119895 =1205770119863120572

120577L119895(119911)|119911=119911(119910119894) 119894 119895 = 0 1 119873 V is defined based on

values of Legendre polynomials at interpolation points 119911(119909119894)such that

V119894119895 = 119875119895lowast (119911 (119909119894)) 119894 119895 = 0 1 119873 (74)

D120572 is defined as fractional derivativeintegral of Legendrepolynomials at collocation points 120577119894 = 119911(119910119894)

D120572119894119895 = 119888

1205770119863120572

120577119875119895lowast (120577119894) 120572 gt 0 (75)

D120572119894119895 = 1205770

119868120572120577119875119895lowast (120577119894) 120572 lt 0 (76)

W119897 andW119903 are the weight matrices defined as follows

W119903119894119895 =

119908(119909119895) 119894 = 1198950 otherwise (77)

W119897119894119895 =

1119908(119910119895) 119894 = 119895 gt 10 otherwise (78)

Theorem 28 For 119906(119909) isin P119873119911119908 vectors defined in (69)ndash(71)

and matrices defined in (72)ndash(78) the following relation holds

U(120572) = M

120572U = (W119897D120572V

minus1W

119903)U (79)

Proof From the definition (69)ndash(71) (72)ndash(78) and (63)-(64) it is easy to obtain

U(120572) = M

120572U (80)

Next we prove thatM120572 = W119897D120572Vminus1W119903From the definition of the interpolation function L119895(119911)

119875119895lowast (119911) = 119873sum119894=0

119875119895lowast (119911 (119909119894)) L119894 (119911) (81)

Suppose 997888rarr119875(119911) = (1198750lowast(119911) 1198751lowast(119911) 119875119873lowast(119911)) 997888rarrL (119911) =(L0(119911) L1(119911) L119873(119911)) for 120572 gt 0 we have997888rarr119875 (119911) = 997888rarrL (119911)V

1205770119863120572

120577

997888rarr119875 (119911) = ( 1205770119863120572

120577

997888rarrL (119911))V (82)

Evaluating the matrices multiplication for each elementthe following relation is derived

(W119897D120572V

minus1W119903)119894119895 = (W119897L

120572W119903)119894119895

= 119908 (119909119895)119908 (119910119894) L(120572)119895 (119911 (119910119894))(83)

= 119873sum119896=0

119908(119909119895) 119897119896119895119908 (119910119894) ( 1205770119863120572

120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119910119894)) = M

120572119894119895 (84)

The case 120572 lt 0 can be proved similarly

Remark 29 Collocation points 119910119895 and the interpolationpoints 119909119894 are not necessary the same To obtain a goodapproximation interpolation points ofGauss-type are usuallyused At the same time collocation points should be chosenproperly to guarantee stability properties of the methodIn the following numerical examples for computation andstability aim both interpolation and collocation points arechosen based on Gauss-type points with respect to 119911(119909)

10 International Journal of Differential Equations

4 Collocation Methods for FractionalDifferential and Integral Equations

41 Fractional Ordinary Differential Equations In this sub-section we consider collocation method for the generalizedfractional ordinary differential equation of the form

1198631205720+[1199111199082]119906 (119909) = 120582 (119909) 119906 (119909) + 119891 (119909)

119909 isin Ω = (0 119887] (85)

119906 (0) = 1199060 (86)

Here 0 lt 120572 lt 1 119908(119909) gt 0 119911(119909) gt 0 and 119911(119909) is a monotonefunction inΩ

We assume 119906119873(119909) isin P119873119911119908 is a numerical solution of the

equation 119909119894(119894 = 0 1 119873) are the chosen interpolationpoints 119906119895119873 = 119906119873(119909119895) The following discretized equation isobtained

[119908 (119909)]minus1 119873sum119895=0

119873sum119896=0

119908(119909119895) 119906119895119873119897119896119895 ( 1198881205770119863120572

120577119875119896lowast (119911 (119909)))

= 120582 (119909) 119906119873 (119909) + 119891 (119909) (87)

Let (87) hold on collocation points 119910119895 (119895 = 1 119873) thematrix form is obtained

(M120572 minus Λ)U = 119865 (88)

Here Λ is a diagonal matrix with Λ 119894119894 = 120582(119910119894) 119865 = (1199060 119891(1199101) 119891(119910119873))119879Considering the initial condition we set S = M120572 minus Λ

with the first row replaced by (1 0 0) 1198651015840 = 119865with its firstelement replaced by 1199060 Then the solution U is obtained bysolving the matrix equation SU = 1198651015840

Example 30 Consider the following example

1198631205720+[1199111199082]119906 (119909) = (1 + 119909) 119906 (119909)

+ Γ (15119903 + 25) 119909119903minus(23)120572Γ (15119903 + 25 minus 120572)minus 119909119903 (1 + 119909) 119909 isin (0 1]

(89)

119906 (0) = 0 (90)

Here 119911(119909) = 11990923119908(119909) = 119909 119903 is an arbitrary positive numberThe exact solution of the ordinary differential equation is119906(119909) = 119909119903 Maximum absolute errors of numerical solutions

for 119903 = 6 7 and 120572 = 03 06 09 are shown in Figures 4 and 5When 119911(119909) = 11990923 119908(119909) = 119909 the scaled polynomial

spaceP119873119911119908 becomes

P119873119911119908 = 119904119901119886119899 11990921198993minus1 119899 = 1 2 119873 (91)

For 119903 = 7 the error converges exponentially and reachesmachine accuracy at 119873 = 12 It is faster than any finitedifference method while for 119903 = 6 solution convergesalgebraically as119873 increases however it still reaches machineaccuracy at 119873 = 23 The major reason for this is that 119906(119909) =1199097 isin P12

119911119908 and 119906(119909) = 1199096 notin P119873119911119908 for any119873 isin N

6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

N

10minus15

10minus10

10minus5

100

max

erro

r

=03=06=09

Figure 4 Log-log plot of the maximum error for 119903 = 6

=03=06=09

10minus15

10minus10

10minus5

100

max

erro

r

15 20 25 30 3510N

Figure 5 Semilog plot of the maximum error for 119903 = 7

Remark 31 Since space P119873119908119911 is transformed from classical

polynomial space with respect to 119911(119909) the convergenceof spectral collocation method for ODEs with generalizedfractional operators depends not only on the smoothness ofthe solution itself but also on the scale function 119911(119905)42 Hadamard-Type Integral Equations We consider thefollowing Hadamard-type boundary value problem

1Γ (120572) int119909

119886( 119904119909)

120583 (log 119909119904 )120572minus1 119891 (119904) 119889119904119904 = 119892 (119909)

119909 isin Ω = (119886 119887] (92)

International Journal of Differential Equations 11

In [42] Kilbas discussed the existence of the solution of(92) Explicit formulas for the solution 119891(119905) were establishedin the following theorem

Theorem 32 (see [42]) If 119909120583119892(119909) isin 119860119862[119886 119887] then theHadamard-type integral equation (92) with 0 lt 120572 lt 1 issolvable in 119883120583(119886 119887) and its solution may be represented in theform

119891 (119909) = 119909minus120583Γ (1 minus 120572) [119886120583119892 (119886) (log 119909119886)minus120572

+ int119909

119886(log 119909119904 )

minus120572 (119904120583119892 (119904))1015840 119889119904] (93)

Here 119860119862[119886 119887] is the set of absolutely continuous functions on[119886 119887] and119883120583(119886 119887) is space of those Lebesgue measurable func-tions 119891 on [119886 119887] for which 119909120583minus1119891(119909) is absolutely integrable

Solution of the Hadamard-type integral equation isexactly the same as generalized fractional derivative ofRiemann-Liouville type with 119911(119904) = log(119904) 119908(119904) = 119904120583 Therelationship between (93) and Caputo type generalized frac-tional derivative is

119891 (119909) = 119886120583119892 (119886) 119909minus120583Γ (1 minus 120572) (log 119909119886)minus120572 + 119863120572

119886+[1199111199082]119892 (119909) (94)

Suppose 119909119895 119895 = 0 1 119873 are interpolation points thediscretized solution of equation (92) is

119891119873 (119909) = 119886120583119892 (119886) 119909minus120583Γ (1 minus 120572) (log 119909119886)minus120572

+ 119909minus120583 119873sum119895=0

119873sum119896=0

119909120583119895119892 (119909119895) 119897119896119895( 119888120577119886119863120572

120577119875119896lowast (log (119909))

(95)

Rewriting (95) in matrix form we have

119865 = 119866119886 +M120572119866 (96)

where M120572 is generalized fractional differential matrix119866 = (119892(1199090) 119892(1199091) 119892(119909119873))119879 119865 = (119891119873(1199090) 119891119873(1199091) 119891119873(119909119873))119879 119866119886 is a vector about the initial condition of theintegral equation defined by

119866119886 = 119886120583119892 (119886)Γ (1 minus 120572) (119909minus1205830 (log 1199090119886 )minus120572 119909minus1205831 (log 1199091119886 )

minus120572 119909minus120583119873 (log 119909119873119886 )minus120572)119879

(97)

Example 33 Assume 119892(119909) = sin(119909 minus 1) 120583 = 13 Ω = [110] Solutions of (92) for 120572 = 03 06 09 are shown inFigure 6

2 4 6 8 10 120X

minus8

minus6

minus4

minus2

0

2

4

6

8

f(X)

=03=06=09

Figure 6 Solution of (92)

43 Erdelyi-Kober Fractional Diffusion Equation In this sub-section we consider the following Erdelyi-Kober fractionaldiffusion equation [12]

119906 (119909 119905) = 1199060 (119909) + 1Γ (120573)sdot int119905

0

120572120573119904120572120573minus1 (119905120572120573 minus 119904120572120573)120573minus1 12059721205971199092 119906 (119909 119904) d119904(98)

Erdelyi-Kober fractional diffusion equation which is alsocalled stretched time-fractional diffusion equation is themaster equation of a kind of generalized grey Brownianmotion (ggBm) The ggBm is a parametric class of stochasticprocesses that provides models for both fast and slow anoma-lous diffusion This class is made up of self-similar processes119861120572120573(119905)with stationary increments and it depends on two realparameters 0 lt 120572 le 2 and 0 lt 120573 le 1 It includes thefractional Brownian motion when 0 lt 120572 le 2 and 120573 = 1 thetime-fractional diffusion stochastic processes when 0 lt 120572 =120573 lt 1 and the standard Brownian motion when 120572 = 120573 = 1About the relationship between stochastic process119861120572120573(119905) andstretched time-fractional diffusion equation the followingproposition is presented in [12]

Proposition 34 The marginal probability density function119891120572120573(119909 119905) of the process 119861120572120573(119905) 119905 ge 0 is the fundamentalsolution of the stretched time-fractional diffusion equation

119906 (119909 119905) = 1199060 (119909) + 1Γ (120573)sdot int119905

0

120572120573119904120572120573minus1 (119905120572120573 minus 119904120572120573)120573minus1 12059721205971199092 119906 (119909 119904) d119904(99)

12 International Journal of Differential Equations

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 7 Standard Brownian motion 120572 = 1 120573 = 1

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 8 Time-fractional diffusion with 120572 = 06 120573 = 06

Recalling the definition of generalized fractional integraland setting 119911(119905) = 119905120572120573 119908(119905) = 1 the equation can be rewrit-ten as

119906 (119909 119905) = 1199060 (119909) + 1198681205730+[119911119908]119906119909119909 (119909 119905) (100)

We use collocation method for both space and timediscretization We choose Legendre-Gauss-Lobatto (L-G-L)points 119909119894 (119894 = 0 1 119872) as the space collocation points andchoose 119905119895 (119895 = 0 1 119873) such that 119911(119905119895) are L-G-L pointsas the time collocation points

Define space collocation matrix M2 such that M2119894119895 =(11988921198891199092)L119895(119909119894) and generalized fractional integral matrix

M120573 Matrix M120573 is computed through Theorem 28 and thespace-time collocation matrices are obtained using Kro-necker product 1015840otimes1015840 Suppose A and B are space-timecollocation matrices with dimension (119872 + 1)(119873 + 1) times (119872 +1)(119873 + 1) for the second order derivative and fractionalintegral of order 120573 separately Then

A = M2 otimesI119873+1

B = I119872+1 otimesM120573 (101)

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 9 Fractional Brownian motion with 120572 = 15 120573 = 1

Suppose 119906119873(119909 119905) is the numerical solution of (98) defin-ing 119906119873119894119895 = 119906119873(119909119894 119905119895) solution vectorU and initial vector 1198800such that

U = [11990611987300 11990611987301 1199061198730119873 11990611987310 1199061198731119873 1199061198731198720 119906119873119872119873]119879

1198800 = [1199060 (1199090) 1199060 (1199090) 1199060 (1199091) 1199060 (1199091) 1199060 (119909119872) 1199060 (119909119872)]119879

(102)

where in the definition of 1198800 each 1199060(119909119895) is repeated 119873 + 1times

The matrix form discretized equation of (98) is obtainedas

U = 1198800 +BAU (103)

In the discretized equation initial condition is explicitlyinvolved After boundary condition added properly numeri-cal solution can be obtained by solving the matrix equation

Example 35 Assume 119909 isin Ω = (minus1 1) 119905 isin (0 05] 1199060(119909) =119890minus101199092 minus 119890minus10 119906(sdot 119905)|120597Ω = 0 Numerical solutions with 119872 =119873 = 50 are shown in Figures 7ndash10

Erdelyi-Kober diffusion equation characterizes the mar-ginal density function of the process 119861120572120573(119905) 119905 ge 0 When120572 = 120573 = 1 we recover the standard diffusion equationWhen0 lt 120572 = 120573 lt 1 we get the time-fractional diffusion equationof order 120573 When 120573 = 1 and 0 lt 120572 lt 2 we have the equationof the fractional Brownian motion marginal density

As shown in Figures 7 and 8 when 1 lt 120572 lt 2 the diffu-sion is fast and the increments exhibit long-range depen-dence when 0 lt 120572 lt 1 the diffusion is slow and theincrements form a stationary process which does not exhibitlong-range dependenceThe results coincide with theoreticalanalysis in [12 14]

International Journal of Differential Equations 13

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 10 Fractional Brownian motion with 120572 = 05 120573 = 1

5 Conclusion

In this paper we propose a spectral collocation method fordifferential and integral equations with generalized fractionaloperators To deal with the difficulty in designing spectralapproximation scheme due to complexity of integral kerneland weight a variable transform technique is applied to thegeneralized fractional operator and a spectral approximationmethod is proposed for the generalized fractional operatorOperational matrices for generalized fractional operatorsare derived Spectral collocation methods are designed forfractional ordinary differential equations Hadamard-typeintegral equations and Erdelyi-Kober diffusion equationsseparately Numerical experiments are carried out to verifythe accuracy and efficiency of the method and characteristicsof the Erdelyi-Kober diffusion equation are analyzed basedon numerical results

Data Availability

(i) The programs used to support the findings of this studyhave been deposited in the GitHub repository (httpsgithubcomqinwuxuSpectralGFPDE ) (ii) No data were used tosupport this study

Disclosure

An earlier version of this workwas presented at the ldquo8th Inter-national Congress on Industrial and Applied Mathematics(ICIAM 2015)rdquo

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The first author is supported by the National Key RampD Pro-gram of China (No 2017YFC0209804) the National ScienceFoundation for Young Scientists of China (No 11701273) andYouth Foundation of Jiangsu Province (No BK20170628)

The second author is supported by the National Key RampDProgram of China (No 2017YFB0305601)

References

[1] B BaeumerDA BensonMMMeerschaert and SWWheat-craft ldquoSubordinated advection-dispersion equation for contam-inant transportrdquo Water Resources Research vol 37 no 6 pp1543ndash1550 2001

[2] E Barkai RMetzler and J Klafter ldquoFrom continuous time ran-dom walks to the fractional Fokker-Planck equationrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 61no 1 pp 132ndash138 2000

[3] A Blumen G Zumofen and J Klafter ldquoTransport aspects inanomalous diffusion Levy walksrdquo Physical Review A AtomicMolecular and Optical Physics vol 40 no 7 pp 3964ndash39731989

[4] J P Bouchaud and A Georges ldquoAnomalous diffusion in dis-ordered media statistical mechanisms models and physicalapplicationsrdquoPhysics Reports vol 195 no 4-5 pp 127ndash293 1990

[5] M Raberto E Scalas and F Mainardi ldquoWaiting-times andreturns in high-frequency financial data an empirical studyrdquoPhysica A Statistical Mechanics and its Applications vol 314 no1ndash4 pp 749ndash755 2002

[6] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations New York NYUSA Elsevier 2006

[7] S G Samko A A Kilbas and O I Marichev Fractional Inte-grals and Derivatives Theory and Applications Gordon andBreach Yverdon Switzerland 1993

[8] V Kiryakova ldquoA brief story about the operators of the general-ized fractional calculusrdquo Fractional CalculusampAppliedAnalysisAn International Journal forTheory and Applications vol 11 no2 pp 203ndash220 2008

[9] S L Kalla ldquoOn operators of fractional integration Irdquo Mathe-maticae Notae vol 22 pp 89ndash93 197071

[10] S L Kalla ldquoOn operators of fractional integration IIrdquo Mathe-maticae Notae vol 25 pp 29ndash35 1976

[11] V S Kiryakova Generalized Fractional Calculus and Applica-tions Long-man amp J Wiley Harlow New York NY USA 1994

[12] A Mura and F Mainardi ldquoA class of self-similar stochasticprocesses with stationary increments to model anomalousdiffusion in physicsrdquo Integral Transforms and Special Functionsvol 20 no 3-4 pp 185ndash198 2009

[13] O P Agrawal ldquoSome generalized fractional calculus operatorsand their applications in integral equationsrdquo Fractional Calculusand Applied Analysis An International Journal for Theory andApplications vol 15 no 4 pp 700ndash711 2012

[14] G Pagnini ldquoErdelyi-Kober fractional diffusionrdquo FractionalCalculus and Applied Analysis An International Journal forTheory and Applications vol 15 no 1 pp 117ndash127 2012

[15] A Mura and G Pagnini ldquoCharacterizations and simulations ofa class of stochastic processes to model anomalous diffusionrdquoJournal of Physics A Mathematical and General vol 41 no 28285003 22 pages 2008

[16] E K Lenzi L R Evangelista M K Lenzi H V Ribeiro and EC de Oliveira ldquoSolutions for a non-Markovian diffusion equa-tionrdquo Physics Letters A vol 374 no 41 pp 4193ndash4198 2010

[17] B Al-Saqabi and V S Kiryakova ldquoExplicit solutions of frac-tional integral and differential equations involving Erderyi-Kober operatorsrdquo Applied Mathematics and Computation vol95 no 1 pp 1ndash13 1998

14 International Journal of Differential Equations

[18] L A Hanna and Y F Luchko ldquoOperational calculus for theCaputo-type fractional Erdelyi-Kober derivative and its appli-cationsrdquo Integral Transforms and Special Functions vol 25 no5 pp 359ndash373 2014

[19] Y Xu Z He and Q Xu ldquoNumerical solutions of fractionaladvection-diffusion equations with a kind of new generalizedfractional derivativerdquo International Journal of Computer Math-ematics vol 91 no 3 pp 588ndash600 2014

[20] Y Xu Z He and O P Agrawal ldquoNumerical and analytical solu-tions of new generalized fractional diffusion equationrdquo Com-puters amp Mathematics with Applications vol 66 no 10 pp2019ndash2029 2013

[21] Y Xu and O P Agrawal ldquoNumerical solutions and analysisof diffusion for new generalized fractional Burgers equationrdquoFractional Calculus and Applied Analysis An International Jour-nal forTheory and Applications vol 16 no 3 pp 709ndash736 2013

[22] X Li and C Xu ldquoA space-time spectral method for the timefractional diffusion equationrdquo SIAM Journal on NumericalAnalysis vol 47 no 3 pp 2108ndash2131 2009

[23] Q Xu and J S Hesthaven ldquoStable multi-domain spectralpenalty methods for fractional partial differential equationsrdquoJournal of Computational Physics vol 257 pp 241ndash258 2014

[24] RMittal and S Pandit ldquoQuasilinearized Scale-3 Haar wavelets-based algorithm for numerical simulation of fractional dynam-ical systemsrdquo Engineering Computations vol 35 no 5 pp 1907ndash1931 2018

[25] C Li F Zeng and F Liu ldquoSpectral approximations to the frac-tional integral and derivativerdquo Fractional Calculus and AppliedAnalysis vol 15 no 3 pp 383ndash406 2012

[26] M Zheng F Liu I Turner and V Anh ldquoA novel high orderspace-time spectral method for the time fractional Fokker-Planck equationrdquo SIAM Journal on Scientific Computing vol 37no 2 pp A701ndashA724 2015

[27] F Zeng F Liu C Li K Burrage I Turner and V Anh ldquoACrank-Nicolson ADI spectral method for a two-dimensionalRiesz space fractional nonlinear reaction-diffusion equationrdquoSIAM Journal on Numerical Analysis vol 52 no 6 pp 2599ndash2622 2014

[28] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical ModellingSimulation andComputation for Engineering and EnvironmentalSystems vol 35 no 12 pp 5662ndash5672 2011

[29] M Zayernouri and G E Karniadakis ldquoExponentially accuratespectral and spectral element methods for fractional ODEsrdquoJournal of Computational Physics vol 257 pp 460ndash480 2014

[30] X Zhao and Z Zhang ldquoSuperconvergence points of fractionalspectral interpolationrdquo SIAM Journal on Scientific Computingvol 38 no 1 pp A598ndashA613 2016

[31] F Chen Q Xu and J S Hesthaven ldquoA multi-domain spectralmethod for time-fractional differential equationsrdquo Journal ofComputational Physics vol 293 pp 157ndash172 2015

[32] Z Mao and J Shen ldquoEfficient spectral-Galerkin methods forfractional partial differential equations with variable coeffi-cientsrdquo Journal of Computational Physics vol 307 pp 243ndash2612016

[33] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[34] J Hadamard ldquoEssai sur lrsquoetude des fonctions donnees par leurdeveloppement de Taylorrdquo Journal de Mathematiques Pures etAppliquees vol 4 pp 101ndash186 1892

[35] A Erdelyi andHKober ldquoSome remarks onHankel transformsrdquoQuarterly Journal of Mathematics vol 11 pp 212ndash221 1940

[36] I Dimovski ldquoOperational calculus for a class of differentialoperatorsrdquo Comptes Rendus De L Academie Bulgare Des Sci-ences vol 19 pp 1111ndash1114 1966

[37] S B Yakubovich and Y F LuchkoThe hypergeometric approachto integral transforms and convolutions vol 287 ofMathematicsand its Applications Kluwer Academic Publishers Dordrecht-Boston-London 1994

[38] Y Luchko ldquoOperational rules for a mixed operator of theErdelyi-Kober typerdquo Fractional Calculus and Applied Analysisvol 7 no 3 pp 339ndash364 2004

[39] G Szego Orthogonal polynomials American MathematicalSociety Providence 1992

[40] M Zayernouri and G E Karniadakis ldquoFractional Sturm-Liou-ville eigen-problems theory and numerical approximationrdquoJournal of Computational Physics vol 252 pp 495ndash517 2013

[41] S Chen J Shen and L-L Wang ldquoGeneralized Jacobi functionsand their applications to fractional differential equationsrdquoMathematics of Computation vol 85 no 300 pp 1603ndash16382016

[42] A A Kilbas ldquoHadamard-type integral equations and fractionalcalculus operatorsrdquo in Singular integral operators factorizationand applications vol 142 ofOperTheory Adv Appl pp 175ndash188Birkhauser Basel 2003

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Submit your manuscripts atwwwhindawicom

8 International Journal of Differential Equations

From (57) and (58) the corresponding nodal form of gener-alized fractional integral and derivative are obtained

(119868120572119886+[119911119908]119906) (119909)= [119908 (119909)]minus1 119873sum

119895=0

119873sum119896=0

119908119895119906119895119897119896119895 ( 1205770119868120572120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119909)) (64)

(119863120572119886+[1199111199082]119906) (119909)= [119908 (119909)]minus1 119873sum

119895=0

119873sum119896=0

119908119895119906119895119897119896119895 ( 1198881205770119863120572

120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119909)) (65)

Example 26 Now we give an example to show the effec-tiveness and accuracy of the method Assuming 119911(119909) =radic1199093 119908(119909) = 1 we consider generalized fractional derivativeof 119910(119909) on the interval [0 1] with the following form

119910 (119909) = 5sum119895=1

(minus1)119895+1 1199093119895(2119895) (66)

The exact generalized fractional derivative of 119910(119909) is119863120572

0+[1199111199082]119910 (119909) = 5sum119895=1

(minus1)119895+1 1199093119895minus31205722Γ (2119895 + 1 minus 120572) (67)

Numerical approximation to generalized fractionalderivative of 119910(119909) can be obtained using (58) We considerthe maximum absolute error

119890 = max119909isin[01]

100381610038161003816100381610038161198631205720+[1199111199082]119910 (119909) minus 119863120572

0+[1199111199082]119876119873119910 (119909)10038161003816100381610038161003816 (68)

of the numerical derivative Results for 120572 = 02 05 08 areshown in Figure 1

For this example it is easy to check that 119910 isin P10119908119911 From

theory of spectral approximation the error would decreaseexponentially when 119873 lt 10 and the numerical fractionalderivative would be exact when119873 ge 10 Our numerical resultcoincides with the theory exactly

Example 27 In this example we test the spectral approxi-mation of Hadamard integral Considering Hadamard-typefractional integral of 119910(119909) = sin(120587119909) on the interval [1 2]when 119908(119909) = 1 and 120572 = 1 the Hadamard integral of 119910(119909) isSine integral function Si(120587119909) minus Si(120587) for more general 119908(119909)and 120572 the exact Hadamard-type integral of 119910(119909) is unknownHere we consider several pairs of 119908(119909) and 120572 NumericalHadamard-type integral of 119910(119909) would be computed using(57) with 119911(119909) = log(119909) To evaluate the approximationaccuracy for the case 119908(119909) = 1 120572 = 1 the exact Hada-mard fractional integral is computed usingMATLAB built-infunction 119904119894119899119894119899119905 for other cases ldquoexactrdquo Hadamard fractionalintegral is computed using (57) with large 119873 (eg 119873 =50) which is treated as reference solution The results fornumerical Hadamard integral and approximation error areshown in Figures 2 and 3

For this example 119906 is not in the spaceP119899119908119911 for any 119899 Max-

imum error of approximated fractional integral convergesexponentially until reaching machine accuracy

10minus16

10minus14

10minus12

10minus10

10minus8

10minus6

10minus4

10minus2

max

imum

erro

r

5 6 7 8 94 11 1210

N

alpha=02alpha=05alpha=08

Figure 1 Error of numerical approximation to generalized frac-tional derivative of 119910(119909)

11 12 13 14 15 16 17 18 19 21x

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Had

amar

d fra

ctio

nal i

nteg

ral

w(x)=1=1w(x)=1=06

w(x)=x=1w(x)=x=06

Figure 2 Hadamard fractional integral for different weight 119908(119909)and 120572

33 Fractional IntegralDifferential Matrices Suppose 119906(119909) isinP119873

119911119908 119909119894 (119894 = 0 1 119873) are the interpolation points and119910119895 (119895 = 0 1 119873) are the collocation points we define thefollowing symbols

U = [119906 (1199090) 1199061199091 119906 (119909119873)]119879 (69)

U(120572) = [119863120572

119886+[119911119908]119906 (1199100) 119863120572119886+[119911119908]119906 (1199101)

119863120572119886+[119911119908]119906 (119910119873)]119879 120572 gt 0 (70)

International Journal of Differential Equations 9

10minus16

10minus14

10minus12

10minus10

10minus8

10minus6

10minus4

max

imum

erro

r

w(x)=1=1w(x)=1=06

w(x)=x=1w(x)=x=06

10 12 14 16 18 20 228x

Figure 3 Approximation error of Hadamard fractional integral fordifferent weight 119908 and 120572

U(120572) = [119868minus120572119886+[119911119908]119906 (1199100) 119868minus120572119886+[119911119908]119906 (1199101) 119868minus120572119886+[119911119908]119906 (119910119873)]119879 120572 lt 0 (71)

We define a generalized fractional differentialintegralmatrixM120572 such that

M120572119894119895 = 119873sum

119896=0

119908(119909119895) 119897119896119895119908 (119910119894) ( 1198881205770119863120572

120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119910119894))

for 120572 gt 0(72)

M120572119894119895 = 119873sum

119896=0

119908(119909119895) 119897119896119895119908 (119910119894) ( 1205770119868120572120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119910119894))

for 120572 lt 0(73)

In order to compute the fractional matrix M120572 moreefficiently we define a few more matrices L and L120572 are(119873 + 1) times (119873 + 1) matrices such thatL119894119895 = L119895(119911(119910119894))L120572

119894119895 =1205770119863120572

120577L119895(119911)|119911=119911(119910119894) 119894 119895 = 0 1 119873 V is defined based on

values of Legendre polynomials at interpolation points 119911(119909119894)such that

V119894119895 = 119875119895lowast (119911 (119909119894)) 119894 119895 = 0 1 119873 (74)

D120572 is defined as fractional derivativeintegral of Legendrepolynomials at collocation points 120577119894 = 119911(119910119894)

D120572119894119895 = 119888

1205770119863120572

120577119875119895lowast (120577119894) 120572 gt 0 (75)

D120572119894119895 = 1205770

119868120572120577119875119895lowast (120577119894) 120572 lt 0 (76)

W119897 andW119903 are the weight matrices defined as follows

W119903119894119895 =

119908(119909119895) 119894 = 1198950 otherwise (77)

W119897119894119895 =

1119908(119910119895) 119894 = 119895 gt 10 otherwise (78)

Theorem 28 For 119906(119909) isin P119873119911119908 vectors defined in (69)ndash(71)

and matrices defined in (72)ndash(78) the following relation holds

U(120572) = M

120572U = (W119897D120572V

minus1W

119903)U (79)

Proof From the definition (69)ndash(71) (72)ndash(78) and (63)-(64) it is easy to obtain

U(120572) = M

120572U (80)

Next we prove thatM120572 = W119897D120572Vminus1W119903From the definition of the interpolation function L119895(119911)

119875119895lowast (119911) = 119873sum119894=0

119875119895lowast (119911 (119909119894)) L119894 (119911) (81)

Suppose 997888rarr119875(119911) = (1198750lowast(119911) 1198751lowast(119911) 119875119873lowast(119911)) 997888rarrL (119911) =(L0(119911) L1(119911) L119873(119911)) for 120572 gt 0 we have997888rarr119875 (119911) = 997888rarrL (119911)V

1205770119863120572

120577

997888rarr119875 (119911) = ( 1205770119863120572

120577

997888rarrL (119911))V (82)

Evaluating the matrices multiplication for each elementthe following relation is derived

(W119897D120572V

minus1W119903)119894119895 = (W119897L

120572W119903)119894119895

= 119908 (119909119895)119908 (119910119894) L(120572)119895 (119911 (119910119894))(83)

= 119873sum119896=0

119908(119909119895) 119897119896119895119908 (119910119894) ( 1205770119863120572

120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119910119894)) = M

120572119894119895 (84)

The case 120572 lt 0 can be proved similarly

Remark 29 Collocation points 119910119895 and the interpolationpoints 119909119894 are not necessary the same To obtain a goodapproximation interpolation points ofGauss-type are usuallyused At the same time collocation points should be chosenproperly to guarantee stability properties of the methodIn the following numerical examples for computation andstability aim both interpolation and collocation points arechosen based on Gauss-type points with respect to 119911(119909)

10 International Journal of Differential Equations

4 Collocation Methods for FractionalDifferential and Integral Equations

41 Fractional Ordinary Differential Equations In this sub-section we consider collocation method for the generalizedfractional ordinary differential equation of the form

1198631205720+[1199111199082]119906 (119909) = 120582 (119909) 119906 (119909) + 119891 (119909)

119909 isin Ω = (0 119887] (85)

119906 (0) = 1199060 (86)

Here 0 lt 120572 lt 1 119908(119909) gt 0 119911(119909) gt 0 and 119911(119909) is a monotonefunction inΩ

We assume 119906119873(119909) isin P119873119911119908 is a numerical solution of the

equation 119909119894(119894 = 0 1 119873) are the chosen interpolationpoints 119906119895119873 = 119906119873(119909119895) The following discretized equation isobtained

[119908 (119909)]minus1 119873sum119895=0

119873sum119896=0

119908(119909119895) 119906119895119873119897119896119895 ( 1198881205770119863120572

120577119875119896lowast (119911 (119909)))

= 120582 (119909) 119906119873 (119909) + 119891 (119909) (87)

Let (87) hold on collocation points 119910119895 (119895 = 1 119873) thematrix form is obtained

(M120572 minus Λ)U = 119865 (88)

Here Λ is a diagonal matrix with Λ 119894119894 = 120582(119910119894) 119865 = (1199060 119891(1199101) 119891(119910119873))119879Considering the initial condition we set S = M120572 minus Λ

with the first row replaced by (1 0 0) 1198651015840 = 119865with its firstelement replaced by 1199060 Then the solution U is obtained bysolving the matrix equation SU = 1198651015840

Example 30 Consider the following example

1198631205720+[1199111199082]119906 (119909) = (1 + 119909) 119906 (119909)

+ Γ (15119903 + 25) 119909119903minus(23)120572Γ (15119903 + 25 minus 120572)minus 119909119903 (1 + 119909) 119909 isin (0 1]

(89)

119906 (0) = 0 (90)

Here 119911(119909) = 11990923119908(119909) = 119909 119903 is an arbitrary positive numberThe exact solution of the ordinary differential equation is119906(119909) = 119909119903 Maximum absolute errors of numerical solutions

for 119903 = 6 7 and 120572 = 03 06 09 are shown in Figures 4 and 5When 119911(119909) = 11990923 119908(119909) = 119909 the scaled polynomial

spaceP119873119911119908 becomes

P119873119911119908 = 119904119901119886119899 11990921198993minus1 119899 = 1 2 119873 (91)

For 119903 = 7 the error converges exponentially and reachesmachine accuracy at 119873 = 12 It is faster than any finitedifference method while for 119903 = 6 solution convergesalgebraically as119873 increases however it still reaches machineaccuracy at 119873 = 23 The major reason for this is that 119906(119909) =1199097 isin P12

119911119908 and 119906(119909) = 1199096 notin P119873119911119908 for any119873 isin N

6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

N

10minus15

10minus10

10minus5

100

max

erro

r

=03=06=09

Figure 4 Log-log plot of the maximum error for 119903 = 6

=03=06=09

10minus15

10minus10

10minus5

100

max

erro

r

15 20 25 30 3510N

Figure 5 Semilog plot of the maximum error for 119903 = 7

Remark 31 Since space P119873119908119911 is transformed from classical

polynomial space with respect to 119911(119909) the convergenceof spectral collocation method for ODEs with generalizedfractional operators depends not only on the smoothness ofthe solution itself but also on the scale function 119911(119905)42 Hadamard-Type Integral Equations We consider thefollowing Hadamard-type boundary value problem

1Γ (120572) int119909

119886( 119904119909)

120583 (log 119909119904 )120572minus1 119891 (119904) 119889119904119904 = 119892 (119909)

119909 isin Ω = (119886 119887] (92)

International Journal of Differential Equations 11

In [42] Kilbas discussed the existence of the solution of(92) Explicit formulas for the solution 119891(119905) were establishedin the following theorem

Theorem 32 (see [42]) If 119909120583119892(119909) isin 119860119862[119886 119887] then theHadamard-type integral equation (92) with 0 lt 120572 lt 1 issolvable in 119883120583(119886 119887) and its solution may be represented in theform

119891 (119909) = 119909minus120583Γ (1 minus 120572) [119886120583119892 (119886) (log 119909119886)minus120572

+ int119909

119886(log 119909119904 )

minus120572 (119904120583119892 (119904))1015840 119889119904] (93)

Here 119860119862[119886 119887] is the set of absolutely continuous functions on[119886 119887] and119883120583(119886 119887) is space of those Lebesgue measurable func-tions 119891 on [119886 119887] for which 119909120583minus1119891(119909) is absolutely integrable

Solution of the Hadamard-type integral equation isexactly the same as generalized fractional derivative ofRiemann-Liouville type with 119911(119904) = log(119904) 119908(119904) = 119904120583 Therelationship between (93) and Caputo type generalized frac-tional derivative is

119891 (119909) = 119886120583119892 (119886) 119909minus120583Γ (1 minus 120572) (log 119909119886)minus120572 + 119863120572

119886+[1199111199082]119892 (119909) (94)

Suppose 119909119895 119895 = 0 1 119873 are interpolation points thediscretized solution of equation (92) is

119891119873 (119909) = 119886120583119892 (119886) 119909minus120583Γ (1 minus 120572) (log 119909119886)minus120572

+ 119909minus120583 119873sum119895=0

119873sum119896=0

119909120583119895119892 (119909119895) 119897119896119895( 119888120577119886119863120572

120577119875119896lowast (log (119909))

(95)

Rewriting (95) in matrix form we have

119865 = 119866119886 +M120572119866 (96)

where M120572 is generalized fractional differential matrix119866 = (119892(1199090) 119892(1199091) 119892(119909119873))119879 119865 = (119891119873(1199090) 119891119873(1199091) 119891119873(119909119873))119879 119866119886 is a vector about the initial condition of theintegral equation defined by

119866119886 = 119886120583119892 (119886)Γ (1 minus 120572) (119909minus1205830 (log 1199090119886 )minus120572 119909minus1205831 (log 1199091119886 )

minus120572 119909minus120583119873 (log 119909119873119886 )minus120572)119879

(97)

Example 33 Assume 119892(119909) = sin(119909 minus 1) 120583 = 13 Ω = [110] Solutions of (92) for 120572 = 03 06 09 are shown inFigure 6

2 4 6 8 10 120X

minus8

minus6

minus4

minus2

0

2

4

6

8

f(X)

=03=06=09

Figure 6 Solution of (92)

43 Erdelyi-Kober Fractional Diffusion Equation In this sub-section we consider the following Erdelyi-Kober fractionaldiffusion equation [12]

119906 (119909 119905) = 1199060 (119909) + 1Γ (120573)sdot int119905

0

120572120573119904120572120573minus1 (119905120572120573 minus 119904120572120573)120573minus1 12059721205971199092 119906 (119909 119904) d119904(98)

Erdelyi-Kober fractional diffusion equation which is alsocalled stretched time-fractional diffusion equation is themaster equation of a kind of generalized grey Brownianmotion (ggBm) The ggBm is a parametric class of stochasticprocesses that provides models for both fast and slow anoma-lous diffusion This class is made up of self-similar processes119861120572120573(119905)with stationary increments and it depends on two realparameters 0 lt 120572 le 2 and 0 lt 120573 le 1 It includes thefractional Brownian motion when 0 lt 120572 le 2 and 120573 = 1 thetime-fractional diffusion stochastic processes when 0 lt 120572 =120573 lt 1 and the standard Brownian motion when 120572 = 120573 = 1About the relationship between stochastic process119861120572120573(119905) andstretched time-fractional diffusion equation the followingproposition is presented in [12]

Proposition 34 The marginal probability density function119891120572120573(119909 119905) of the process 119861120572120573(119905) 119905 ge 0 is the fundamentalsolution of the stretched time-fractional diffusion equation

119906 (119909 119905) = 1199060 (119909) + 1Γ (120573)sdot int119905

0

120572120573119904120572120573minus1 (119905120572120573 minus 119904120572120573)120573minus1 12059721205971199092 119906 (119909 119904) d119904(99)

12 International Journal of Differential Equations

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 7 Standard Brownian motion 120572 = 1 120573 = 1

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 8 Time-fractional diffusion with 120572 = 06 120573 = 06

Recalling the definition of generalized fractional integraland setting 119911(119905) = 119905120572120573 119908(119905) = 1 the equation can be rewrit-ten as

119906 (119909 119905) = 1199060 (119909) + 1198681205730+[119911119908]119906119909119909 (119909 119905) (100)

We use collocation method for both space and timediscretization We choose Legendre-Gauss-Lobatto (L-G-L)points 119909119894 (119894 = 0 1 119872) as the space collocation points andchoose 119905119895 (119895 = 0 1 119873) such that 119911(119905119895) are L-G-L pointsas the time collocation points

Define space collocation matrix M2 such that M2119894119895 =(11988921198891199092)L119895(119909119894) and generalized fractional integral matrix

M120573 Matrix M120573 is computed through Theorem 28 and thespace-time collocation matrices are obtained using Kro-necker product 1015840otimes1015840 Suppose A and B are space-timecollocation matrices with dimension (119872 + 1)(119873 + 1) times (119872 +1)(119873 + 1) for the second order derivative and fractionalintegral of order 120573 separately Then

A = M2 otimesI119873+1

B = I119872+1 otimesM120573 (101)

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 9 Fractional Brownian motion with 120572 = 15 120573 = 1

Suppose 119906119873(119909 119905) is the numerical solution of (98) defin-ing 119906119873119894119895 = 119906119873(119909119894 119905119895) solution vectorU and initial vector 1198800such that

U = [11990611987300 11990611987301 1199061198730119873 11990611987310 1199061198731119873 1199061198731198720 119906119873119872119873]119879

1198800 = [1199060 (1199090) 1199060 (1199090) 1199060 (1199091) 1199060 (1199091) 1199060 (119909119872) 1199060 (119909119872)]119879

(102)

where in the definition of 1198800 each 1199060(119909119895) is repeated 119873 + 1times

The matrix form discretized equation of (98) is obtainedas

U = 1198800 +BAU (103)

In the discretized equation initial condition is explicitlyinvolved After boundary condition added properly numeri-cal solution can be obtained by solving the matrix equation

Example 35 Assume 119909 isin Ω = (minus1 1) 119905 isin (0 05] 1199060(119909) =119890minus101199092 minus 119890minus10 119906(sdot 119905)|120597Ω = 0 Numerical solutions with 119872 =119873 = 50 are shown in Figures 7ndash10

Erdelyi-Kober diffusion equation characterizes the mar-ginal density function of the process 119861120572120573(119905) 119905 ge 0 When120572 = 120573 = 1 we recover the standard diffusion equationWhen0 lt 120572 = 120573 lt 1 we get the time-fractional diffusion equationof order 120573 When 120573 = 1 and 0 lt 120572 lt 2 we have the equationof the fractional Brownian motion marginal density

As shown in Figures 7 and 8 when 1 lt 120572 lt 2 the diffu-sion is fast and the increments exhibit long-range depen-dence when 0 lt 120572 lt 1 the diffusion is slow and theincrements form a stationary process which does not exhibitlong-range dependenceThe results coincide with theoreticalanalysis in [12 14]

International Journal of Differential Equations 13

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 10 Fractional Brownian motion with 120572 = 05 120573 = 1

5 Conclusion

In this paper we propose a spectral collocation method fordifferential and integral equations with generalized fractionaloperators To deal with the difficulty in designing spectralapproximation scheme due to complexity of integral kerneland weight a variable transform technique is applied to thegeneralized fractional operator and a spectral approximationmethod is proposed for the generalized fractional operatorOperational matrices for generalized fractional operatorsare derived Spectral collocation methods are designed forfractional ordinary differential equations Hadamard-typeintegral equations and Erdelyi-Kober diffusion equationsseparately Numerical experiments are carried out to verifythe accuracy and efficiency of the method and characteristicsof the Erdelyi-Kober diffusion equation are analyzed basedon numerical results

Data Availability

(i) The programs used to support the findings of this studyhave been deposited in the GitHub repository (httpsgithubcomqinwuxuSpectralGFPDE ) (ii) No data were used tosupport this study

Disclosure

An earlier version of this workwas presented at the ldquo8th Inter-national Congress on Industrial and Applied Mathematics(ICIAM 2015)rdquo

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The first author is supported by the National Key RampD Pro-gram of China (No 2017YFC0209804) the National ScienceFoundation for Young Scientists of China (No 11701273) andYouth Foundation of Jiangsu Province (No BK20170628)

The second author is supported by the National Key RampDProgram of China (No 2017YFB0305601)

References

[1] B BaeumerDA BensonMMMeerschaert and SWWheat-craft ldquoSubordinated advection-dispersion equation for contam-inant transportrdquo Water Resources Research vol 37 no 6 pp1543ndash1550 2001

[2] E Barkai RMetzler and J Klafter ldquoFrom continuous time ran-dom walks to the fractional Fokker-Planck equationrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 61no 1 pp 132ndash138 2000

[3] A Blumen G Zumofen and J Klafter ldquoTransport aspects inanomalous diffusion Levy walksrdquo Physical Review A AtomicMolecular and Optical Physics vol 40 no 7 pp 3964ndash39731989

[4] J P Bouchaud and A Georges ldquoAnomalous diffusion in dis-ordered media statistical mechanisms models and physicalapplicationsrdquoPhysics Reports vol 195 no 4-5 pp 127ndash293 1990

[5] M Raberto E Scalas and F Mainardi ldquoWaiting-times andreturns in high-frequency financial data an empirical studyrdquoPhysica A Statistical Mechanics and its Applications vol 314 no1ndash4 pp 749ndash755 2002

[6] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations New York NYUSA Elsevier 2006

[7] S G Samko A A Kilbas and O I Marichev Fractional Inte-grals and Derivatives Theory and Applications Gordon andBreach Yverdon Switzerland 1993

[8] V Kiryakova ldquoA brief story about the operators of the general-ized fractional calculusrdquo Fractional CalculusampAppliedAnalysisAn International Journal forTheory and Applications vol 11 no2 pp 203ndash220 2008

[9] S L Kalla ldquoOn operators of fractional integration Irdquo Mathe-maticae Notae vol 22 pp 89ndash93 197071

[10] S L Kalla ldquoOn operators of fractional integration IIrdquo Mathe-maticae Notae vol 25 pp 29ndash35 1976

[11] V S Kiryakova Generalized Fractional Calculus and Applica-tions Long-man amp J Wiley Harlow New York NY USA 1994

[12] A Mura and F Mainardi ldquoA class of self-similar stochasticprocesses with stationary increments to model anomalousdiffusion in physicsrdquo Integral Transforms and Special Functionsvol 20 no 3-4 pp 185ndash198 2009

[13] O P Agrawal ldquoSome generalized fractional calculus operatorsand their applications in integral equationsrdquo Fractional Calculusand Applied Analysis An International Journal for Theory andApplications vol 15 no 4 pp 700ndash711 2012

[14] G Pagnini ldquoErdelyi-Kober fractional diffusionrdquo FractionalCalculus and Applied Analysis An International Journal forTheory and Applications vol 15 no 1 pp 117ndash127 2012

[15] A Mura and G Pagnini ldquoCharacterizations and simulations ofa class of stochastic processes to model anomalous diffusionrdquoJournal of Physics A Mathematical and General vol 41 no 28285003 22 pages 2008

[16] E K Lenzi L R Evangelista M K Lenzi H V Ribeiro and EC de Oliveira ldquoSolutions for a non-Markovian diffusion equa-tionrdquo Physics Letters A vol 374 no 41 pp 4193ndash4198 2010

[17] B Al-Saqabi and V S Kiryakova ldquoExplicit solutions of frac-tional integral and differential equations involving Erderyi-Kober operatorsrdquo Applied Mathematics and Computation vol95 no 1 pp 1ndash13 1998

14 International Journal of Differential Equations

[18] L A Hanna and Y F Luchko ldquoOperational calculus for theCaputo-type fractional Erdelyi-Kober derivative and its appli-cationsrdquo Integral Transforms and Special Functions vol 25 no5 pp 359ndash373 2014

[19] Y Xu Z He and Q Xu ldquoNumerical solutions of fractionaladvection-diffusion equations with a kind of new generalizedfractional derivativerdquo International Journal of Computer Math-ematics vol 91 no 3 pp 588ndash600 2014

[20] Y Xu Z He and O P Agrawal ldquoNumerical and analytical solu-tions of new generalized fractional diffusion equationrdquo Com-puters amp Mathematics with Applications vol 66 no 10 pp2019ndash2029 2013

[21] Y Xu and O P Agrawal ldquoNumerical solutions and analysisof diffusion for new generalized fractional Burgers equationrdquoFractional Calculus and Applied Analysis An International Jour-nal forTheory and Applications vol 16 no 3 pp 709ndash736 2013

[22] X Li and C Xu ldquoA space-time spectral method for the timefractional diffusion equationrdquo SIAM Journal on NumericalAnalysis vol 47 no 3 pp 2108ndash2131 2009

[23] Q Xu and J S Hesthaven ldquoStable multi-domain spectralpenalty methods for fractional partial differential equationsrdquoJournal of Computational Physics vol 257 pp 241ndash258 2014

[24] RMittal and S Pandit ldquoQuasilinearized Scale-3 Haar wavelets-based algorithm for numerical simulation of fractional dynam-ical systemsrdquo Engineering Computations vol 35 no 5 pp 1907ndash1931 2018

[25] C Li F Zeng and F Liu ldquoSpectral approximations to the frac-tional integral and derivativerdquo Fractional Calculus and AppliedAnalysis vol 15 no 3 pp 383ndash406 2012

[26] M Zheng F Liu I Turner and V Anh ldquoA novel high orderspace-time spectral method for the time fractional Fokker-Planck equationrdquo SIAM Journal on Scientific Computing vol 37no 2 pp A701ndashA724 2015

[27] F Zeng F Liu C Li K Burrage I Turner and V Anh ldquoACrank-Nicolson ADI spectral method for a two-dimensionalRiesz space fractional nonlinear reaction-diffusion equationrdquoSIAM Journal on Numerical Analysis vol 52 no 6 pp 2599ndash2622 2014

[28] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical ModellingSimulation andComputation for Engineering and EnvironmentalSystems vol 35 no 12 pp 5662ndash5672 2011

[29] M Zayernouri and G E Karniadakis ldquoExponentially accuratespectral and spectral element methods for fractional ODEsrdquoJournal of Computational Physics vol 257 pp 460ndash480 2014

[30] X Zhao and Z Zhang ldquoSuperconvergence points of fractionalspectral interpolationrdquo SIAM Journal on Scientific Computingvol 38 no 1 pp A598ndashA613 2016

[31] F Chen Q Xu and J S Hesthaven ldquoA multi-domain spectralmethod for time-fractional differential equationsrdquo Journal ofComputational Physics vol 293 pp 157ndash172 2015

[32] Z Mao and J Shen ldquoEfficient spectral-Galerkin methods forfractional partial differential equations with variable coeffi-cientsrdquo Journal of Computational Physics vol 307 pp 243ndash2612016

[33] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[34] J Hadamard ldquoEssai sur lrsquoetude des fonctions donnees par leurdeveloppement de Taylorrdquo Journal de Mathematiques Pures etAppliquees vol 4 pp 101ndash186 1892

[35] A Erdelyi andHKober ldquoSome remarks onHankel transformsrdquoQuarterly Journal of Mathematics vol 11 pp 212ndash221 1940

[36] I Dimovski ldquoOperational calculus for a class of differentialoperatorsrdquo Comptes Rendus De L Academie Bulgare Des Sci-ences vol 19 pp 1111ndash1114 1966

[37] S B Yakubovich and Y F LuchkoThe hypergeometric approachto integral transforms and convolutions vol 287 ofMathematicsand its Applications Kluwer Academic Publishers Dordrecht-Boston-London 1994

[38] Y Luchko ldquoOperational rules for a mixed operator of theErdelyi-Kober typerdquo Fractional Calculus and Applied Analysisvol 7 no 3 pp 339ndash364 2004

[39] G Szego Orthogonal polynomials American MathematicalSociety Providence 1992

[40] M Zayernouri and G E Karniadakis ldquoFractional Sturm-Liou-ville eigen-problems theory and numerical approximationrdquoJournal of Computational Physics vol 252 pp 495ndash517 2013

[41] S Chen J Shen and L-L Wang ldquoGeneralized Jacobi functionsand their applications to fractional differential equationsrdquoMathematics of Computation vol 85 no 300 pp 1603ndash16382016

[42] A A Kilbas ldquoHadamard-type integral equations and fractionalcalculus operatorsrdquo in Singular integral operators factorizationand applications vol 142 ofOperTheory Adv Appl pp 175ndash188Birkhauser Basel 2003

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Submit your manuscripts atwwwhindawicom

International Journal of Differential Equations 9

10minus16

10minus14

10minus12

10minus10

10minus8

10minus6

10minus4

max

imum

erro

r

w(x)=1=1w(x)=1=06

w(x)=x=1w(x)=x=06

10 12 14 16 18 20 228x

Figure 3 Approximation error of Hadamard fractional integral fordifferent weight 119908 and 120572

U(120572) = [119868minus120572119886+[119911119908]119906 (1199100) 119868minus120572119886+[119911119908]119906 (1199101) 119868minus120572119886+[119911119908]119906 (119910119873)]119879 120572 lt 0 (71)

We define a generalized fractional differentialintegralmatrixM120572 such that

M120572119894119895 = 119873sum

119896=0

119908(119909119895) 119897119896119895119908 (119910119894) ( 1198881205770119863120572

120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119910119894))

for 120572 gt 0(72)

M120572119894119895 = 119873sum

119896=0

119908(119909119895) 119897119896119895119908 (119910119894) ( 1205770119868120572120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119910119894))

for 120572 lt 0(73)

In order to compute the fractional matrix M120572 moreefficiently we define a few more matrices L and L120572 are(119873 + 1) times (119873 + 1) matrices such thatL119894119895 = L119895(119911(119910119894))L120572

119894119895 =1205770119863120572

120577L119895(119911)|119911=119911(119910119894) 119894 119895 = 0 1 119873 V is defined based on

values of Legendre polynomials at interpolation points 119911(119909119894)such that

V119894119895 = 119875119895lowast (119911 (119909119894)) 119894 119895 = 0 1 119873 (74)

D120572 is defined as fractional derivativeintegral of Legendrepolynomials at collocation points 120577119894 = 119911(119910119894)

D120572119894119895 = 119888

1205770119863120572

120577119875119895lowast (120577119894) 120572 gt 0 (75)

D120572119894119895 = 1205770

119868120572120577119875119895lowast (120577119894) 120572 lt 0 (76)

W119897 andW119903 are the weight matrices defined as follows

W119903119894119895 =

119908(119909119895) 119894 = 1198950 otherwise (77)

W119897119894119895 =

1119908(119910119895) 119894 = 119895 gt 10 otherwise (78)

Theorem 28 For 119906(119909) isin P119873119911119908 vectors defined in (69)ndash(71)

and matrices defined in (72)ndash(78) the following relation holds

U(120572) = M

120572U = (W119897D120572V

minus1W

119903)U (79)

Proof From the definition (69)ndash(71) (72)ndash(78) and (63)-(64) it is easy to obtain

U(120572) = M

120572U (80)

Next we prove thatM120572 = W119897D120572Vminus1W119903From the definition of the interpolation function L119895(119911)

119875119895lowast (119911) = 119873sum119894=0

119875119895lowast (119911 (119909119894)) L119894 (119911) (81)

Suppose 997888rarr119875(119911) = (1198750lowast(119911) 1198751lowast(119911) 119875119873lowast(119911)) 997888rarrL (119911) =(L0(119911) L1(119911) L119873(119911)) for 120572 gt 0 we have997888rarr119875 (119911) = 997888rarrL (119911)V

1205770119863120572

120577

997888rarr119875 (119911) = ( 1205770119863120572

120577

997888rarrL (119911))V (82)

Evaluating the matrices multiplication for each elementthe following relation is derived

(W119897D120572V

minus1W119903)119894119895 = (W119897L

120572W119903)119894119895

= 119908 (119909119895)119908 (119910119894) L(120572)119895 (119911 (119910119894))(83)

= 119873sum119896=0

119908(119909119895) 119897119896119895119908 (119910119894) ( 1205770119863120572

120577119875119896lowast (120577)1003816100381610038161003816120577=119911(119910119894)) = M

120572119894119895 (84)

The case 120572 lt 0 can be proved similarly

Remark 29 Collocation points 119910119895 and the interpolationpoints 119909119894 are not necessary the same To obtain a goodapproximation interpolation points ofGauss-type are usuallyused At the same time collocation points should be chosenproperly to guarantee stability properties of the methodIn the following numerical examples for computation andstability aim both interpolation and collocation points arechosen based on Gauss-type points with respect to 119911(119909)

10 International Journal of Differential Equations

4 Collocation Methods for FractionalDifferential and Integral Equations

41 Fractional Ordinary Differential Equations In this sub-section we consider collocation method for the generalizedfractional ordinary differential equation of the form

1198631205720+[1199111199082]119906 (119909) = 120582 (119909) 119906 (119909) + 119891 (119909)

119909 isin Ω = (0 119887] (85)

119906 (0) = 1199060 (86)

Here 0 lt 120572 lt 1 119908(119909) gt 0 119911(119909) gt 0 and 119911(119909) is a monotonefunction inΩ

We assume 119906119873(119909) isin P119873119911119908 is a numerical solution of the

equation 119909119894(119894 = 0 1 119873) are the chosen interpolationpoints 119906119895119873 = 119906119873(119909119895) The following discretized equation isobtained

[119908 (119909)]minus1 119873sum119895=0

119873sum119896=0

119908(119909119895) 119906119895119873119897119896119895 ( 1198881205770119863120572

120577119875119896lowast (119911 (119909)))

= 120582 (119909) 119906119873 (119909) + 119891 (119909) (87)

Let (87) hold on collocation points 119910119895 (119895 = 1 119873) thematrix form is obtained

(M120572 minus Λ)U = 119865 (88)

Here Λ is a diagonal matrix with Λ 119894119894 = 120582(119910119894) 119865 = (1199060 119891(1199101) 119891(119910119873))119879Considering the initial condition we set S = M120572 minus Λ

with the first row replaced by (1 0 0) 1198651015840 = 119865with its firstelement replaced by 1199060 Then the solution U is obtained bysolving the matrix equation SU = 1198651015840

Example 30 Consider the following example

1198631205720+[1199111199082]119906 (119909) = (1 + 119909) 119906 (119909)

+ Γ (15119903 + 25) 119909119903minus(23)120572Γ (15119903 + 25 minus 120572)minus 119909119903 (1 + 119909) 119909 isin (0 1]

(89)

119906 (0) = 0 (90)

Here 119911(119909) = 11990923119908(119909) = 119909 119903 is an arbitrary positive numberThe exact solution of the ordinary differential equation is119906(119909) = 119909119903 Maximum absolute errors of numerical solutions

for 119903 = 6 7 and 120572 = 03 06 09 are shown in Figures 4 and 5When 119911(119909) = 11990923 119908(119909) = 119909 the scaled polynomial

spaceP119873119911119908 becomes

P119873119911119908 = 119904119901119886119899 11990921198993minus1 119899 = 1 2 119873 (91)

For 119903 = 7 the error converges exponentially and reachesmachine accuracy at 119873 = 12 It is faster than any finitedifference method while for 119903 = 6 solution convergesalgebraically as119873 increases however it still reaches machineaccuracy at 119873 = 23 The major reason for this is that 119906(119909) =1199097 isin P12

119911119908 and 119906(119909) = 1199096 notin P119873119911119908 for any119873 isin N

6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

N

10minus15

10minus10

10minus5

100

max

erro

r

=03=06=09

Figure 4 Log-log plot of the maximum error for 119903 = 6

=03=06=09

10minus15

10minus10

10minus5

100

max

erro

r

15 20 25 30 3510N

Figure 5 Semilog plot of the maximum error for 119903 = 7

Remark 31 Since space P119873119908119911 is transformed from classical

polynomial space with respect to 119911(119909) the convergenceof spectral collocation method for ODEs with generalizedfractional operators depends not only on the smoothness ofthe solution itself but also on the scale function 119911(119905)42 Hadamard-Type Integral Equations We consider thefollowing Hadamard-type boundary value problem

1Γ (120572) int119909

119886( 119904119909)

120583 (log 119909119904 )120572minus1 119891 (119904) 119889119904119904 = 119892 (119909)

119909 isin Ω = (119886 119887] (92)

International Journal of Differential Equations 11

In [42] Kilbas discussed the existence of the solution of(92) Explicit formulas for the solution 119891(119905) were establishedin the following theorem

Theorem 32 (see [42]) If 119909120583119892(119909) isin 119860119862[119886 119887] then theHadamard-type integral equation (92) with 0 lt 120572 lt 1 issolvable in 119883120583(119886 119887) and its solution may be represented in theform

119891 (119909) = 119909minus120583Γ (1 minus 120572) [119886120583119892 (119886) (log 119909119886)minus120572

+ int119909

119886(log 119909119904 )

minus120572 (119904120583119892 (119904))1015840 119889119904] (93)

Here 119860119862[119886 119887] is the set of absolutely continuous functions on[119886 119887] and119883120583(119886 119887) is space of those Lebesgue measurable func-tions 119891 on [119886 119887] for which 119909120583minus1119891(119909) is absolutely integrable

Solution of the Hadamard-type integral equation isexactly the same as generalized fractional derivative ofRiemann-Liouville type with 119911(119904) = log(119904) 119908(119904) = 119904120583 Therelationship between (93) and Caputo type generalized frac-tional derivative is

119891 (119909) = 119886120583119892 (119886) 119909minus120583Γ (1 minus 120572) (log 119909119886)minus120572 + 119863120572

119886+[1199111199082]119892 (119909) (94)

Suppose 119909119895 119895 = 0 1 119873 are interpolation points thediscretized solution of equation (92) is

119891119873 (119909) = 119886120583119892 (119886) 119909minus120583Γ (1 minus 120572) (log 119909119886)minus120572

+ 119909minus120583 119873sum119895=0

119873sum119896=0

119909120583119895119892 (119909119895) 119897119896119895( 119888120577119886119863120572

120577119875119896lowast (log (119909))

(95)

Rewriting (95) in matrix form we have

119865 = 119866119886 +M120572119866 (96)

where M120572 is generalized fractional differential matrix119866 = (119892(1199090) 119892(1199091) 119892(119909119873))119879 119865 = (119891119873(1199090) 119891119873(1199091) 119891119873(119909119873))119879 119866119886 is a vector about the initial condition of theintegral equation defined by

119866119886 = 119886120583119892 (119886)Γ (1 minus 120572) (119909minus1205830 (log 1199090119886 )minus120572 119909minus1205831 (log 1199091119886 )

minus120572 119909minus120583119873 (log 119909119873119886 )minus120572)119879

(97)

Example 33 Assume 119892(119909) = sin(119909 minus 1) 120583 = 13 Ω = [110] Solutions of (92) for 120572 = 03 06 09 are shown inFigure 6

2 4 6 8 10 120X

minus8

minus6

minus4

minus2

0

2

4

6

8

f(X)

=03=06=09

Figure 6 Solution of (92)

43 Erdelyi-Kober Fractional Diffusion Equation In this sub-section we consider the following Erdelyi-Kober fractionaldiffusion equation [12]

119906 (119909 119905) = 1199060 (119909) + 1Γ (120573)sdot int119905

0

120572120573119904120572120573minus1 (119905120572120573 minus 119904120572120573)120573minus1 12059721205971199092 119906 (119909 119904) d119904(98)

Erdelyi-Kober fractional diffusion equation which is alsocalled stretched time-fractional diffusion equation is themaster equation of a kind of generalized grey Brownianmotion (ggBm) The ggBm is a parametric class of stochasticprocesses that provides models for both fast and slow anoma-lous diffusion This class is made up of self-similar processes119861120572120573(119905)with stationary increments and it depends on two realparameters 0 lt 120572 le 2 and 0 lt 120573 le 1 It includes thefractional Brownian motion when 0 lt 120572 le 2 and 120573 = 1 thetime-fractional diffusion stochastic processes when 0 lt 120572 =120573 lt 1 and the standard Brownian motion when 120572 = 120573 = 1About the relationship between stochastic process119861120572120573(119905) andstretched time-fractional diffusion equation the followingproposition is presented in [12]

Proposition 34 The marginal probability density function119891120572120573(119909 119905) of the process 119861120572120573(119905) 119905 ge 0 is the fundamentalsolution of the stretched time-fractional diffusion equation

119906 (119909 119905) = 1199060 (119909) + 1Γ (120573)sdot int119905

0

120572120573119904120572120573minus1 (119905120572120573 minus 119904120572120573)120573minus1 12059721205971199092 119906 (119909 119904) d119904(99)

12 International Journal of Differential Equations

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 7 Standard Brownian motion 120572 = 1 120573 = 1

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 8 Time-fractional diffusion with 120572 = 06 120573 = 06

Recalling the definition of generalized fractional integraland setting 119911(119905) = 119905120572120573 119908(119905) = 1 the equation can be rewrit-ten as

119906 (119909 119905) = 1199060 (119909) + 1198681205730+[119911119908]119906119909119909 (119909 119905) (100)

We use collocation method for both space and timediscretization We choose Legendre-Gauss-Lobatto (L-G-L)points 119909119894 (119894 = 0 1 119872) as the space collocation points andchoose 119905119895 (119895 = 0 1 119873) such that 119911(119905119895) are L-G-L pointsas the time collocation points

Define space collocation matrix M2 such that M2119894119895 =(11988921198891199092)L119895(119909119894) and generalized fractional integral matrix

M120573 Matrix M120573 is computed through Theorem 28 and thespace-time collocation matrices are obtained using Kro-necker product 1015840otimes1015840 Suppose A and B are space-timecollocation matrices with dimension (119872 + 1)(119873 + 1) times (119872 +1)(119873 + 1) for the second order derivative and fractionalintegral of order 120573 separately Then

A = M2 otimesI119873+1

B = I119872+1 otimesM120573 (101)

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 9 Fractional Brownian motion with 120572 = 15 120573 = 1

Suppose 119906119873(119909 119905) is the numerical solution of (98) defin-ing 119906119873119894119895 = 119906119873(119909119894 119905119895) solution vectorU and initial vector 1198800such that

U = [11990611987300 11990611987301 1199061198730119873 11990611987310 1199061198731119873 1199061198731198720 119906119873119872119873]119879

1198800 = [1199060 (1199090) 1199060 (1199090) 1199060 (1199091) 1199060 (1199091) 1199060 (119909119872) 1199060 (119909119872)]119879

(102)

where in the definition of 1198800 each 1199060(119909119895) is repeated 119873 + 1times

The matrix form discretized equation of (98) is obtainedas

U = 1198800 +BAU (103)

In the discretized equation initial condition is explicitlyinvolved After boundary condition added properly numeri-cal solution can be obtained by solving the matrix equation

Example 35 Assume 119909 isin Ω = (minus1 1) 119905 isin (0 05] 1199060(119909) =119890minus101199092 minus 119890minus10 119906(sdot 119905)|120597Ω = 0 Numerical solutions with 119872 =119873 = 50 are shown in Figures 7ndash10

Erdelyi-Kober diffusion equation characterizes the mar-ginal density function of the process 119861120572120573(119905) 119905 ge 0 When120572 = 120573 = 1 we recover the standard diffusion equationWhen0 lt 120572 = 120573 lt 1 we get the time-fractional diffusion equationof order 120573 When 120573 = 1 and 0 lt 120572 lt 2 we have the equationof the fractional Brownian motion marginal density

As shown in Figures 7 and 8 when 1 lt 120572 lt 2 the diffu-sion is fast and the increments exhibit long-range depen-dence when 0 lt 120572 lt 1 the diffusion is slow and theincrements form a stationary process which does not exhibitlong-range dependenceThe results coincide with theoreticalanalysis in [12 14]

International Journal of Differential Equations 13

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 10 Fractional Brownian motion with 120572 = 05 120573 = 1

5 Conclusion

In this paper we propose a spectral collocation method fordifferential and integral equations with generalized fractionaloperators To deal with the difficulty in designing spectralapproximation scheme due to complexity of integral kerneland weight a variable transform technique is applied to thegeneralized fractional operator and a spectral approximationmethod is proposed for the generalized fractional operatorOperational matrices for generalized fractional operatorsare derived Spectral collocation methods are designed forfractional ordinary differential equations Hadamard-typeintegral equations and Erdelyi-Kober diffusion equationsseparately Numerical experiments are carried out to verifythe accuracy and efficiency of the method and characteristicsof the Erdelyi-Kober diffusion equation are analyzed basedon numerical results

Data Availability

(i) The programs used to support the findings of this studyhave been deposited in the GitHub repository (httpsgithubcomqinwuxuSpectralGFPDE ) (ii) No data were used tosupport this study

Disclosure

An earlier version of this workwas presented at the ldquo8th Inter-national Congress on Industrial and Applied Mathematics(ICIAM 2015)rdquo

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The first author is supported by the National Key RampD Pro-gram of China (No 2017YFC0209804) the National ScienceFoundation for Young Scientists of China (No 11701273) andYouth Foundation of Jiangsu Province (No BK20170628)

The second author is supported by the National Key RampDProgram of China (No 2017YFB0305601)

References

[1] B BaeumerDA BensonMMMeerschaert and SWWheat-craft ldquoSubordinated advection-dispersion equation for contam-inant transportrdquo Water Resources Research vol 37 no 6 pp1543ndash1550 2001

[2] E Barkai RMetzler and J Klafter ldquoFrom continuous time ran-dom walks to the fractional Fokker-Planck equationrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 61no 1 pp 132ndash138 2000

[3] A Blumen G Zumofen and J Klafter ldquoTransport aspects inanomalous diffusion Levy walksrdquo Physical Review A AtomicMolecular and Optical Physics vol 40 no 7 pp 3964ndash39731989

[4] J P Bouchaud and A Georges ldquoAnomalous diffusion in dis-ordered media statistical mechanisms models and physicalapplicationsrdquoPhysics Reports vol 195 no 4-5 pp 127ndash293 1990

[5] M Raberto E Scalas and F Mainardi ldquoWaiting-times andreturns in high-frequency financial data an empirical studyrdquoPhysica A Statistical Mechanics and its Applications vol 314 no1ndash4 pp 749ndash755 2002

[6] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations New York NYUSA Elsevier 2006

[7] S G Samko A A Kilbas and O I Marichev Fractional Inte-grals and Derivatives Theory and Applications Gordon andBreach Yverdon Switzerland 1993

[8] V Kiryakova ldquoA brief story about the operators of the general-ized fractional calculusrdquo Fractional CalculusampAppliedAnalysisAn International Journal forTheory and Applications vol 11 no2 pp 203ndash220 2008

[9] S L Kalla ldquoOn operators of fractional integration Irdquo Mathe-maticae Notae vol 22 pp 89ndash93 197071

[10] S L Kalla ldquoOn operators of fractional integration IIrdquo Mathe-maticae Notae vol 25 pp 29ndash35 1976

[11] V S Kiryakova Generalized Fractional Calculus and Applica-tions Long-man amp J Wiley Harlow New York NY USA 1994

[12] A Mura and F Mainardi ldquoA class of self-similar stochasticprocesses with stationary increments to model anomalousdiffusion in physicsrdquo Integral Transforms and Special Functionsvol 20 no 3-4 pp 185ndash198 2009

[13] O P Agrawal ldquoSome generalized fractional calculus operatorsand their applications in integral equationsrdquo Fractional Calculusand Applied Analysis An International Journal for Theory andApplications vol 15 no 4 pp 700ndash711 2012

[14] G Pagnini ldquoErdelyi-Kober fractional diffusionrdquo FractionalCalculus and Applied Analysis An International Journal forTheory and Applications vol 15 no 1 pp 117ndash127 2012

[15] A Mura and G Pagnini ldquoCharacterizations and simulations ofa class of stochastic processes to model anomalous diffusionrdquoJournal of Physics A Mathematical and General vol 41 no 28285003 22 pages 2008

[16] E K Lenzi L R Evangelista M K Lenzi H V Ribeiro and EC de Oliveira ldquoSolutions for a non-Markovian diffusion equa-tionrdquo Physics Letters A vol 374 no 41 pp 4193ndash4198 2010

[17] B Al-Saqabi and V S Kiryakova ldquoExplicit solutions of frac-tional integral and differential equations involving Erderyi-Kober operatorsrdquo Applied Mathematics and Computation vol95 no 1 pp 1ndash13 1998

14 International Journal of Differential Equations

[18] L A Hanna and Y F Luchko ldquoOperational calculus for theCaputo-type fractional Erdelyi-Kober derivative and its appli-cationsrdquo Integral Transforms and Special Functions vol 25 no5 pp 359ndash373 2014

[19] Y Xu Z He and Q Xu ldquoNumerical solutions of fractionaladvection-diffusion equations with a kind of new generalizedfractional derivativerdquo International Journal of Computer Math-ematics vol 91 no 3 pp 588ndash600 2014

[20] Y Xu Z He and O P Agrawal ldquoNumerical and analytical solu-tions of new generalized fractional diffusion equationrdquo Com-puters amp Mathematics with Applications vol 66 no 10 pp2019ndash2029 2013

[21] Y Xu and O P Agrawal ldquoNumerical solutions and analysisof diffusion for new generalized fractional Burgers equationrdquoFractional Calculus and Applied Analysis An International Jour-nal forTheory and Applications vol 16 no 3 pp 709ndash736 2013

[22] X Li and C Xu ldquoA space-time spectral method for the timefractional diffusion equationrdquo SIAM Journal on NumericalAnalysis vol 47 no 3 pp 2108ndash2131 2009

[23] Q Xu and J S Hesthaven ldquoStable multi-domain spectralpenalty methods for fractional partial differential equationsrdquoJournal of Computational Physics vol 257 pp 241ndash258 2014

[24] RMittal and S Pandit ldquoQuasilinearized Scale-3 Haar wavelets-based algorithm for numerical simulation of fractional dynam-ical systemsrdquo Engineering Computations vol 35 no 5 pp 1907ndash1931 2018

[25] C Li F Zeng and F Liu ldquoSpectral approximations to the frac-tional integral and derivativerdquo Fractional Calculus and AppliedAnalysis vol 15 no 3 pp 383ndash406 2012

[26] M Zheng F Liu I Turner and V Anh ldquoA novel high orderspace-time spectral method for the time fractional Fokker-Planck equationrdquo SIAM Journal on Scientific Computing vol 37no 2 pp A701ndashA724 2015

[27] F Zeng F Liu C Li K Burrage I Turner and V Anh ldquoACrank-Nicolson ADI spectral method for a two-dimensionalRiesz space fractional nonlinear reaction-diffusion equationrdquoSIAM Journal on Numerical Analysis vol 52 no 6 pp 2599ndash2622 2014

[28] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical ModellingSimulation andComputation for Engineering and EnvironmentalSystems vol 35 no 12 pp 5662ndash5672 2011

[29] M Zayernouri and G E Karniadakis ldquoExponentially accuratespectral and spectral element methods for fractional ODEsrdquoJournal of Computational Physics vol 257 pp 460ndash480 2014

[30] X Zhao and Z Zhang ldquoSuperconvergence points of fractionalspectral interpolationrdquo SIAM Journal on Scientific Computingvol 38 no 1 pp A598ndashA613 2016

[31] F Chen Q Xu and J S Hesthaven ldquoA multi-domain spectralmethod for time-fractional differential equationsrdquo Journal ofComputational Physics vol 293 pp 157ndash172 2015

[32] Z Mao and J Shen ldquoEfficient spectral-Galerkin methods forfractional partial differential equations with variable coeffi-cientsrdquo Journal of Computational Physics vol 307 pp 243ndash2612016

[33] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[34] J Hadamard ldquoEssai sur lrsquoetude des fonctions donnees par leurdeveloppement de Taylorrdquo Journal de Mathematiques Pures etAppliquees vol 4 pp 101ndash186 1892

[35] A Erdelyi andHKober ldquoSome remarks onHankel transformsrdquoQuarterly Journal of Mathematics vol 11 pp 212ndash221 1940

[36] I Dimovski ldquoOperational calculus for a class of differentialoperatorsrdquo Comptes Rendus De L Academie Bulgare Des Sci-ences vol 19 pp 1111ndash1114 1966

[37] S B Yakubovich and Y F LuchkoThe hypergeometric approachto integral transforms and convolutions vol 287 ofMathematicsand its Applications Kluwer Academic Publishers Dordrecht-Boston-London 1994

[38] Y Luchko ldquoOperational rules for a mixed operator of theErdelyi-Kober typerdquo Fractional Calculus and Applied Analysisvol 7 no 3 pp 339ndash364 2004

[39] G Szego Orthogonal polynomials American MathematicalSociety Providence 1992

[40] M Zayernouri and G E Karniadakis ldquoFractional Sturm-Liou-ville eigen-problems theory and numerical approximationrdquoJournal of Computational Physics vol 252 pp 495ndash517 2013

[41] S Chen J Shen and L-L Wang ldquoGeneralized Jacobi functionsand their applications to fractional differential equationsrdquoMathematics of Computation vol 85 no 300 pp 1603ndash16382016

[42] A A Kilbas ldquoHadamard-type integral equations and fractionalcalculus operatorsrdquo in Singular integral operators factorizationand applications vol 142 ofOperTheory Adv Appl pp 175ndash188Birkhauser Basel 2003

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Submit your manuscripts atwwwhindawicom

10 International Journal of Differential Equations

4 Collocation Methods for FractionalDifferential and Integral Equations

41 Fractional Ordinary Differential Equations In this sub-section we consider collocation method for the generalizedfractional ordinary differential equation of the form

1198631205720+[1199111199082]119906 (119909) = 120582 (119909) 119906 (119909) + 119891 (119909)

119909 isin Ω = (0 119887] (85)

119906 (0) = 1199060 (86)

Here 0 lt 120572 lt 1 119908(119909) gt 0 119911(119909) gt 0 and 119911(119909) is a monotonefunction inΩ

We assume 119906119873(119909) isin P119873119911119908 is a numerical solution of the

equation 119909119894(119894 = 0 1 119873) are the chosen interpolationpoints 119906119895119873 = 119906119873(119909119895) The following discretized equation isobtained

[119908 (119909)]minus1 119873sum119895=0

119873sum119896=0

119908(119909119895) 119906119895119873119897119896119895 ( 1198881205770119863120572

120577119875119896lowast (119911 (119909)))

= 120582 (119909) 119906119873 (119909) + 119891 (119909) (87)

Let (87) hold on collocation points 119910119895 (119895 = 1 119873) thematrix form is obtained

(M120572 minus Λ)U = 119865 (88)

Here Λ is a diagonal matrix with Λ 119894119894 = 120582(119910119894) 119865 = (1199060 119891(1199101) 119891(119910119873))119879Considering the initial condition we set S = M120572 minus Λ

with the first row replaced by (1 0 0) 1198651015840 = 119865with its firstelement replaced by 1199060 Then the solution U is obtained bysolving the matrix equation SU = 1198651015840

Example 30 Consider the following example

1198631205720+[1199111199082]119906 (119909) = (1 + 119909) 119906 (119909)

+ Γ (15119903 + 25) 119909119903minus(23)120572Γ (15119903 + 25 minus 120572)minus 119909119903 (1 + 119909) 119909 isin (0 1]

(89)

119906 (0) = 0 (90)

Here 119911(119909) = 11990923119908(119909) = 119909 119903 is an arbitrary positive numberThe exact solution of the ordinary differential equation is119906(119909) = 119909119903 Maximum absolute errors of numerical solutions

for 119903 = 6 7 and 120572 = 03 06 09 are shown in Figures 4 and 5When 119911(119909) = 11990923 119908(119909) = 119909 the scaled polynomial

spaceP119873119911119908 becomes

P119873119911119908 = 119904119901119886119899 11990921198993minus1 119899 = 1 2 119873 (91)

For 119903 = 7 the error converges exponentially and reachesmachine accuracy at 119873 = 12 It is faster than any finitedifference method while for 119903 = 6 solution convergesalgebraically as119873 increases however it still reaches machineaccuracy at 119873 = 23 The major reason for this is that 119906(119909) =1199097 isin P12

119911119908 and 119906(119909) = 1199096 notin P119873119911119908 for any119873 isin N

6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

N

10minus15

10minus10

10minus5

100

max

erro

r

=03=06=09

Figure 4 Log-log plot of the maximum error for 119903 = 6

=03=06=09

10minus15

10minus10

10minus5

100

max

erro

r

15 20 25 30 3510N

Figure 5 Semilog plot of the maximum error for 119903 = 7

Remark 31 Since space P119873119908119911 is transformed from classical

polynomial space with respect to 119911(119909) the convergenceof spectral collocation method for ODEs with generalizedfractional operators depends not only on the smoothness ofthe solution itself but also on the scale function 119911(119905)42 Hadamard-Type Integral Equations We consider thefollowing Hadamard-type boundary value problem

1Γ (120572) int119909

119886( 119904119909)

120583 (log 119909119904 )120572minus1 119891 (119904) 119889119904119904 = 119892 (119909)

119909 isin Ω = (119886 119887] (92)

International Journal of Differential Equations 11

In [42] Kilbas discussed the existence of the solution of(92) Explicit formulas for the solution 119891(119905) were establishedin the following theorem

Theorem 32 (see [42]) If 119909120583119892(119909) isin 119860119862[119886 119887] then theHadamard-type integral equation (92) with 0 lt 120572 lt 1 issolvable in 119883120583(119886 119887) and its solution may be represented in theform

119891 (119909) = 119909minus120583Γ (1 minus 120572) [119886120583119892 (119886) (log 119909119886)minus120572

+ int119909

119886(log 119909119904 )

minus120572 (119904120583119892 (119904))1015840 119889119904] (93)

Here 119860119862[119886 119887] is the set of absolutely continuous functions on[119886 119887] and119883120583(119886 119887) is space of those Lebesgue measurable func-tions 119891 on [119886 119887] for which 119909120583minus1119891(119909) is absolutely integrable

Solution of the Hadamard-type integral equation isexactly the same as generalized fractional derivative ofRiemann-Liouville type with 119911(119904) = log(119904) 119908(119904) = 119904120583 Therelationship between (93) and Caputo type generalized frac-tional derivative is

119891 (119909) = 119886120583119892 (119886) 119909minus120583Γ (1 minus 120572) (log 119909119886)minus120572 + 119863120572

119886+[1199111199082]119892 (119909) (94)

Suppose 119909119895 119895 = 0 1 119873 are interpolation points thediscretized solution of equation (92) is

119891119873 (119909) = 119886120583119892 (119886) 119909minus120583Γ (1 minus 120572) (log 119909119886)minus120572

+ 119909minus120583 119873sum119895=0

119873sum119896=0

119909120583119895119892 (119909119895) 119897119896119895( 119888120577119886119863120572

120577119875119896lowast (log (119909))

(95)

Rewriting (95) in matrix form we have

119865 = 119866119886 +M120572119866 (96)

where M120572 is generalized fractional differential matrix119866 = (119892(1199090) 119892(1199091) 119892(119909119873))119879 119865 = (119891119873(1199090) 119891119873(1199091) 119891119873(119909119873))119879 119866119886 is a vector about the initial condition of theintegral equation defined by

119866119886 = 119886120583119892 (119886)Γ (1 minus 120572) (119909minus1205830 (log 1199090119886 )minus120572 119909minus1205831 (log 1199091119886 )

minus120572 119909minus120583119873 (log 119909119873119886 )minus120572)119879

(97)

Example 33 Assume 119892(119909) = sin(119909 minus 1) 120583 = 13 Ω = [110] Solutions of (92) for 120572 = 03 06 09 are shown inFigure 6

2 4 6 8 10 120X

minus8

minus6

minus4

minus2

0

2

4

6

8

f(X)

=03=06=09

Figure 6 Solution of (92)

43 Erdelyi-Kober Fractional Diffusion Equation In this sub-section we consider the following Erdelyi-Kober fractionaldiffusion equation [12]

119906 (119909 119905) = 1199060 (119909) + 1Γ (120573)sdot int119905

0

120572120573119904120572120573minus1 (119905120572120573 minus 119904120572120573)120573minus1 12059721205971199092 119906 (119909 119904) d119904(98)

Erdelyi-Kober fractional diffusion equation which is alsocalled stretched time-fractional diffusion equation is themaster equation of a kind of generalized grey Brownianmotion (ggBm) The ggBm is a parametric class of stochasticprocesses that provides models for both fast and slow anoma-lous diffusion This class is made up of self-similar processes119861120572120573(119905)with stationary increments and it depends on two realparameters 0 lt 120572 le 2 and 0 lt 120573 le 1 It includes thefractional Brownian motion when 0 lt 120572 le 2 and 120573 = 1 thetime-fractional diffusion stochastic processes when 0 lt 120572 =120573 lt 1 and the standard Brownian motion when 120572 = 120573 = 1About the relationship between stochastic process119861120572120573(119905) andstretched time-fractional diffusion equation the followingproposition is presented in [12]

Proposition 34 The marginal probability density function119891120572120573(119909 119905) of the process 119861120572120573(119905) 119905 ge 0 is the fundamentalsolution of the stretched time-fractional diffusion equation

119906 (119909 119905) = 1199060 (119909) + 1Γ (120573)sdot int119905

0

120572120573119904120572120573minus1 (119905120572120573 minus 119904120572120573)120573minus1 12059721205971199092 119906 (119909 119904) d119904(99)

12 International Journal of Differential Equations

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 7 Standard Brownian motion 120572 = 1 120573 = 1

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 8 Time-fractional diffusion with 120572 = 06 120573 = 06

Recalling the definition of generalized fractional integraland setting 119911(119905) = 119905120572120573 119908(119905) = 1 the equation can be rewrit-ten as

119906 (119909 119905) = 1199060 (119909) + 1198681205730+[119911119908]119906119909119909 (119909 119905) (100)

We use collocation method for both space and timediscretization We choose Legendre-Gauss-Lobatto (L-G-L)points 119909119894 (119894 = 0 1 119872) as the space collocation points andchoose 119905119895 (119895 = 0 1 119873) such that 119911(119905119895) are L-G-L pointsas the time collocation points

Define space collocation matrix M2 such that M2119894119895 =(11988921198891199092)L119895(119909119894) and generalized fractional integral matrix

M120573 Matrix M120573 is computed through Theorem 28 and thespace-time collocation matrices are obtained using Kro-necker product 1015840otimes1015840 Suppose A and B are space-timecollocation matrices with dimension (119872 + 1)(119873 + 1) times (119872 +1)(119873 + 1) for the second order derivative and fractionalintegral of order 120573 separately Then

A = M2 otimesI119873+1

B = I119872+1 otimesM120573 (101)

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 9 Fractional Brownian motion with 120572 = 15 120573 = 1

Suppose 119906119873(119909 119905) is the numerical solution of (98) defin-ing 119906119873119894119895 = 119906119873(119909119894 119905119895) solution vectorU and initial vector 1198800such that

U = [11990611987300 11990611987301 1199061198730119873 11990611987310 1199061198731119873 1199061198731198720 119906119873119872119873]119879

1198800 = [1199060 (1199090) 1199060 (1199090) 1199060 (1199091) 1199060 (1199091) 1199060 (119909119872) 1199060 (119909119872)]119879

(102)

where in the definition of 1198800 each 1199060(119909119895) is repeated 119873 + 1times

The matrix form discretized equation of (98) is obtainedas

U = 1198800 +BAU (103)

In the discretized equation initial condition is explicitlyinvolved After boundary condition added properly numeri-cal solution can be obtained by solving the matrix equation

Example 35 Assume 119909 isin Ω = (minus1 1) 119905 isin (0 05] 1199060(119909) =119890minus101199092 minus 119890minus10 119906(sdot 119905)|120597Ω = 0 Numerical solutions with 119872 =119873 = 50 are shown in Figures 7ndash10

Erdelyi-Kober diffusion equation characterizes the mar-ginal density function of the process 119861120572120573(119905) 119905 ge 0 When120572 = 120573 = 1 we recover the standard diffusion equationWhen0 lt 120572 = 120573 lt 1 we get the time-fractional diffusion equationof order 120573 When 120573 = 1 and 0 lt 120572 lt 2 we have the equationof the fractional Brownian motion marginal density

As shown in Figures 7 and 8 when 1 lt 120572 lt 2 the diffu-sion is fast and the increments exhibit long-range depen-dence when 0 lt 120572 lt 1 the diffusion is slow and theincrements form a stationary process which does not exhibitlong-range dependenceThe results coincide with theoreticalanalysis in [12 14]

International Journal of Differential Equations 13

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 10 Fractional Brownian motion with 120572 = 05 120573 = 1

5 Conclusion

In this paper we propose a spectral collocation method fordifferential and integral equations with generalized fractionaloperators To deal with the difficulty in designing spectralapproximation scheme due to complexity of integral kerneland weight a variable transform technique is applied to thegeneralized fractional operator and a spectral approximationmethod is proposed for the generalized fractional operatorOperational matrices for generalized fractional operatorsare derived Spectral collocation methods are designed forfractional ordinary differential equations Hadamard-typeintegral equations and Erdelyi-Kober diffusion equationsseparately Numerical experiments are carried out to verifythe accuracy and efficiency of the method and characteristicsof the Erdelyi-Kober diffusion equation are analyzed basedon numerical results

Data Availability

(i) The programs used to support the findings of this studyhave been deposited in the GitHub repository (httpsgithubcomqinwuxuSpectralGFPDE ) (ii) No data were used tosupport this study

Disclosure

An earlier version of this workwas presented at the ldquo8th Inter-national Congress on Industrial and Applied Mathematics(ICIAM 2015)rdquo

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The first author is supported by the National Key RampD Pro-gram of China (No 2017YFC0209804) the National ScienceFoundation for Young Scientists of China (No 11701273) andYouth Foundation of Jiangsu Province (No BK20170628)

The second author is supported by the National Key RampDProgram of China (No 2017YFB0305601)

References

[1] B BaeumerDA BensonMMMeerschaert and SWWheat-craft ldquoSubordinated advection-dispersion equation for contam-inant transportrdquo Water Resources Research vol 37 no 6 pp1543ndash1550 2001

[2] E Barkai RMetzler and J Klafter ldquoFrom continuous time ran-dom walks to the fractional Fokker-Planck equationrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 61no 1 pp 132ndash138 2000

[3] A Blumen G Zumofen and J Klafter ldquoTransport aspects inanomalous diffusion Levy walksrdquo Physical Review A AtomicMolecular and Optical Physics vol 40 no 7 pp 3964ndash39731989

[4] J P Bouchaud and A Georges ldquoAnomalous diffusion in dis-ordered media statistical mechanisms models and physicalapplicationsrdquoPhysics Reports vol 195 no 4-5 pp 127ndash293 1990

[5] M Raberto E Scalas and F Mainardi ldquoWaiting-times andreturns in high-frequency financial data an empirical studyrdquoPhysica A Statistical Mechanics and its Applications vol 314 no1ndash4 pp 749ndash755 2002

[6] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations New York NYUSA Elsevier 2006

[7] S G Samko A A Kilbas and O I Marichev Fractional Inte-grals and Derivatives Theory and Applications Gordon andBreach Yverdon Switzerland 1993

[8] V Kiryakova ldquoA brief story about the operators of the general-ized fractional calculusrdquo Fractional CalculusampAppliedAnalysisAn International Journal forTheory and Applications vol 11 no2 pp 203ndash220 2008

[9] S L Kalla ldquoOn operators of fractional integration Irdquo Mathe-maticae Notae vol 22 pp 89ndash93 197071

[10] S L Kalla ldquoOn operators of fractional integration IIrdquo Mathe-maticae Notae vol 25 pp 29ndash35 1976

[11] V S Kiryakova Generalized Fractional Calculus and Applica-tions Long-man amp J Wiley Harlow New York NY USA 1994

[12] A Mura and F Mainardi ldquoA class of self-similar stochasticprocesses with stationary increments to model anomalousdiffusion in physicsrdquo Integral Transforms and Special Functionsvol 20 no 3-4 pp 185ndash198 2009

[13] O P Agrawal ldquoSome generalized fractional calculus operatorsand their applications in integral equationsrdquo Fractional Calculusand Applied Analysis An International Journal for Theory andApplications vol 15 no 4 pp 700ndash711 2012

[14] G Pagnini ldquoErdelyi-Kober fractional diffusionrdquo FractionalCalculus and Applied Analysis An International Journal forTheory and Applications vol 15 no 1 pp 117ndash127 2012

[15] A Mura and G Pagnini ldquoCharacterizations and simulations ofa class of stochastic processes to model anomalous diffusionrdquoJournal of Physics A Mathematical and General vol 41 no 28285003 22 pages 2008

[16] E K Lenzi L R Evangelista M K Lenzi H V Ribeiro and EC de Oliveira ldquoSolutions for a non-Markovian diffusion equa-tionrdquo Physics Letters A vol 374 no 41 pp 4193ndash4198 2010

[17] B Al-Saqabi and V S Kiryakova ldquoExplicit solutions of frac-tional integral and differential equations involving Erderyi-Kober operatorsrdquo Applied Mathematics and Computation vol95 no 1 pp 1ndash13 1998

14 International Journal of Differential Equations

[18] L A Hanna and Y F Luchko ldquoOperational calculus for theCaputo-type fractional Erdelyi-Kober derivative and its appli-cationsrdquo Integral Transforms and Special Functions vol 25 no5 pp 359ndash373 2014

[19] Y Xu Z He and Q Xu ldquoNumerical solutions of fractionaladvection-diffusion equations with a kind of new generalizedfractional derivativerdquo International Journal of Computer Math-ematics vol 91 no 3 pp 588ndash600 2014

[20] Y Xu Z He and O P Agrawal ldquoNumerical and analytical solu-tions of new generalized fractional diffusion equationrdquo Com-puters amp Mathematics with Applications vol 66 no 10 pp2019ndash2029 2013

[21] Y Xu and O P Agrawal ldquoNumerical solutions and analysisof diffusion for new generalized fractional Burgers equationrdquoFractional Calculus and Applied Analysis An International Jour-nal forTheory and Applications vol 16 no 3 pp 709ndash736 2013

[22] X Li and C Xu ldquoA space-time spectral method for the timefractional diffusion equationrdquo SIAM Journal on NumericalAnalysis vol 47 no 3 pp 2108ndash2131 2009

[23] Q Xu and J S Hesthaven ldquoStable multi-domain spectralpenalty methods for fractional partial differential equationsrdquoJournal of Computational Physics vol 257 pp 241ndash258 2014

[24] RMittal and S Pandit ldquoQuasilinearized Scale-3 Haar wavelets-based algorithm for numerical simulation of fractional dynam-ical systemsrdquo Engineering Computations vol 35 no 5 pp 1907ndash1931 2018

[25] C Li F Zeng and F Liu ldquoSpectral approximations to the frac-tional integral and derivativerdquo Fractional Calculus and AppliedAnalysis vol 15 no 3 pp 383ndash406 2012

[26] M Zheng F Liu I Turner and V Anh ldquoA novel high orderspace-time spectral method for the time fractional Fokker-Planck equationrdquo SIAM Journal on Scientific Computing vol 37no 2 pp A701ndashA724 2015

[27] F Zeng F Liu C Li K Burrage I Turner and V Anh ldquoACrank-Nicolson ADI spectral method for a two-dimensionalRiesz space fractional nonlinear reaction-diffusion equationrdquoSIAM Journal on Numerical Analysis vol 52 no 6 pp 2599ndash2622 2014

[28] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical ModellingSimulation andComputation for Engineering and EnvironmentalSystems vol 35 no 12 pp 5662ndash5672 2011

[29] M Zayernouri and G E Karniadakis ldquoExponentially accuratespectral and spectral element methods for fractional ODEsrdquoJournal of Computational Physics vol 257 pp 460ndash480 2014

[30] X Zhao and Z Zhang ldquoSuperconvergence points of fractionalspectral interpolationrdquo SIAM Journal on Scientific Computingvol 38 no 1 pp A598ndashA613 2016

[31] F Chen Q Xu and J S Hesthaven ldquoA multi-domain spectralmethod for time-fractional differential equationsrdquo Journal ofComputational Physics vol 293 pp 157ndash172 2015

[32] Z Mao and J Shen ldquoEfficient spectral-Galerkin methods forfractional partial differential equations with variable coeffi-cientsrdquo Journal of Computational Physics vol 307 pp 243ndash2612016

[33] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[34] J Hadamard ldquoEssai sur lrsquoetude des fonctions donnees par leurdeveloppement de Taylorrdquo Journal de Mathematiques Pures etAppliquees vol 4 pp 101ndash186 1892

[35] A Erdelyi andHKober ldquoSome remarks onHankel transformsrdquoQuarterly Journal of Mathematics vol 11 pp 212ndash221 1940

[36] I Dimovski ldquoOperational calculus for a class of differentialoperatorsrdquo Comptes Rendus De L Academie Bulgare Des Sci-ences vol 19 pp 1111ndash1114 1966

[37] S B Yakubovich and Y F LuchkoThe hypergeometric approachto integral transforms and convolutions vol 287 ofMathematicsand its Applications Kluwer Academic Publishers Dordrecht-Boston-London 1994

[38] Y Luchko ldquoOperational rules for a mixed operator of theErdelyi-Kober typerdquo Fractional Calculus and Applied Analysisvol 7 no 3 pp 339ndash364 2004

[39] G Szego Orthogonal polynomials American MathematicalSociety Providence 1992

[40] M Zayernouri and G E Karniadakis ldquoFractional Sturm-Liou-ville eigen-problems theory and numerical approximationrdquoJournal of Computational Physics vol 252 pp 495ndash517 2013

[41] S Chen J Shen and L-L Wang ldquoGeneralized Jacobi functionsand their applications to fractional differential equationsrdquoMathematics of Computation vol 85 no 300 pp 1603ndash16382016

[42] A A Kilbas ldquoHadamard-type integral equations and fractionalcalculus operatorsrdquo in Singular integral operators factorizationand applications vol 142 ofOperTheory Adv Appl pp 175ndash188Birkhauser Basel 2003

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Submit your manuscripts atwwwhindawicom

International Journal of Differential Equations 11

In [42] Kilbas discussed the existence of the solution of(92) Explicit formulas for the solution 119891(119905) were establishedin the following theorem

Theorem 32 (see [42]) If 119909120583119892(119909) isin 119860119862[119886 119887] then theHadamard-type integral equation (92) with 0 lt 120572 lt 1 issolvable in 119883120583(119886 119887) and its solution may be represented in theform

119891 (119909) = 119909minus120583Γ (1 minus 120572) [119886120583119892 (119886) (log 119909119886)minus120572

+ int119909

119886(log 119909119904 )

minus120572 (119904120583119892 (119904))1015840 119889119904] (93)

Here 119860119862[119886 119887] is the set of absolutely continuous functions on[119886 119887] and119883120583(119886 119887) is space of those Lebesgue measurable func-tions 119891 on [119886 119887] for which 119909120583minus1119891(119909) is absolutely integrable

Solution of the Hadamard-type integral equation isexactly the same as generalized fractional derivative ofRiemann-Liouville type with 119911(119904) = log(119904) 119908(119904) = 119904120583 Therelationship between (93) and Caputo type generalized frac-tional derivative is

119891 (119909) = 119886120583119892 (119886) 119909minus120583Γ (1 minus 120572) (log 119909119886)minus120572 + 119863120572

119886+[1199111199082]119892 (119909) (94)

Suppose 119909119895 119895 = 0 1 119873 are interpolation points thediscretized solution of equation (92) is

119891119873 (119909) = 119886120583119892 (119886) 119909minus120583Γ (1 minus 120572) (log 119909119886)minus120572

+ 119909minus120583 119873sum119895=0

119873sum119896=0

119909120583119895119892 (119909119895) 119897119896119895( 119888120577119886119863120572

120577119875119896lowast (log (119909))

(95)

Rewriting (95) in matrix form we have

119865 = 119866119886 +M120572119866 (96)

where M120572 is generalized fractional differential matrix119866 = (119892(1199090) 119892(1199091) 119892(119909119873))119879 119865 = (119891119873(1199090) 119891119873(1199091) 119891119873(119909119873))119879 119866119886 is a vector about the initial condition of theintegral equation defined by

119866119886 = 119886120583119892 (119886)Γ (1 minus 120572) (119909minus1205830 (log 1199090119886 )minus120572 119909minus1205831 (log 1199091119886 )

minus120572 119909minus120583119873 (log 119909119873119886 )minus120572)119879

(97)

Example 33 Assume 119892(119909) = sin(119909 minus 1) 120583 = 13 Ω = [110] Solutions of (92) for 120572 = 03 06 09 are shown inFigure 6

2 4 6 8 10 120X

minus8

minus6

minus4

minus2

0

2

4

6

8

f(X)

=03=06=09

Figure 6 Solution of (92)

43 Erdelyi-Kober Fractional Diffusion Equation In this sub-section we consider the following Erdelyi-Kober fractionaldiffusion equation [12]

119906 (119909 119905) = 1199060 (119909) + 1Γ (120573)sdot int119905

0

120572120573119904120572120573minus1 (119905120572120573 minus 119904120572120573)120573minus1 12059721205971199092 119906 (119909 119904) d119904(98)

Erdelyi-Kober fractional diffusion equation which is alsocalled stretched time-fractional diffusion equation is themaster equation of a kind of generalized grey Brownianmotion (ggBm) The ggBm is a parametric class of stochasticprocesses that provides models for both fast and slow anoma-lous diffusion This class is made up of self-similar processes119861120572120573(119905)with stationary increments and it depends on two realparameters 0 lt 120572 le 2 and 0 lt 120573 le 1 It includes thefractional Brownian motion when 0 lt 120572 le 2 and 120573 = 1 thetime-fractional diffusion stochastic processes when 0 lt 120572 =120573 lt 1 and the standard Brownian motion when 120572 = 120573 = 1About the relationship between stochastic process119861120572120573(119905) andstretched time-fractional diffusion equation the followingproposition is presented in [12]

Proposition 34 The marginal probability density function119891120572120573(119909 119905) of the process 119861120572120573(119905) 119905 ge 0 is the fundamentalsolution of the stretched time-fractional diffusion equation

119906 (119909 119905) = 1199060 (119909) + 1Γ (120573)sdot int119905

0

120572120573119904120572120573minus1 (119905120572120573 minus 119904120572120573)120573minus1 12059721205971199092 119906 (119909 119904) d119904(99)

12 International Journal of Differential Equations

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 7 Standard Brownian motion 120572 = 1 120573 = 1

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 8 Time-fractional diffusion with 120572 = 06 120573 = 06

Recalling the definition of generalized fractional integraland setting 119911(119905) = 119905120572120573 119908(119905) = 1 the equation can be rewrit-ten as

119906 (119909 119905) = 1199060 (119909) + 1198681205730+[119911119908]119906119909119909 (119909 119905) (100)

We use collocation method for both space and timediscretization We choose Legendre-Gauss-Lobatto (L-G-L)points 119909119894 (119894 = 0 1 119872) as the space collocation points andchoose 119905119895 (119895 = 0 1 119873) such that 119911(119905119895) are L-G-L pointsas the time collocation points

Define space collocation matrix M2 such that M2119894119895 =(11988921198891199092)L119895(119909119894) and generalized fractional integral matrix

M120573 Matrix M120573 is computed through Theorem 28 and thespace-time collocation matrices are obtained using Kro-necker product 1015840otimes1015840 Suppose A and B are space-timecollocation matrices with dimension (119872 + 1)(119873 + 1) times (119872 +1)(119873 + 1) for the second order derivative and fractionalintegral of order 120573 separately Then

A = M2 otimesI119873+1

B = I119872+1 otimesM120573 (101)

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 9 Fractional Brownian motion with 120572 = 15 120573 = 1

Suppose 119906119873(119909 119905) is the numerical solution of (98) defin-ing 119906119873119894119895 = 119906119873(119909119894 119905119895) solution vectorU and initial vector 1198800such that

U = [11990611987300 11990611987301 1199061198730119873 11990611987310 1199061198731119873 1199061198731198720 119906119873119872119873]119879

1198800 = [1199060 (1199090) 1199060 (1199090) 1199060 (1199091) 1199060 (1199091) 1199060 (119909119872) 1199060 (119909119872)]119879

(102)

where in the definition of 1198800 each 1199060(119909119895) is repeated 119873 + 1times

The matrix form discretized equation of (98) is obtainedas

U = 1198800 +BAU (103)

In the discretized equation initial condition is explicitlyinvolved After boundary condition added properly numeri-cal solution can be obtained by solving the matrix equation

Example 35 Assume 119909 isin Ω = (minus1 1) 119905 isin (0 05] 1199060(119909) =119890minus101199092 minus 119890minus10 119906(sdot 119905)|120597Ω = 0 Numerical solutions with 119872 =119873 = 50 are shown in Figures 7ndash10

Erdelyi-Kober diffusion equation characterizes the mar-ginal density function of the process 119861120572120573(119905) 119905 ge 0 When120572 = 120573 = 1 we recover the standard diffusion equationWhen0 lt 120572 = 120573 lt 1 we get the time-fractional diffusion equationof order 120573 When 120573 = 1 and 0 lt 120572 lt 2 we have the equationof the fractional Brownian motion marginal density

As shown in Figures 7 and 8 when 1 lt 120572 lt 2 the diffu-sion is fast and the increments exhibit long-range depen-dence when 0 lt 120572 lt 1 the diffusion is slow and theincrements form a stationary process which does not exhibitlong-range dependenceThe results coincide with theoreticalanalysis in [12 14]

International Journal of Differential Equations 13

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 10 Fractional Brownian motion with 120572 = 05 120573 = 1

5 Conclusion

In this paper we propose a spectral collocation method fordifferential and integral equations with generalized fractionaloperators To deal with the difficulty in designing spectralapproximation scheme due to complexity of integral kerneland weight a variable transform technique is applied to thegeneralized fractional operator and a spectral approximationmethod is proposed for the generalized fractional operatorOperational matrices for generalized fractional operatorsare derived Spectral collocation methods are designed forfractional ordinary differential equations Hadamard-typeintegral equations and Erdelyi-Kober diffusion equationsseparately Numerical experiments are carried out to verifythe accuracy and efficiency of the method and characteristicsof the Erdelyi-Kober diffusion equation are analyzed basedon numerical results

Data Availability

(i) The programs used to support the findings of this studyhave been deposited in the GitHub repository (httpsgithubcomqinwuxuSpectralGFPDE ) (ii) No data were used tosupport this study

Disclosure

An earlier version of this workwas presented at the ldquo8th Inter-national Congress on Industrial and Applied Mathematics(ICIAM 2015)rdquo

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The first author is supported by the National Key RampD Pro-gram of China (No 2017YFC0209804) the National ScienceFoundation for Young Scientists of China (No 11701273) andYouth Foundation of Jiangsu Province (No BK20170628)

The second author is supported by the National Key RampDProgram of China (No 2017YFB0305601)

References

[1] B BaeumerDA BensonMMMeerschaert and SWWheat-craft ldquoSubordinated advection-dispersion equation for contam-inant transportrdquo Water Resources Research vol 37 no 6 pp1543ndash1550 2001

[2] E Barkai RMetzler and J Klafter ldquoFrom continuous time ran-dom walks to the fractional Fokker-Planck equationrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 61no 1 pp 132ndash138 2000

[3] A Blumen G Zumofen and J Klafter ldquoTransport aspects inanomalous diffusion Levy walksrdquo Physical Review A AtomicMolecular and Optical Physics vol 40 no 7 pp 3964ndash39731989

[4] J P Bouchaud and A Georges ldquoAnomalous diffusion in dis-ordered media statistical mechanisms models and physicalapplicationsrdquoPhysics Reports vol 195 no 4-5 pp 127ndash293 1990

[5] M Raberto E Scalas and F Mainardi ldquoWaiting-times andreturns in high-frequency financial data an empirical studyrdquoPhysica A Statistical Mechanics and its Applications vol 314 no1ndash4 pp 749ndash755 2002

[6] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations New York NYUSA Elsevier 2006

[7] S G Samko A A Kilbas and O I Marichev Fractional Inte-grals and Derivatives Theory and Applications Gordon andBreach Yverdon Switzerland 1993

[8] V Kiryakova ldquoA brief story about the operators of the general-ized fractional calculusrdquo Fractional CalculusampAppliedAnalysisAn International Journal forTheory and Applications vol 11 no2 pp 203ndash220 2008

[9] S L Kalla ldquoOn operators of fractional integration Irdquo Mathe-maticae Notae vol 22 pp 89ndash93 197071

[10] S L Kalla ldquoOn operators of fractional integration IIrdquo Mathe-maticae Notae vol 25 pp 29ndash35 1976

[11] V S Kiryakova Generalized Fractional Calculus and Applica-tions Long-man amp J Wiley Harlow New York NY USA 1994

[12] A Mura and F Mainardi ldquoA class of self-similar stochasticprocesses with stationary increments to model anomalousdiffusion in physicsrdquo Integral Transforms and Special Functionsvol 20 no 3-4 pp 185ndash198 2009

[13] O P Agrawal ldquoSome generalized fractional calculus operatorsand their applications in integral equationsrdquo Fractional Calculusand Applied Analysis An International Journal for Theory andApplications vol 15 no 4 pp 700ndash711 2012

[14] G Pagnini ldquoErdelyi-Kober fractional diffusionrdquo FractionalCalculus and Applied Analysis An International Journal forTheory and Applications vol 15 no 1 pp 117ndash127 2012

[15] A Mura and G Pagnini ldquoCharacterizations and simulations ofa class of stochastic processes to model anomalous diffusionrdquoJournal of Physics A Mathematical and General vol 41 no 28285003 22 pages 2008

[16] E K Lenzi L R Evangelista M K Lenzi H V Ribeiro and EC de Oliveira ldquoSolutions for a non-Markovian diffusion equa-tionrdquo Physics Letters A vol 374 no 41 pp 4193ndash4198 2010

[17] B Al-Saqabi and V S Kiryakova ldquoExplicit solutions of frac-tional integral and differential equations involving Erderyi-Kober operatorsrdquo Applied Mathematics and Computation vol95 no 1 pp 1ndash13 1998

14 International Journal of Differential Equations

[18] L A Hanna and Y F Luchko ldquoOperational calculus for theCaputo-type fractional Erdelyi-Kober derivative and its appli-cationsrdquo Integral Transforms and Special Functions vol 25 no5 pp 359ndash373 2014

[19] Y Xu Z He and Q Xu ldquoNumerical solutions of fractionaladvection-diffusion equations with a kind of new generalizedfractional derivativerdquo International Journal of Computer Math-ematics vol 91 no 3 pp 588ndash600 2014

[20] Y Xu Z He and O P Agrawal ldquoNumerical and analytical solu-tions of new generalized fractional diffusion equationrdquo Com-puters amp Mathematics with Applications vol 66 no 10 pp2019ndash2029 2013

[21] Y Xu and O P Agrawal ldquoNumerical solutions and analysisof diffusion for new generalized fractional Burgers equationrdquoFractional Calculus and Applied Analysis An International Jour-nal forTheory and Applications vol 16 no 3 pp 709ndash736 2013

[22] X Li and C Xu ldquoA space-time spectral method for the timefractional diffusion equationrdquo SIAM Journal on NumericalAnalysis vol 47 no 3 pp 2108ndash2131 2009

[23] Q Xu and J S Hesthaven ldquoStable multi-domain spectralpenalty methods for fractional partial differential equationsrdquoJournal of Computational Physics vol 257 pp 241ndash258 2014

[24] RMittal and S Pandit ldquoQuasilinearized Scale-3 Haar wavelets-based algorithm for numerical simulation of fractional dynam-ical systemsrdquo Engineering Computations vol 35 no 5 pp 1907ndash1931 2018

[25] C Li F Zeng and F Liu ldquoSpectral approximations to the frac-tional integral and derivativerdquo Fractional Calculus and AppliedAnalysis vol 15 no 3 pp 383ndash406 2012

[26] M Zheng F Liu I Turner and V Anh ldquoA novel high orderspace-time spectral method for the time fractional Fokker-Planck equationrdquo SIAM Journal on Scientific Computing vol 37no 2 pp A701ndashA724 2015

[27] F Zeng F Liu C Li K Burrage I Turner and V Anh ldquoACrank-Nicolson ADI spectral method for a two-dimensionalRiesz space fractional nonlinear reaction-diffusion equationrdquoSIAM Journal on Numerical Analysis vol 52 no 6 pp 2599ndash2622 2014

[28] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical ModellingSimulation andComputation for Engineering and EnvironmentalSystems vol 35 no 12 pp 5662ndash5672 2011

[29] M Zayernouri and G E Karniadakis ldquoExponentially accuratespectral and spectral element methods for fractional ODEsrdquoJournal of Computational Physics vol 257 pp 460ndash480 2014

[30] X Zhao and Z Zhang ldquoSuperconvergence points of fractionalspectral interpolationrdquo SIAM Journal on Scientific Computingvol 38 no 1 pp A598ndashA613 2016

[31] F Chen Q Xu and J S Hesthaven ldquoA multi-domain spectralmethod for time-fractional differential equationsrdquo Journal ofComputational Physics vol 293 pp 157ndash172 2015

[32] Z Mao and J Shen ldquoEfficient spectral-Galerkin methods forfractional partial differential equations with variable coeffi-cientsrdquo Journal of Computational Physics vol 307 pp 243ndash2612016

[33] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[34] J Hadamard ldquoEssai sur lrsquoetude des fonctions donnees par leurdeveloppement de Taylorrdquo Journal de Mathematiques Pures etAppliquees vol 4 pp 101ndash186 1892

[35] A Erdelyi andHKober ldquoSome remarks onHankel transformsrdquoQuarterly Journal of Mathematics vol 11 pp 212ndash221 1940

[36] I Dimovski ldquoOperational calculus for a class of differentialoperatorsrdquo Comptes Rendus De L Academie Bulgare Des Sci-ences vol 19 pp 1111ndash1114 1966

[37] S B Yakubovich and Y F LuchkoThe hypergeometric approachto integral transforms and convolutions vol 287 ofMathematicsand its Applications Kluwer Academic Publishers Dordrecht-Boston-London 1994

[38] Y Luchko ldquoOperational rules for a mixed operator of theErdelyi-Kober typerdquo Fractional Calculus and Applied Analysisvol 7 no 3 pp 339ndash364 2004

[39] G Szego Orthogonal polynomials American MathematicalSociety Providence 1992

[40] M Zayernouri and G E Karniadakis ldquoFractional Sturm-Liou-ville eigen-problems theory and numerical approximationrdquoJournal of Computational Physics vol 252 pp 495ndash517 2013

[41] S Chen J Shen and L-L Wang ldquoGeneralized Jacobi functionsand their applications to fractional differential equationsrdquoMathematics of Computation vol 85 no 300 pp 1603ndash16382016

[42] A A Kilbas ldquoHadamard-type integral equations and fractionalcalculus operatorsrdquo in Singular integral operators factorizationand applications vol 142 ofOperTheory Adv Appl pp 175ndash188Birkhauser Basel 2003

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

12 International Journal of Differential Equations

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 7 Standard Brownian motion 120572 = 1 120573 = 1

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 8 Time-fractional diffusion with 120572 = 06 120573 = 06

Recalling the definition of generalized fractional integraland setting 119911(119905) = 119905120572120573 119908(119905) = 1 the equation can be rewrit-ten as

119906 (119909 119905) = 1199060 (119909) + 1198681205730+[119911119908]119906119909119909 (119909 119905) (100)

We use collocation method for both space and timediscretization We choose Legendre-Gauss-Lobatto (L-G-L)points 119909119894 (119894 = 0 1 119872) as the space collocation points andchoose 119905119895 (119895 = 0 1 119873) such that 119911(119905119895) are L-G-L pointsas the time collocation points

Define space collocation matrix M2 such that M2119894119895 =(11988921198891199092)L119895(119909119894) and generalized fractional integral matrix

M120573 Matrix M120573 is computed through Theorem 28 and thespace-time collocation matrices are obtained using Kro-necker product 1015840otimes1015840 Suppose A and B are space-timecollocation matrices with dimension (119872 + 1)(119873 + 1) times (119872 +1)(119873 + 1) for the second order derivative and fractionalintegral of order 120573 separately Then

A = M2 otimesI119873+1

B = I119872+1 otimesM120573 (101)

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 9 Fractional Brownian motion with 120572 = 15 120573 = 1

Suppose 119906119873(119909 119905) is the numerical solution of (98) defin-ing 119906119873119894119895 = 119906119873(119909119894 119905119895) solution vectorU and initial vector 1198800such that

U = [11990611987300 11990611987301 1199061198730119873 11990611987310 1199061198731119873 1199061198731198720 119906119873119872119873]119879

1198800 = [1199060 (1199090) 1199060 (1199090) 1199060 (1199091) 1199060 (1199091) 1199060 (119909119872) 1199060 (119909119872)]119879

(102)

where in the definition of 1198800 each 1199060(119909119895) is repeated 119873 + 1times

The matrix form discretized equation of (98) is obtainedas

U = 1198800 +BAU (103)

In the discretized equation initial condition is explicitlyinvolved After boundary condition added properly numeri-cal solution can be obtained by solving the matrix equation

Example 35 Assume 119909 isin Ω = (minus1 1) 119905 isin (0 05] 1199060(119909) =119890minus101199092 minus 119890minus10 119906(sdot 119905)|120597Ω = 0 Numerical solutions with 119872 =119873 = 50 are shown in Figures 7ndash10

Erdelyi-Kober diffusion equation characterizes the mar-ginal density function of the process 119861120572120573(119905) 119905 ge 0 When120572 = 120573 = 1 we recover the standard diffusion equationWhen0 lt 120572 = 120573 lt 1 we get the time-fractional diffusion equationof order 120573 When 120573 = 1 and 0 lt 120572 lt 2 we have the equationof the fractional Brownian motion marginal density

As shown in Figures 7 and 8 when 1 lt 120572 lt 2 the diffu-sion is fast and the increments exhibit long-range depen-dence when 0 lt 120572 lt 1 the diffusion is slow and theincrements form a stationary process which does not exhibitlong-range dependenceThe results coincide with theoreticalanalysis in [12 14]

International Journal of Differential Equations 13

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 10 Fractional Brownian motion with 120572 = 05 120573 = 1

5 Conclusion

In this paper we propose a spectral collocation method fordifferential and integral equations with generalized fractionaloperators To deal with the difficulty in designing spectralapproximation scheme due to complexity of integral kerneland weight a variable transform technique is applied to thegeneralized fractional operator and a spectral approximationmethod is proposed for the generalized fractional operatorOperational matrices for generalized fractional operatorsare derived Spectral collocation methods are designed forfractional ordinary differential equations Hadamard-typeintegral equations and Erdelyi-Kober diffusion equationsseparately Numerical experiments are carried out to verifythe accuracy and efficiency of the method and characteristicsof the Erdelyi-Kober diffusion equation are analyzed basedon numerical results

Data Availability

(i) The programs used to support the findings of this studyhave been deposited in the GitHub repository (httpsgithubcomqinwuxuSpectralGFPDE ) (ii) No data were used tosupport this study

Disclosure

An earlier version of this workwas presented at the ldquo8th Inter-national Congress on Industrial and Applied Mathematics(ICIAM 2015)rdquo

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The first author is supported by the National Key RampD Pro-gram of China (No 2017YFC0209804) the National ScienceFoundation for Young Scientists of China (No 11701273) andYouth Foundation of Jiangsu Province (No BK20170628)

The second author is supported by the National Key RampDProgram of China (No 2017YFB0305601)

References

[1] B BaeumerDA BensonMMMeerschaert and SWWheat-craft ldquoSubordinated advection-dispersion equation for contam-inant transportrdquo Water Resources Research vol 37 no 6 pp1543ndash1550 2001

[2] E Barkai RMetzler and J Klafter ldquoFrom continuous time ran-dom walks to the fractional Fokker-Planck equationrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 61no 1 pp 132ndash138 2000

[3] A Blumen G Zumofen and J Klafter ldquoTransport aspects inanomalous diffusion Levy walksrdquo Physical Review A AtomicMolecular and Optical Physics vol 40 no 7 pp 3964ndash39731989

[4] J P Bouchaud and A Georges ldquoAnomalous diffusion in dis-ordered media statistical mechanisms models and physicalapplicationsrdquoPhysics Reports vol 195 no 4-5 pp 127ndash293 1990

[5] M Raberto E Scalas and F Mainardi ldquoWaiting-times andreturns in high-frequency financial data an empirical studyrdquoPhysica A Statistical Mechanics and its Applications vol 314 no1ndash4 pp 749ndash755 2002

[6] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations New York NYUSA Elsevier 2006

[7] S G Samko A A Kilbas and O I Marichev Fractional Inte-grals and Derivatives Theory and Applications Gordon andBreach Yverdon Switzerland 1993

[8] V Kiryakova ldquoA brief story about the operators of the general-ized fractional calculusrdquo Fractional CalculusampAppliedAnalysisAn International Journal forTheory and Applications vol 11 no2 pp 203ndash220 2008

[9] S L Kalla ldquoOn operators of fractional integration Irdquo Mathe-maticae Notae vol 22 pp 89ndash93 197071

[10] S L Kalla ldquoOn operators of fractional integration IIrdquo Mathe-maticae Notae vol 25 pp 29ndash35 1976

[11] V S Kiryakova Generalized Fractional Calculus and Applica-tions Long-man amp J Wiley Harlow New York NY USA 1994

[12] A Mura and F Mainardi ldquoA class of self-similar stochasticprocesses with stationary increments to model anomalousdiffusion in physicsrdquo Integral Transforms and Special Functionsvol 20 no 3-4 pp 185ndash198 2009

[13] O P Agrawal ldquoSome generalized fractional calculus operatorsand their applications in integral equationsrdquo Fractional Calculusand Applied Analysis An International Journal for Theory andApplications vol 15 no 4 pp 700ndash711 2012

[14] G Pagnini ldquoErdelyi-Kober fractional diffusionrdquo FractionalCalculus and Applied Analysis An International Journal forTheory and Applications vol 15 no 1 pp 117ndash127 2012

[15] A Mura and G Pagnini ldquoCharacterizations and simulations ofa class of stochastic processes to model anomalous diffusionrdquoJournal of Physics A Mathematical and General vol 41 no 28285003 22 pages 2008

[16] E K Lenzi L R Evangelista M K Lenzi H V Ribeiro and EC de Oliveira ldquoSolutions for a non-Markovian diffusion equa-tionrdquo Physics Letters A vol 374 no 41 pp 4193ndash4198 2010

[17] B Al-Saqabi and V S Kiryakova ldquoExplicit solutions of frac-tional integral and differential equations involving Erderyi-Kober operatorsrdquo Applied Mathematics and Computation vol95 no 1 pp 1ndash13 1998

14 International Journal of Differential Equations

[18] L A Hanna and Y F Luchko ldquoOperational calculus for theCaputo-type fractional Erdelyi-Kober derivative and its appli-cationsrdquo Integral Transforms and Special Functions vol 25 no5 pp 359ndash373 2014

[19] Y Xu Z He and Q Xu ldquoNumerical solutions of fractionaladvection-diffusion equations with a kind of new generalizedfractional derivativerdquo International Journal of Computer Math-ematics vol 91 no 3 pp 588ndash600 2014

[20] Y Xu Z He and O P Agrawal ldquoNumerical and analytical solu-tions of new generalized fractional diffusion equationrdquo Com-puters amp Mathematics with Applications vol 66 no 10 pp2019ndash2029 2013

[21] Y Xu and O P Agrawal ldquoNumerical solutions and analysisof diffusion for new generalized fractional Burgers equationrdquoFractional Calculus and Applied Analysis An International Jour-nal forTheory and Applications vol 16 no 3 pp 709ndash736 2013

[22] X Li and C Xu ldquoA space-time spectral method for the timefractional diffusion equationrdquo SIAM Journal on NumericalAnalysis vol 47 no 3 pp 2108ndash2131 2009

[23] Q Xu and J S Hesthaven ldquoStable multi-domain spectralpenalty methods for fractional partial differential equationsrdquoJournal of Computational Physics vol 257 pp 241ndash258 2014

[24] RMittal and S Pandit ldquoQuasilinearized Scale-3 Haar wavelets-based algorithm for numerical simulation of fractional dynam-ical systemsrdquo Engineering Computations vol 35 no 5 pp 1907ndash1931 2018

[25] C Li F Zeng and F Liu ldquoSpectral approximations to the frac-tional integral and derivativerdquo Fractional Calculus and AppliedAnalysis vol 15 no 3 pp 383ndash406 2012

[26] M Zheng F Liu I Turner and V Anh ldquoA novel high orderspace-time spectral method for the time fractional Fokker-Planck equationrdquo SIAM Journal on Scientific Computing vol 37no 2 pp A701ndashA724 2015

[27] F Zeng F Liu C Li K Burrage I Turner and V Anh ldquoACrank-Nicolson ADI spectral method for a two-dimensionalRiesz space fractional nonlinear reaction-diffusion equationrdquoSIAM Journal on Numerical Analysis vol 52 no 6 pp 2599ndash2622 2014

[28] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical ModellingSimulation andComputation for Engineering and EnvironmentalSystems vol 35 no 12 pp 5662ndash5672 2011

[29] M Zayernouri and G E Karniadakis ldquoExponentially accuratespectral and spectral element methods for fractional ODEsrdquoJournal of Computational Physics vol 257 pp 460ndash480 2014

[30] X Zhao and Z Zhang ldquoSuperconvergence points of fractionalspectral interpolationrdquo SIAM Journal on Scientific Computingvol 38 no 1 pp A598ndashA613 2016

[31] F Chen Q Xu and J S Hesthaven ldquoA multi-domain spectralmethod for time-fractional differential equationsrdquo Journal ofComputational Physics vol 293 pp 157ndash172 2015

[32] Z Mao and J Shen ldquoEfficient spectral-Galerkin methods forfractional partial differential equations with variable coeffi-cientsrdquo Journal of Computational Physics vol 307 pp 243ndash2612016

[33] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[34] J Hadamard ldquoEssai sur lrsquoetude des fonctions donnees par leurdeveloppement de Taylorrdquo Journal de Mathematiques Pures etAppliquees vol 4 pp 101ndash186 1892

[35] A Erdelyi andHKober ldquoSome remarks onHankel transformsrdquoQuarterly Journal of Mathematics vol 11 pp 212ndash221 1940

[36] I Dimovski ldquoOperational calculus for a class of differentialoperatorsrdquo Comptes Rendus De L Academie Bulgare Des Sci-ences vol 19 pp 1111ndash1114 1966

[37] S B Yakubovich and Y F LuchkoThe hypergeometric approachto integral transforms and convolutions vol 287 ofMathematicsand its Applications Kluwer Academic Publishers Dordrecht-Boston-London 1994

[38] Y Luchko ldquoOperational rules for a mixed operator of theErdelyi-Kober typerdquo Fractional Calculus and Applied Analysisvol 7 no 3 pp 339ndash364 2004

[39] G Szego Orthogonal polynomials American MathematicalSociety Providence 1992

[40] M Zayernouri and G E Karniadakis ldquoFractional Sturm-Liou-ville eigen-problems theory and numerical approximationrdquoJournal of Computational Physics vol 252 pp 495ndash517 2013

[41] S Chen J Shen and L-L Wang ldquoGeneralized Jacobi functionsand their applications to fractional differential equationsrdquoMathematics of Computation vol 85 no 300 pp 1603ndash16382016

[42] A A Kilbas ldquoHadamard-type integral equations and fractionalcalculus operatorsrdquo in Singular integral operators factorizationand applications vol 142 ofOperTheory Adv Appl pp 175ndash188Birkhauser Basel 2003

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

International Journal of Differential Equations 13

minus1minus05

005

1

001

0203

0405

minus02

0

02

04

06

08

1

xt

u(x

t)

Figure 10 Fractional Brownian motion with 120572 = 05 120573 = 1

5 Conclusion

In this paper we propose a spectral collocation method fordifferential and integral equations with generalized fractionaloperators To deal with the difficulty in designing spectralapproximation scheme due to complexity of integral kerneland weight a variable transform technique is applied to thegeneralized fractional operator and a spectral approximationmethod is proposed for the generalized fractional operatorOperational matrices for generalized fractional operatorsare derived Spectral collocation methods are designed forfractional ordinary differential equations Hadamard-typeintegral equations and Erdelyi-Kober diffusion equationsseparately Numerical experiments are carried out to verifythe accuracy and efficiency of the method and characteristicsof the Erdelyi-Kober diffusion equation are analyzed basedon numerical results

Data Availability

(i) The programs used to support the findings of this studyhave been deposited in the GitHub repository (httpsgithubcomqinwuxuSpectralGFPDE ) (ii) No data were used tosupport this study

Disclosure

An earlier version of this workwas presented at the ldquo8th Inter-national Congress on Industrial and Applied Mathematics(ICIAM 2015)rdquo

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The first author is supported by the National Key RampD Pro-gram of China (No 2017YFC0209804) the National ScienceFoundation for Young Scientists of China (No 11701273) andYouth Foundation of Jiangsu Province (No BK20170628)

The second author is supported by the National Key RampDProgram of China (No 2017YFB0305601)

References

[1] B BaeumerDA BensonMMMeerschaert and SWWheat-craft ldquoSubordinated advection-dispersion equation for contam-inant transportrdquo Water Resources Research vol 37 no 6 pp1543ndash1550 2001

[2] E Barkai RMetzler and J Klafter ldquoFrom continuous time ran-dom walks to the fractional Fokker-Planck equationrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 61no 1 pp 132ndash138 2000

[3] A Blumen G Zumofen and J Klafter ldquoTransport aspects inanomalous diffusion Levy walksrdquo Physical Review A AtomicMolecular and Optical Physics vol 40 no 7 pp 3964ndash39731989

[4] J P Bouchaud and A Georges ldquoAnomalous diffusion in dis-ordered media statistical mechanisms models and physicalapplicationsrdquoPhysics Reports vol 195 no 4-5 pp 127ndash293 1990

[5] M Raberto E Scalas and F Mainardi ldquoWaiting-times andreturns in high-frequency financial data an empirical studyrdquoPhysica A Statistical Mechanics and its Applications vol 314 no1ndash4 pp 749ndash755 2002

[6] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations New York NYUSA Elsevier 2006

[7] S G Samko A A Kilbas and O I Marichev Fractional Inte-grals and Derivatives Theory and Applications Gordon andBreach Yverdon Switzerland 1993

[8] V Kiryakova ldquoA brief story about the operators of the general-ized fractional calculusrdquo Fractional CalculusampAppliedAnalysisAn International Journal forTheory and Applications vol 11 no2 pp 203ndash220 2008

[9] S L Kalla ldquoOn operators of fractional integration Irdquo Mathe-maticae Notae vol 22 pp 89ndash93 197071

[10] S L Kalla ldquoOn operators of fractional integration IIrdquo Mathe-maticae Notae vol 25 pp 29ndash35 1976

[11] V S Kiryakova Generalized Fractional Calculus and Applica-tions Long-man amp J Wiley Harlow New York NY USA 1994

[12] A Mura and F Mainardi ldquoA class of self-similar stochasticprocesses with stationary increments to model anomalousdiffusion in physicsrdquo Integral Transforms and Special Functionsvol 20 no 3-4 pp 185ndash198 2009

[13] O P Agrawal ldquoSome generalized fractional calculus operatorsand their applications in integral equationsrdquo Fractional Calculusand Applied Analysis An International Journal for Theory andApplications vol 15 no 4 pp 700ndash711 2012

[14] G Pagnini ldquoErdelyi-Kober fractional diffusionrdquo FractionalCalculus and Applied Analysis An International Journal forTheory and Applications vol 15 no 1 pp 117ndash127 2012

[15] A Mura and G Pagnini ldquoCharacterizations and simulations ofa class of stochastic processes to model anomalous diffusionrdquoJournal of Physics A Mathematical and General vol 41 no 28285003 22 pages 2008

[16] E K Lenzi L R Evangelista M K Lenzi H V Ribeiro and EC de Oliveira ldquoSolutions for a non-Markovian diffusion equa-tionrdquo Physics Letters A vol 374 no 41 pp 4193ndash4198 2010

[17] B Al-Saqabi and V S Kiryakova ldquoExplicit solutions of frac-tional integral and differential equations involving Erderyi-Kober operatorsrdquo Applied Mathematics and Computation vol95 no 1 pp 1ndash13 1998

14 International Journal of Differential Equations

[18] L A Hanna and Y F Luchko ldquoOperational calculus for theCaputo-type fractional Erdelyi-Kober derivative and its appli-cationsrdquo Integral Transforms and Special Functions vol 25 no5 pp 359ndash373 2014

[19] Y Xu Z He and Q Xu ldquoNumerical solutions of fractionaladvection-diffusion equations with a kind of new generalizedfractional derivativerdquo International Journal of Computer Math-ematics vol 91 no 3 pp 588ndash600 2014

[20] Y Xu Z He and O P Agrawal ldquoNumerical and analytical solu-tions of new generalized fractional diffusion equationrdquo Com-puters amp Mathematics with Applications vol 66 no 10 pp2019ndash2029 2013

[21] Y Xu and O P Agrawal ldquoNumerical solutions and analysisof diffusion for new generalized fractional Burgers equationrdquoFractional Calculus and Applied Analysis An International Jour-nal forTheory and Applications vol 16 no 3 pp 709ndash736 2013

[22] X Li and C Xu ldquoA space-time spectral method for the timefractional diffusion equationrdquo SIAM Journal on NumericalAnalysis vol 47 no 3 pp 2108ndash2131 2009

[23] Q Xu and J S Hesthaven ldquoStable multi-domain spectralpenalty methods for fractional partial differential equationsrdquoJournal of Computational Physics vol 257 pp 241ndash258 2014

[24] RMittal and S Pandit ldquoQuasilinearized Scale-3 Haar wavelets-based algorithm for numerical simulation of fractional dynam-ical systemsrdquo Engineering Computations vol 35 no 5 pp 1907ndash1931 2018

[25] C Li F Zeng and F Liu ldquoSpectral approximations to the frac-tional integral and derivativerdquo Fractional Calculus and AppliedAnalysis vol 15 no 3 pp 383ndash406 2012

[26] M Zheng F Liu I Turner and V Anh ldquoA novel high orderspace-time spectral method for the time fractional Fokker-Planck equationrdquo SIAM Journal on Scientific Computing vol 37no 2 pp A701ndashA724 2015

[27] F Zeng F Liu C Li K Burrage I Turner and V Anh ldquoACrank-Nicolson ADI spectral method for a two-dimensionalRiesz space fractional nonlinear reaction-diffusion equationrdquoSIAM Journal on Numerical Analysis vol 52 no 6 pp 2599ndash2622 2014

[28] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical ModellingSimulation andComputation for Engineering and EnvironmentalSystems vol 35 no 12 pp 5662ndash5672 2011

[29] M Zayernouri and G E Karniadakis ldquoExponentially accuratespectral and spectral element methods for fractional ODEsrdquoJournal of Computational Physics vol 257 pp 460ndash480 2014

[30] X Zhao and Z Zhang ldquoSuperconvergence points of fractionalspectral interpolationrdquo SIAM Journal on Scientific Computingvol 38 no 1 pp A598ndashA613 2016

[31] F Chen Q Xu and J S Hesthaven ldquoA multi-domain spectralmethod for time-fractional differential equationsrdquo Journal ofComputational Physics vol 293 pp 157ndash172 2015

[32] Z Mao and J Shen ldquoEfficient spectral-Galerkin methods forfractional partial differential equations with variable coeffi-cientsrdquo Journal of Computational Physics vol 307 pp 243ndash2612016

[33] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[34] J Hadamard ldquoEssai sur lrsquoetude des fonctions donnees par leurdeveloppement de Taylorrdquo Journal de Mathematiques Pures etAppliquees vol 4 pp 101ndash186 1892

[35] A Erdelyi andHKober ldquoSome remarks onHankel transformsrdquoQuarterly Journal of Mathematics vol 11 pp 212ndash221 1940

[36] I Dimovski ldquoOperational calculus for a class of differentialoperatorsrdquo Comptes Rendus De L Academie Bulgare Des Sci-ences vol 19 pp 1111ndash1114 1966

[37] S B Yakubovich and Y F LuchkoThe hypergeometric approachto integral transforms and convolutions vol 287 ofMathematicsand its Applications Kluwer Academic Publishers Dordrecht-Boston-London 1994

[38] Y Luchko ldquoOperational rules for a mixed operator of theErdelyi-Kober typerdquo Fractional Calculus and Applied Analysisvol 7 no 3 pp 339ndash364 2004

[39] G Szego Orthogonal polynomials American MathematicalSociety Providence 1992

[40] M Zayernouri and G E Karniadakis ldquoFractional Sturm-Liou-ville eigen-problems theory and numerical approximationrdquoJournal of Computational Physics vol 252 pp 495ndash517 2013

[41] S Chen J Shen and L-L Wang ldquoGeneralized Jacobi functionsand their applications to fractional differential equationsrdquoMathematics of Computation vol 85 no 300 pp 1603ndash16382016

[42] A A Kilbas ldquoHadamard-type integral equations and fractionalcalculus operatorsrdquo in Singular integral operators factorizationand applications vol 142 ofOperTheory Adv Appl pp 175ndash188Birkhauser Basel 2003

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

14 International Journal of Differential Equations

[18] L A Hanna and Y F Luchko ldquoOperational calculus for theCaputo-type fractional Erdelyi-Kober derivative and its appli-cationsrdquo Integral Transforms and Special Functions vol 25 no5 pp 359ndash373 2014

[19] Y Xu Z He and Q Xu ldquoNumerical solutions of fractionaladvection-diffusion equations with a kind of new generalizedfractional derivativerdquo International Journal of Computer Math-ematics vol 91 no 3 pp 588ndash600 2014

[20] Y Xu Z He and O P Agrawal ldquoNumerical and analytical solu-tions of new generalized fractional diffusion equationrdquo Com-puters amp Mathematics with Applications vol 66 no 10 pp2019ndash2029 2013

[21] Y Xu and O P Agrawal ldquoNumerical solutions and analysisof diffusion for new generalized fractional Burgers equationrdquoFractional Calculus and Applied Analysis An International Jour-nal forTheory and Applications vol 16 no 3 pp 709ndash736 2013

[22] X Li and C Xu ldquoA space-time spectral method for the timefractional diffusion equationrdquo SIAM Journal on NumericalAnalysis vol 47 no 3 pp 2108ndash2131 2009

[23] Q Xu and J S Hesthaven ldquoStable multi-domain spectralpenalty methods for fractional partial differential equationsrdquoJournal of Computational Physics vol 257 pp 241ndash258 2014

[24] RMittal and S Pandit ldquoQuasilinearized Scale-3 Haar wavelets-based algorithm for numerical simulation of fractional dynam-ical systemsrdquo Engineering Computations vol 35 no 5 pp 1907ndash1931 2018

[25] C Li F Zeng and F Liu ldquoSpectral approximations to the frac-tional integral and derivativerdquo Fractional Calculus and AppliedAnalysis vol 15 no 3 pp 383ndash406 2012

[26] M Zheng F Liu I Turner and V Anh ldquoA novel high orderspace-time spectral method for the time fractional Fokker-Planck equationrdquo SIAM Journal on Scientific Computing vol 37no 2 pp A701ndashA724 2015

[27] F Zeng F Liu C Li K Burrage I Turner and V Anh ldquoACrank-Nicolson ADI spectral method for a two-dimensionalRiesz space fractional nonlinear reaction-diffusion equationrdquoSIAM Journal on Numerical Analysis vol 52 no 6 pp 2599ndash2622 2014

[28] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical ModellingSimulation andComputation for Engineering and EnvironmentalSystems vol 35 no 12 pp 5662ndash5672 2011

[29] M Zayernouri and G E Karniadakis ldquoExponentially accuratespectral and spectral element methods for fractional ODEsrdquoJournal of Computational Physics vol 257 pp 460ndash480 2014

[30] X Zhao and Z Zhang ldquoSuperconvergence points of fractionalspectral interpolationrdquo SIAM Journal on Scientific Computingvol 38 no 1 pp A598ndashA613 2016

[31] F Chen Q Xu and J S Hesthaven ldquoA multi-domain spectralmethod for time-fractional differential equationsrdquo Journal ofComputational Physics vol 293 pp 157ndash172 2015

[32] Z Mao and J Shen ldquoEfficient spectral-Galerkin methods forfractional partial differential equations with variable coeffi-cientsrdquo Journal of Computational Physics vol 307 pp 243ndash2612016

[33] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[34] J Hadamard ldquoEssai sur lrsquoetude des fonctions donnees par leurdeveloppement de Taylorrdquo Journal de Mathematiques Pures etAppliquees vol 4 pp 101ndash186 1892

[35] A Erdelyi andHKober ldquoSome remarks onHankel transformsrdquoQuarterly Journal of Mathematics vol 11 pp 212ndash221 1940

[36] I Dimovski ldquoOperational calculus for a class of differentialoperatorsrdquo Comptes Rendus De L Academie Bulgare Des Sci-ences vol 19 pp 1111ndash1114 1966

[37] S B Yakubovich and Y F LuchkoThe hypergeometric approachto integral transforms and convolutions vol 287 ofMathematicsand its Applications Kluwer Academic Publishers Dordrecht-Boston-London 1994

[38] Y Luchko ldquoOperational rules for a mixed operator of theErdelyi-Kober typerdquo Fractional Calculus and Applied Analysisvol 7 no 3 pp 339ndash364 2004

[39] G Szego Orthogonal polynomials American MathematicalSociety Providence 1992

[40] M Zayernouri and G E Karniadakis ldquoFractional Sturm-Liou-ville eigen-problems theory and numerical approximationrdquoJournal of Computational Physics vol 252 pp 495ndash517 2013

[41] S Chen J Shen and L-L Wang ldquoGeneralized Jacobi functionsand their applications to fractional differential equationsrdquoMathematics of Computation vol 85 no 300 pp 1603ndash16382016

[42] A A Kilbas ldquoHadamard-type integral equations and fractionalcalculus operatorsrdquo in Singular integral operators factorizationand applications vol 142 ofOperTheory Adv Appl pp 175ndash188Birkhauser Basel 2003

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom