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Yang, Qianqian, Liu, Fawang, & Turner, Ian(2010)Numerical methods for fractional partial differential equations with Rieszspace fractional derivatives.Applied Mathematical Modelling, 34(1), pp. 200-218.

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https://doi.org/10.1016/j.apm.2009.04.006

Numerical methods for the fractional partial

differential equations with the Riesz space

fractional derivatives ?

Q. Yang a, F. Liu a,b,∗, I. Turner a

aSchool of Mathematical Sciences, Queensland University of Technology, GPO Box2434, Brisbane, QLD. 4001, Australia

bSchool of Mathematical Sciences, Xiamen University, Xiamen 361005, China

Abstract

In this paper, we consider the numerical solutions of a fractional partial differ-ential equation with Riesz space fractional derivatives (FPDE-RSFD) on a finitedomain. Two kinds of FPDE-RSFD are considered: the Riesz fractional diffusionequation (RFDE) and the Riesz fractional advection-dispersion equation (RFADE).RFDE is obtained from the standard diffusion equation by replacing the second-order space derivative with the Riesz fractional derivative of order α ∈ (1, 2];RFADE is obtained from the standard advection-dispersion equation by replac-ing the the first-order and second-order space derivatives with the Riesz fractionalderivatives of order β ∈ (0, 1) and of order α ∈ (1, 2], respectively. Firstly, ana-lytic solutions of both RFDE and RFADE are derived. Secondly, three numericalmethods are provided to deal with the Riesz space fractional derivatives, the L1/L2-approximation method, the standard/shifted Grunwald method, and a new matrixtransform method (MTM). Thirdly, the RFDE and RFADE are transformed into asystem of ordinary differential equations (ODE), which is then solved by the methodof lines (MOL). Finally, numerical results are given, which are in good agreementwith the analytic solutions.

Key words: fractional advection-dispersion equation, Riesz space fractionalderivative, L1/L2-approximation method, standard/shifted Grunwald method,matrix transform method, method of linesMSC(2000) 26A33, 33F05, 40C05, 65M20

Preprint submitted to Elsevier Science 29 March 2009

1 INTRODUCTION

The concept of fractional derivatives is by no means new. In fact, they arealmost as old as their more familiar integer-order counterparts [30,32,33,36].Until recently, however, fractional derivatives have been successfully appliedto problems in system biology [42], physics [1,28,29,35,43], chemistry andbiochemistry [41], hydrology [2,3,15,16], and finance [9,34,37,40]. These newfractional-order models are more adequate than the previously used integer-order models, because fractional order derivatives and integrals enables thedescription of the memory and hereditary properties of different substances[33]. This is the most significant advantage of the fractional order models incomparison with integer order models, in which such effects are neglected. Inthe area of physics, fractional space derivatives are used to model anomalousdiffusion or dispersion, where a particle spreads at a rate inconsistent with theclassical Brownian motion model [29]. Especially, the Riesz fractional deriva-tive includes a left Riemann-Liouville derivative and a right Riemann-Liouvillederivative that allow the modelling of flow regime impacts from either side ofthe domain [43]. Fractional advection-dispersion equation (FADE) is used ingroundwater hydrology to model the transport of passive tracers carried byfluid flow in a porous medium [15]. Benson et al. [2,3] also studied the FADEfor solute transport in subsurface material. However, the analytical solutions ofthe fractional order partial differential equations are usually derived in termsof Green funciton or Fox funtion [10,24], and hence are difficult to evaluate.Therefore, numerical treatments and analysis of the fractional order differen-tial equations become a fruitful research topic and have an extensive field andgreat potentiality.

A number of authors have discussed the numerical methods for solving theFADE with nonsymmetric fractional derivatives. Liu et al. [16] considered thetime FADE and the solution was obtained using a variable transformation,together with Mellin and Laplace transforms, and H-functions. Huang andLiu [11] also considered the space-time FADE and the solution was obtainedin terms of Green functions and representations of the Green function byapplying the Fourier-Laplace transforms. Meerschaert and Tadjeran [26] pre-sented practical numerical methods to solve the one-dimensional space FADEwith variable coefficients on a finite domain. Liu et al. [15] transformed thespace fractional Fokker-Planck equation into a system of ordinary differentialequations (Method of Lines), which was then solved using backward differen-

? This research has been supported by the National Natural Science Foundationof China grant 10271098, the Australian Research Council grant LP0348653, thePhD Fee Waiver Scholarship and the School of Mathematical Sciences Scholarshipat QUT.∗ Corresponding author.

Email address: f.liu@qut.edu.au or fwliu@xmu.edu.cn (F. Liu).

2

tiation formulas. Momani and Odibat [31] developed a reliable algorithm ofthe Adomian decomposition method to construct a numerical solution of thespace-time FADE in the form of a rapidly convergent series with easily com-putable components. However, they did not give its theoretical analysis. Liuet al. [23] proposed an approximation of the Levy-Feller advection-dispersionprocess by employing a random walk and finite difference methods.

The FADE with a symmetric fractional derivative, namely the Riesz fractionalderivative, is derived from the kinetics of chaotic dynamics by Saichev and Za-slavsky in [35] and summerised by Zaslavsky in [43]. Ciesielski and Leszczynski[6] present a numerical solution for the Riesz FADE (without the advectionterm) based on the finite differences method. Meerschaert and Tadjeran [27]propose a shifted Grunwald estimate for the two-sided space fractional partialdifferential equations. In the light of their paper, Shen et al. [38] present ex-plicit and implicit difference approximations for the Riesz FADE with initialand boundary conditions on a finite domain, and derive the stability and con-vergence of the proposed numerical methods. To the authors’ best knowledge,there is no other research on the numerial methods for solving Riesz FADE.Therefore, the numerical treatments for the Riesz FADE are still spare. Thismotivate us to investigate more computationally efficient numerical techniquesfor solving Riesz FADE.

In this paper, we consider the following fractional partial differential equationswith the Riesz space fractional derivatives (FPDE-RSFD):

∂u(x, t)

∂t= Kα

∂α

∂|x|α u(x, t) + Kβ∂β

∂|x|β u(x, t), 0 < t ≤ T, 0 < x < L (1)

subject to the boundary and initial conditions given by

u(0, t) = u(L, t) = 0, (2)

u(x, 0) = g(x), (3)

where u is, for example, a solute concentration; Kα and Kβ represent thedispersion coefficient and the average fluid velocity. The Riesz space fractionalderivatives of order α(1 < α ≤ 2) and β(0 < β < 1) are defined respectivelyas

∂α

∂|x|α u(x, t) = −cα(0Dαx +x Dα

L)u(x, t), (4)

∂β

∂|x|β u(x, t) = −cβ(0Dβx +x Dβ

L)u(x, t), (5)

3

where

cα =1

2 cos(πα2

), α 6= 1, cβ =

1

2 cos(πβ2

), β 6= 1,

0Dαxu(x, t) =

1

Γ(2− α)

∂2

∂x2

∫ x

0

u(ξ, t)dξ

(x− ξ)α−1,

xDαLu(x, t) =

1

Γ(2− α)

∂2

∂x2

∫ L

x

u(ξ, t)dξ

(ξ − x)α−1,

0Dβxu(x, t) =

1

Γ(1− β)

∂

∂x

∫ x

0

u(ξ, t)dξ

(x− ξ)β,

xDβLu(x, t) = − 1

Γ(1− β)

∂

∂x

∫ L

x

u(ξ, t)dξ

(ξ − x)β.

Here 0Dαx (0D

βx) and xD

αL (xD

βL) are defined as the left- and right-side Riemann-

Liouville derivatives.

The fractional kinetic equation (1) has a physical meaning (see [35,43] forfurther details). Physical considerations of a fractional advection-dispersiontransport model restrict 1 < α ≤ 2, 0 < β < 1, and we assume Kα > 0 andKβ ≥ 0 so that the flow is from left to right. The physical meaning of usinghomogeneous Dirichlet boundary conditions is that the boundary is set farenough away from an evolving plume such that no significant concentrationsreach that boundary [25,27]. In the case of α = 2 and β = 1, Eq. (1) reducesto the classical advection-dispersion equation (ADE). In this paper, we onlyconsider the fractional cases: when Kβ = 0, Eq. (1) reduces to the Rieszfractional diffusion equation (RFDE) [35]; when Kβ 6= 0, the Riesz fractionaladvection-dispersion equation (RFADE) is obtained [43].

There are different techniques for approximating different fractional deriva-tives [33,36,32]. People may get confused on this issue. One contribution ofthis paper is that we illustrate how to choose different techniques for differentfractional derivatives. For simulation of Riemann-Liouville fractional deriva-tive, there exists a link between the Riemann-Liouville and Grunwald-Letnikovfractional derivatives. This allows the use of the Riemann-Liouville defini-tion during problem formulation, and then the use of the Grunwald-Letnikovdefinition for obtaining the numerical solution [33]. When using the stan-dard Grunwald approximation for the Riemann-Liouville fractional derivative,Meerschaert and Tadjeran [26] found that the standard Grunwald approxima-tion to discretize the fractional diffusion equation results in an unstable finitedifference scheme regardless of whether the resulting finite difference methodis an explicit or an implicit system. In order to show how to use the standardGrunwald approximation and Shifted Grunwald approximation, we considerthe the Riesz fractional advection-dispersion equation 1. We approximate thediffusion term by the shifted Grunwald method and approximate the advec-tion term by the standard Grunwald method. In our opinion, this is a new

4

and novel approach.

Another contribution of this paper is that we clarify the two existing defini-tions of the operator ∂α

∂|x|α . One is defined on the Fourier transform on infinite

domain [36], and the other one is defined through eigenfuntion expansion ona finite domain [12]. On infinite domain, we derived the equivalence of the op-erators ∂α

∂|x|α and −(−∆)α2 . We notice that this equivalence also holds on finite

domain when the function u(x) is subject to homogeneous Dirichlet boundaryconditions. Here, we propose two numerical methods to approximate the Rieszfractional derivative: the standard/shifted Grunwald method and the L1/L2approximation. For ∂α

∂|x|α defined through eigenfuntion expansion, we derivethe analytic solutions of the RFDE and RFADE 1 using a spectral represen-tation, and develop the matrix transform method (MTM) to slove the RFDEand RFADE 1. Though the definitions of fractional Laplacian are different oninfinite domain and finite domain, the numerical results shows that the MTMis still an efficient approximation of the Riesz fractional derivative.

After introducing three numerical methods to estimate the Riesz fractinalderivative, we use the fractional method of lines proposed by Liu et al. [15]to transform the RFDE and RFADE into a system of time ordinary differen-tial equations (TODEs). Then, the TODEs system is solved using a differen-tial/algebraic system solver (DASSL) [5].

The rest of this paper is organized as follows. Section 2 presents some im-portant definitions and lemmas used in this paper. Analytic and numericalsolutions for the RFDE and RFADE are derived in Sections 3 and 4, respec-tively. Section 5 provides the numerical results for solving the RFDE andRFADE. Finally, the main results are summarised in Section 6.

2 PRELIMINARY KNOWLEDGE

In this section, we outline important definitions and lemma used throughoutthe remaining sections of this paper.

Definition 1. [8] The Riesz fractional operator for n− 1 < α ≤ n on a finiteinterval 0 ≤ x ≤ L is defined as

RDαxu(x, t) =

∂α

∂|x|α u(x, t) = −cα(0Dαx +x Dα

L)u(x, t)

where

cα =1

2 cos(πα2

), α 6= 1,

5

0Dαxu(x, t) =

1

Γ(n− α)

∂n

∂xn

∫ x

0

u(ξ, t)dξ

(x− ξ)α+1−n,

xDαLu(x, t) =

(−1)n

Γ(n− α)

∂n

∂xn

∫ L

x

u(ξ, t)dξ

(ξ − x)α+1−n.

Lemma 1. For a function u(x) defined on the infinite domain [−∞ < x < ∞],the following equality holds

−(−∆)α2 u(x) = − 1

2 cos πα2

[−∞Dαxu(x) +x Dα

∞u(x)] =∂α

∂|x|α u(x).

Proof. According to Samko et al. [36], a fractional power of the Laplaceoperator is defined as follows:

−(−∆)α2 u(x) = −F−1|x|αFu(x). (6)

where F and F−1 denote the Fourier transform and inverse Fourier transformof u(x), respectively. Hence, we have

−(−∆)α2 u(x) = − 1

2π

∫ ∞

−∞e−ixξ|ξ|α

∫ ∞

−∞eiξηu(η)dηdξ.

Supposing that u(x) vanishes at x = ±∞, we perform integration by parts,

∫ ∞

−∞eiξηu(η)dη = − 1

iξ

∫ ∞

−∞eiξηu′(η)dη.

Thus, we obtain

−(−∆)α2 u(x) = − 1

2π

∫ ∞

−∞u′(η)

[i∫ ∞

−∞eiξ(η−x) |ξ|α

ξdξ

]dη.

Let I = i∫∞−∞ eiξ(η−x) |ξ|α

ξdξ, then

I = i

[−

∫ ∞

0eiξ(x−η)ξα−1dξ +

∫ ∞

0eiξ(η−x)ξα−1dξ

].

Noting that

L(tv−1) =∫ ∞

0e−sttv−1dt =

Γ(v)

sv, Re(v) > 0,

for 0 < α < 1, we have

I = i

[ −Γ(α)

[i(η − x)]α+

Γ(α)

[i(x− η)]α

]=

sign(x− η)Γ(α)Γ(1− α)

|x− η|αΓ(1− α)[iα−1 + (−i)α−1].

6

Using Γ(α)Γ(1− α) = πsin πα

and iα−1 + (−i)α−1 = 2 sin πα2

, we obtain

I =sign(x− η)π

cos πα2|x− η|αΓ(1− α)

.

Hence, for 0 < α < 1,

−(−∆)α2 u(x) = − 1

2π

∫ ∞

−∞u′(η)

sign(x− η)π

cos πα2|x− η|αΓ(1− α)

dη

= − 1

2 cos πα2

[1

Γ(1− α)

∫ x

−∞u′(η)

(x− η)αdη − 1

Γ(1− α)

∫ ∞

x

u′(η)

(η − x)αdη

].

Following [33], for 0 < α < 1, the Grunwald-Letnikov fractional derivative in[a, x] is given by

aDαxu(x) =

u(a)(x− a)−α

Γ(1− α)+

1

Γ(1− α)

∫ x

a

u′(η)

(x− η)αdη.

Therefore, if u(x) tends to be zero for a→−∞, then we have

−∞Dαxu(x) =

1

Γ(1− α)

∫ x

−∞u′(η)

(x− η)αdη.

Similarly, if u(x) tends to be zero for b→∞, then we have

xDα∞u(x) =

−1

Γ(1− α)

∫ ∞

x

u′(η)

(η − x)αdη.

Hence, if u(x) is continuous and u′(x) is integrable for x ≥ a, then for everyα (0 < α < 1), the Riemann-Liouville derivative exists and coincides with theGrunwald-Letnikov derivative. Finally, for 0 < α < 1, we have

−(−∆)α2 u(x) = − 1

2 cos πα2

[−∞Dαxu(x) +x Dα

∞u(x)] =∂α

∂|x|α u(x),

where

−∞Dαxu(x) =

1

Γ(1− α)

∂

∂x

∫ x

−∞u(η)dη

(x− η)α,

xDα∞u(x) =

−1

Γ(1− α)

∂

∂x

∫ ∞

x

u(η)dη

(η − x)α.

Next, we present here a similar derivation for the case of 1 < α < 2. Supposingthat u(x),u′(x), vanishes at x = ±∞, we perform integration by parts twice,to obtain ∫ ∞

−∞eiξηu(η)dη = −ξ−2

∫ ∞

−∞eiξηu′′(η)dη.

7

Hence, we have

−(−∆)α2 u(x) = − 1

2π

∫ ∞

−∞e−ixξ|ξ|α

[−ξ−2

∫ ∞

−∞eiξηu′′(η)dη

]dξ

=1

2π

∫ ∞

−∞u′′(η)

[∫ ∞

−∞eiξ(η−x)|ξ|α−2dξ

]dη.

Let I =∫∞−∞ eiξ(η−x)|ξ|α−2dξ, then

I =∫ ∞

0eiξ(x−η)ξα−2dξ +

∫ ∞

0eiξ(η−x)ξα−2dξ.

Noting that

L(tv−2) =∫ ∞

0e−sttv−2dt =

Γ(v − 1)

sv−1, Re(v) > 1,

we have

I =Γ(α− 1)

[i(η − x)]α−1+

Γ(α− 1)

[i(x− η)]α−1=

Γ(α− 1)Γ(2− α)

|x− η|α−1Γ(2− α)[iα−1 + (−i)α−1].

Using Γ(α− 1)Γ(2− α) = πsin π(α−1)

= −πsin πα

and iα−1 + (−i)α−1 = 2 sin πα2

, wehave

I =−π

cos πα2|x− η|α−1Γ(2− α)

.

Hence, for 1 < α < 2,

−(−∆)α2 u(x) =

1

2π

∫ ∞

−∞u′′(η)

−π

cos πα2|x− η|α−1Γ(2− α)

dη

= − 1

2 cos πα2

[1

Γ(2− α)

∫ x

−∞u′′(η)

(x− η)α−1dη +

1

Γ(2− α)

∫ ∞

x

u′′(η)

(η − x)α−1dη

].

Following [33], for 1 < α < 2, the Grunwald-Letnikov fractional derivative in[a, x] is given by

aDαxu(x) =

u(a)(x− a)−α

Γ(1− α)+

u′(a)(x− a)1−α

Γ(2− α)+

1

Γ(2− α)

∫ x

a

u′′(η)

(x− η)α−1dη.

Therefore, if u(x) and u′(x) tend to be zero for a→−∞, then we have

−∞Dαxu(x) =

1

Γ(2− α)

∫ x

a

u′′(η)

(x− η)α−1dη.

Similarly, if u(x) and u′(x) tend to be zero for b→∞, then we have

xDα∞u(x) =

1

Γ(2− α)

∫ ∞

x

u′′(η)

(η − x)α−1dη.

8

Hence, if u(x) and u′(x) are continuous and u′′(x) is integrable for x ≥ a, thenfor every α (1 < α < 2), the Riemann-Liouville derivative exists and coincideswith the Grunwald-Letnikov derivative. Finally, for 1 < α < 2, we have

−(−∆)α2 u(x) = − 1

2 cos πα2

[−∞Dαxu(x) +x Dα

∞u(x)] =∂α

∂|x|α u(x),

where

−∞Dαxu(x) =

1

Γ(2− α)

∂2

∂x2

∫ x

−∞u(η)dη

(x− η)α−1,

xDα∞u(x) =

1

Γ(2− α)

∂2

∂x2

∫ ∞

x

u(η)dη

(η − x)α−1.

Further, if n− 1 < α < n, then

−(−∆)α2 u(x) = − 1

2 cos πα2

[−∞Dαxu(x) +x Dα

∞u(x)] =∂α

∂|x|α u(x),

where

−∞Dαxu(x) =

1

Γ(n− α)

∂n

∂xn

∫ x

−∞u(η)dη

(x− η)α+1−n,

xDα∞u(x) =

(−1)n

Γ(n− α)

∂n

∂xn

∫ ∞

x

u(η)dη

(η − x)α+1−n.

Remark 1. For a function u(x) given in a finite interval [0, L], the aboveequality also holds when setting

u∗(x) =

u(x) x ∈ (0, L),

0 x /∈ (0, L),

i.e., u∗(x) = 0 on the boundary points and beyond the boundary points.

Definition 2. [13] Suppose the Laplacian (−∆) has a complete set of or-thonormal eigenfunctions ϕn corresponding to eigenvalues λ2

n on a boundedregion D, i.e., (−∆)ϕn = λ2

nϕn on a bounded region D; B(u) = 0 on ∂D,where B(u) represents homogeneous Dirichlet boundary conditions. Let

Fγ = {f =∞∑

n=1

cnϕn, cn = 〈f, ϕn〉|∞∑

n=1

|cn|2|λ|γn < ∞, γ = max(α, 0)},

then for any f ∈ Fγ, (−∆)α2 is defined by

(−∆)α2 f =

∞∑

n=1

cn(λ2n)

α2 ϕn.

9

3 ANALYTIC SOLUTION AND NUMERICAL METHODS OFTHE RFDE

In this section, we consider the following Riesz fractional diffusion equation(RFDE):

∂u(x, t)

∂t= Kα

∂α

∂|x|α u(x, t), 0 < t ≤ T, 0 < x < L, 1 < α ≤ 2 (7)

with the boundary and initial conditions given by

u(0, t) = u(L, t) = 0, (8)

u(x, 0) = g(x). (9)

Firstly, we derive the analytic solution of the RFDE in Section 3.1, and thenintroduce three different numerical methods for solving the RFDE in Sections3.2-3.4.

3.1 Analytic solution for the RFDE

In this subsection, we present a spectral representation for the RFDE. Follow-ing Lemma 1, we first recognize that

∂αu

∂|x|α = −(−∆)α2 u,

for the function u(x) on a finite domain [0, L] with homogeneous Dirichletboundary values. Hence, the RFDE (7) can be expressed in the form

∂u(x, t)

∂t= −Kα(−∆)

α2 u(x, t). (10)

The spectral representation of the Laplacian operator (−∆) is obtained bysolving the eigenvalue problem:

−∆ϕ = λϕ,

ϕ(0) = ϕ(L) = 0.

The eigenvalues are λn = n2π2

L2 for n = 1, 2, · · · , and the corresponding eigen-functions are nonzero constant multiples of ϕn(x) = sin(nπx

L). Next, the solu-

10

tion is given by

u(x, t) =∞∑

n=1

cn(t) sin(nπx

L),

which automatically satisfies the boundary condition (8). Using Definition 2and substituting u(x, t) into Eq.(10), we obtain

∞∑

n=1

{dcn

dt+ Kα(λn)

α2 cn} sin(

nπx

L) = 0.

The problem for cn becomes into a system of ordinary differential equations

dcn

dt+ Kα(λn)

α2 cn = 0,

which has the general solution

cn(t) = cn(0) exp(−Kα(λn)α2 t).

To obtain cn(0) we use the initial condition (9)

u(x, 0) =∞∑

n=1

cn(0) sin(nπx

L) = g(x),

from which we deduce that

cn(0) =2

L

∫ L

0g(ξ) sin(

nπξ

L)dξ = bn.

With this choice of the coefficients we have the solution for the distributionfunction:

u(x, t) =∞∑

n=1

bn sin(nπx

L) exp(−Kα(

n2π2

L2)

α2 t). (11)

3.2 L2-approximation method for the RFDE

In this section, we provide a numerical solution of the RFDE by the L2-approximation method. In Section 1, the Riesz fractional derivative of order1 < α ≤ 2 on a finite interval [0, L] was defined in the sense of the Riemann-Liouville definition. There is another very important definition for the frac-tional derivative, namely, the Grunwald-Letnikov definition. For 1 < α ≤ 2,the left- and right-handed Grunwald-Letnikov fractional derivatives on [0, L]

11

are, respectively, given by

0Dαxu(x) =

u(0)x−α

Γ(1− α)+

u′(0)x1−α

Γ(2− α)+

1

Γ(2− α)

∫ x

0

u(2)(ξ)dξ

(x− ξ)α−1, (12)

xDαLu(x) =

u(L)(L− x)−α

Γ(1− α)+

u′(L)(L− x)1−α

Γ(2− α)+

1

Γ(2− α)

∫ L

x

u(2)(ξ)dξ

(ξ − x)α−1.

(13)

Following [33], there exists a link between these two approaches to differenti-ation of arbitrary real order. Let us suppose that the function u(x) is n − 1times continuously differentiable in the interval [0, L] and that u(n)(x) is inte-grable in [0, L]. Then for every α (0 ≤ n− 1 < α < n) the Riemann-Liouvillederivative exists and coincides with the Grunwald-Letnikov derivative. This re-lationship between the Riemann-Liouville and Grunwald-Letnikov definitionsallows the use of the Riemann-Liouville definition during the problem formu-lation, and then the Grunwald-Letnikov definition for obtaining the numericalsolution [15]. Therefore, using the link between these two definitions and theL2-algorithm proposed by [15,32], we can easily obtain the following numericaldiscretization scheme for the RFDE.

Assume that the spatial domain is [0, L]. The mesh is N equal intervals ofh = L/N and xl = lh for 0 ≤ l ≤ N .

The second term of the right-hand side of Eq. (12) can be approximated by

u′(0)x1−α

Γ(2− α)≈ h−α

Γ(2− α)lα−1(u1 − u0).

The third term of the right-hand side of Eq. (12) can be approximated by

1

Γ(2− α)

∫ x

0

u(2)(ξ)dξ

(x− ξ)α−1=

1

Γ(2− α)

∫ x

0

u(2)(x− ξ)dξ

ξα−1

=1

Γ(2− α)

l−1∑

j=0

∫ (j+1)h

jh

u(2)(x− ξ)dξ

ξα−1

≈ 1

Γ(2− α)

l−1∑

j=0

u(x− (j − 1)h)− 2u(x− jh) + u(x− (j + 1)h)

h2

∫ (j+1)h

jh

dξ

ξα−1

=h−α

Γ(3− α)

l−1∑

j=0

(ul−j+1 − 2ul−j + ul−j−1)[(j + 1)2−α − j2−α].

Hence, we obtain an approximation of the left-handed fractional derivative

12

(12) with 1 < α ≤ 2 as

0Dαxu(xl) ≈ h−α

Γ(3− α)

{(1− α)(2− α)u0

lα+

(2− α)(u1 − u0)

lα−1

+l−1∑

j=0

(ul−j+1 − 2ul−j + ul−j−1)[(j + 1)2−α − j2−α]

}. (14)

Similarly, we can derive an approximation of the right-handed fractional deriva-tive (13) with 1 < α ≤ 2 as

xDαLu(xl) ≈ h−α

Γ(3− α)

{(1− α)(2− α)uN

(N − l)α+

(2− α)(uN − uN−1)

(N − l)α−1

+N−l−1∑

j=0

(ul+j−1 − 2ul+j + ul+j+1)[(j + 1)2−α − j2−α]

}. (15)

Therefore, using the fractional Grunwald-Letnikov definitions (12) and (13),together with the numerical approximations (14) and (15), the RFDE (7)can be cast into the following system of time ordinary differential equations(TODEs):

dul

dt≈− Kαh−α

2 cos(πα2

)Γ(3− α)

{(1− α)(2− α)u0

lα+

(2− α)

lα−1(u1 − u0)

+l−1∑

j=0

(ul−j+1 − 2ul−j + ul−j−1)[(j + 1)2−α − j2−α]

+(1− α)(2− α)uN

(N − l)α+

(2− α)

(N − l)(α−1)(uN − uN−1)

+N−l−1∑

j=0

(ul+j−1 − 2ul+j + ul+j+1)[(j + 1)2−α − j2−α]

}. (16)

A number of efficient techniques for implementing Eq.(16) have been previ-ously proposed [5,7,14,39]. Brenan et al. [5] developed the differential/algebraicsystem solver (DASSL), which is based on the backward difference formulas(BDF). DASSL approximates the derivatives using the k−th order BDF, wherek ranges from one to five. At every step, it chooses the order k and step sizebased on the behaviour of the solution. In this work, we use DASSL as ourTODEs solver. This technique has been used by many researchers when solv-ing adsorption problems involving step gradients in bidisperse solids [4,17,19],hyperbolic models of transport in bidisperse solids [18], transport problems in-volving steep concentration gradients [20], and modelling saltwater intrusioninto coastal aquifers [21,22].

13

3.3 Shifted Grunwald approximation method for the RFDE

The shifted Grunwald formula for discretizing the two-sided fractional deriva-tive is proposed by Meerschaert and Tadjeran [27], and it was shown thatthe standard (i.e., unshifted) Grunwald formula to discretize the fractionaldiffusion equation results in an unstable finite difference scheme regardless ofwhether the resulting finite difference method is an explicit or an implicit sys-tem. Hence, in this section, we discretise the Riesz fractional derivative ∂α

∂|x|α u

by the shifted Grunwald formulae [27]:

0Dαxu(xl) =

1

hα

l+1∑

j=0

gjul−j+1 + O(h), (17)

xDαLu(xl) =

1

hα

N−l+1∑

j=0

gjul+j−1 + O(h), (18)

where the coefficients are defined by

g0 = 1, gj = (−1)j α(α− 1)...(α− j + 1)

j!

for j = 1, 2, ..., N. Then the RFDE (7) can be cast into the following systemof time ordinary differential equations (TODEs):

dul

dt≈ − Kα

2 cos(πα2

)hα

[l+1∑

j=0

gjul−j+1 +N−l+1∑

j=0

gjul+j−1

], (19)

which can also be solved by DASSL.

3.4 Matrix transform method for the RFDE

Ilic et al. [12] proposed the Matrix transform method (MTM) for a spacefractional diffusion equation with homogeneous boundary conditions. Theyhave shown that the MTM provides the best approximation to the analyticalsolution. In this section, we apply this new technique to the RFDE. Following[12], we approximate Eq.(10) by the matrix representation:

∂U

∂t= −ηA

α2 U, (20)

14

where η = Kα

hα , h is the discrete spatial step defined as h = LN

, and U ∈ RN−1,A ∈ RN−1×N−1 are given respectively by

U =

u1

...

uN−1

,A =

2 −1

−1 2 −1

−1 2 −1. . . . . . −1

−1 2

.

The matrix A is symmetric positive definite (SPD), and therefore there exitsa nonsingular matrix P ∈ RN−1×N−1 such that

A = PΛPT,

where Λ = diag(λ1, λ2, · · · , λN−1), λi(i = 1, 2, · · · , N − 1) being the eigenval-ues of A.

Hence, Eq.(20) becomes the following system of TODEs:

∂U

∂t= −η(PΛ

α2 PT)U, (21)

and initially, we have

U(0) = [g(h), g(2h), . . . , g((N − 1)h)]T . (22)

DASSL can again be used as the solver for this TODEs system.

4 ANALYTIC SOLUTION AND NUMERICAL METHODS FORTHE RFADE

In this section, we consider the following Riesz fractional advection-dispersionequation (RFADE):

∂u(x, t)

∂t= Kα

∂α

∂|x|α u(x, t) + Kβ∂β

∂|x|β u(x, t), 0 < t ≤ T, 0 < x < L (23)

with the boundary and initial conditions given by

u(0, t) = u(L, t) = 0, (24)

u(x, 0) = g(x), (25)

15

where Kα 6= 0, Kβ 6= 0, 1 < α ≤ 2, 0 < β < 1. Firstly, the analytic solutionis provided in Section 4.1, and then we introduce three different numericalmethods for solving the RFADE in Sections 4.2-4.4.

4.1 Analytic solution for the RFADE

In this subsection, we present a spectral representation for the RFADE (23).Following Lemma 1, we first recognize that

∂αu

∂|x|α = −(−∆)α2 u and

∂βu

∂|x|β = −(−∆)β2 u

for the function u(x) on a finite domain [0, L] with homogeneous Dirichletboundary values. Hence, the RFADE (23) can be expressed in the form

∂u(x, t)

∂t= −Kα(−∆)

α2 u(x, t)−Kβ(−∆)

β2 u(x, t). (26)

Next, set

u(x, t) =∞∑

n=1

cn(t) sin(nπx

L),

which automatically satisfies the boundary conditions (24). Using Definition2 and substituting u(x, t) into equation (26), we obtain

∞∑

n=1

{dcn

dt+ [Kα(λn)

α2 + Kβ(λn)

β2 ]cn} sin(

nπx

L) = 0.

The problem for cn becomes

dcn

dt+ [Kα(λn)

α2 + Kβ(λn)

β2 ]cn = 0,

which has the general solution

cn(t) = cn(0) exp(−[Kα(λn)α2 + Kβ(λn)

β2 ]t).

To obtain cn(0), we use the initial condition (25)

u(x, 0) =∞∑

n=1

cn(0) sin(nπx

L) = g(x),

which gives

cn(0) =2

L

∫ L

0g(ξ) sin(

nπξ

L)dξ = bn.

16

Hence, the solution for the distribution function is given as:

u(x, t) =∞∑

n=1

bn sin(nπx

L) exp(−[Kα(λn)

α2 + Kβ(λn)

β2 ]t), (27)

where λn = n2π2

L2 .

4.2 L1- and L2-approximation method for the RFADE

In this section, we approximate the diffusion term by the L2-algorithm aspresented in Section 3.2, and estimate the advection term by the L1-algorithmas follows.

For 0 < β < 1, the left- and right-handed Grunwald-Letnikov fractional deriva-tives on [0, L] are, respectively, given by

0Dβxu(x) =

u(0)x−β

Γ(1− β)+

1

Γ(1− β)

∫ x

0

u′(ξ)dξ

(x− ξ)β, (28)

xDβLu(x) =

u(L)(L− x)−β

Γ(1− β)+

1

Γ(1− β)

∫ L

x

u′(ξ)dξ

(ξ − x)β. (29)

Using the link between the Riemann-Liouville and Grunwald-Letnikov frac-tional derivatives [33], the second term of the right-hand side of Eq.(28) canbe approximated by

1

Γ(1− β)

∫ x

0

u′(ξ)dξ

(x− ξ)β=

1

Γ(1− β)

∫ x

0

u′(x− ξ)dξ

ξβ

=1

Γ(1− β)

l−1∑

j=0

∫ (j+1)h

jh

u′(x− ξ)dξ

ξβ

≈ 1

Γ(1− β)

l−1∑

j=0

u(x− jh)− u(x− (j + 1)h)

h

∫ (j+1)h

jh

dξ

ξβ

=h−β

Γ(2− β)

l−1∑

j=0

(ul−j − ul−j−1)[(j + 1)1−β − j1−β].

Hence, we obtain an approximation of the left-handed fractional derivative

17

(28) with 0 < β < 1 as

0Dβxu(xl) ≈ h−β

Γ(2− β)

{(1− β)u0

lβ+

l−1∑

j=0

(ul−j − ul−j−1)[(j + 1)1−β − j1−β]

}.

(30)

Similarly, we can derive an approximation of the right-handed fractional deriva-tive (29) with 0 < β < 1 as

xDβLu(xl) ≈ h−β

Γ(2− β)

{(1− β)uN

(N − l)β+

N−l−1∑

j=0

(ul+j − ul+j+1)[(j + 1)1−β − j1−β]

}.

(31)

Using the numerical approximations (30) and (31), the advection term of theRFADE (23) can be approximated by

Kβ∂β

∂|x|β u(xl) ≈− Kβh−β

2 cos(πβ2

)Γ(2− β)

{(1− β)u0

lβ

+l−1∑

j=0

(ul−j−1 − ul−j)[(j + 1)1−β − j1−β] +(1− β)uN

(N − l)β

+N−l−1∑

j=0

(ul+j+1 − ul+j)[(j + 1)1−β − j1−β]

}. (32)

Therefore, using the numerical approximation of the diffusion term (16) andthe numerical approximation of the advection term (32), we transform theRFADE (23) into the following system of TODEs

dul

dt≈− Kαh−α

2 cos(πα2

)Γ(3− α)

{(1− α)(2− α)u0

lα+

(2− α)

lα−1(u1 − u0)

+l−1∑

j=0

(ul−j+1 − 2ul−j + ul−j−1)[(j + 1)2−α − j2−α]

+(1− α)(2− α)uN

(N − l)α+

(2− α)

(N − l)(α−1)(uN − uN−1)

+N−l−1∑

j=0

(ul+j−1 − 2ul+j + ul+j+1)[(j + 1)2−α − j2−α]

}

− Kβh−β

2 cos(πβ2

)Γ(2− β)

{(1− β)u0

lβ

+l−1∑

j=0

(ul−j−1 − ul−j)[(j + 1)1−β − j1−β] +(1− β)uN

(N − l)β

+N−l−1∑

j=0

(ul+j+1 − ul+j)[(j + 1)1−β − j1−β]

}. (33)

18

This system of TODEs is then solved using the standard ODE solver, DASSL.

4.3 Shifted and standard Grunwald approximation method for the RFADE

In this section, we approximate the diffusion term by the shifted Grunwaldmethod as presented in Section 3.3, and estimate the advection term by thestandard Grunwald formulae [33] as follows:

0Dβxu(xl) =

1

hβ

l∑

j=0

wjul−j + O(h), xDβLu(xl) =

1

hβ

N−l∑

j=0

wjul+j + O(h),

where the coefficients are defined by

w0 = 1, wj = (−1)j β(β − 1)...(β − j + 1)

j!,

for j = 1, 2, ..., N. If we approximate both the diffusion and advection termsby the shifted Grnwald method or by the standard Grnwald method, thenunstable solutions will be obtained.

Then, the RFADE (23) can be cast into the following system of time ordinarydifferential equations (TODEs):

dul

dt≈− Kα

2 cos(πα2

)hα

[l+1∑

j=0

gjul−j+1 +N−l+1∑

j=0

gjul+j−1

]

− Kβ

2 cos(πβ2

)hβ

[l∑

j=0

wjul−j +N−l∑

j=0

wjul+j

], (34)

which can again be solved using DASSL.

4.4 Matrix transform method for the RFADE

The matrix transform technique presented in Section 3.4 can be easily ap-plied to the advection term ∂βu

∂|x|β (0 < β < 1). Hence, we obtain the matrix

representation for the RFADE (23) as follows:

∂U

∂t= −ηαA

α2 U− γβA

β2 U = −ηα(PΛ

α2 PT)U− γβ(PΛ

β2 PT)U,

19

where ηα = Kα

hα , γβ =Kβ

hβ , h = LN

; U ∈ RN−1, A ∈ RN−1×N−1 and P ∈RN−1×N−1 are defined in Section 3.4; and U(0) is as given in Eq.(22). DASSLcan again be used as the solver for this TODEs system.

5 NUMERICAL RESULTS OF THE RFDE AND RFADE

In this section, we provide numerical examples for the RFDE and RFADE todetermine the accuracy of the numerical methods in comparison to the analyticsolutions presented throughout Sections 3 and 4. We also use our solutionmethods to demonstrate the changes in solution behaviour that arise whenthe exponent is varied from integer order to fractional order, and to identifythe differences between solutions with and without the advection term.

5.1 Numerical examples of the RFDE

In this subsection, we present two examples of the RFDE on a finite domain[0, π]. Firstly, we consider the following equation

∂u(x, t)

∂t= Kα

∂α

∂|x|α u(x, t), 0 < t ≤ T, 0 < x < π (35)

with the initial and boundary conditions given by

u(x, 0) = x2(π − x), (36)

u(0, t) = u(π, t) = 0. (37)

According to Section 3.1,

bn = cn(0) =2

π

∫ π

0g(ξ) sin(

nπξ

π)dξ =

2

π

∫ π

0ξ2(π − ξ) sin(nξ)dξ

=8

n3(−1)n+1 − 4

n3.

Hence, the analytic solution of Eq.(35)-(37) is given by

u(x, t) =∞∑

n=1

(8(−1)n+1 − 4

n3) sin(nx) exp(−(n2)

α2 Kαt).

In Figure 1, this analytic solution is compared with the three numerical meth-ods proposed in Sections 3.2-3.4, for t = 0.4, α = 1.8 and Kα = 0.25. It canbe seen that all three numerical methods perform well. However, from Table

20

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

4

4.5

x

u(x,

t=0.

4) a

t α=

1.8

L2Shifted GrunwaldMTMAnalytical

Fig. 1. Comparison of the numerical solutions with the analytic solution at t = 0.4for the RFDE (35)-(37) with Kα = 0.25 and α = 1.8.

Table 1Maximum errors for 3 methods when 0 < x < π

h = π/N L2 approximation Matrix transform Shifted Grunwald

π/10 9.471E-2 2.217E-2 6.726E-2

π/20 9.472E-2 5.759E-3 7.343E-2

π/40 9.041E-2 1.481E-3 8.693E-2

π/80 8.604E-2 3.727E-4 9.535E-2

1, we see that, among three methods, only the matrix transform method isstable and convergent. Using the matrix method from Section 3.4, The so-lution profiles of RFDE for different values of α = 1.2, 1.4, 1.6, 1.8, 2.0 whent = 0.5, 1.5, 2.5, 3.5 are shown in Figure 2. It can be seen that the process de-scribed by the RFDE is slightly more skewed to the right than that modelledby the standard diffusion equation.

To further demonstrate the impact of fractional order to the solution be-haviour, another example of RFDE with a different initial condition is nowconsidered:

∂u(x, t)

∂t= Kα

∂α

∂|x|α u(x, t), 0 < t ≤ T, 0 < x < π, (38)

u(x, 0) = sin 4x, (39)

u(0, t) = u(π, t) = 0. (40)

21

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

4

4.5

x

u(x,

t)

α=1.2α=1.4α=1.6α=1.8α=2.0

t=0.5

t=1.5

t=2.5

t=3.5

Fig. 2. The numerical approximation of u(x, t) for the RFDE (35)-(37) with differentvalues of α = 1.2, 1.4, 1.6, 1.8, 2.0 when t = 0.5, 1.5, 2.5, 3.5.

11.2

1.41.6

1.82

0

1

2

3

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

αx

u(x,

t=0.

5)

Fig. 3. The numerical approximation of u(x, t) for the RFDE (38)-(40) when t = 0.5and Kα = 0.25.

Figure 3 presents the solution profiles of RFDE over space for 1 < α ≤ 2 whent = 0.5. It can be observed that as α is decreased from 2 to 1 the amplitude ofthe sinusoidal solution behaviour is increased. Figure 4 displays the solutionprofiles of RFDE over space for 0 < t < 1 when α = 2.0 (left) and α = 1.5(right), respectively. We can see that the fractional diffusion (α = 1.5) is slowerthan the standard diffusion (α = 2.0). From Figures 1-4, we conclude that the

22

0

0.5

10

1

2

3

−1.5

−1

−0.5

0

0.5

1

tx

u(x,

t) fo

r α=

2.0

0

0.5

10

1

2

3

−1.5

−1

−0.5

0

0.5

1

tx

u(x,

t) fo

r α=

1.5

Fig. 4. A comparison of solutions for the RFDE (38)-(40) with Kα = 0.25 whenα = 2 (left) and α = 1.5 (right).

solution continuously depends on the Riesz space fractional derivatives.

5.2 Numerical examples of the RFADE

In this subsection, we present two examples of the RFADE on a finite domain[0, π]. Firstly, consider the following problem

∂u(x, t)

∂t= Kα

∂αu(x, t)

∂|x|α + Kβ∂βu(x, t)

∂|x|β , 0 < x < π, t > 0, (41)

with the initial and boundary conditions given by

u(x, 0) = x2(π − x), (42)

u(0, t) = u(π, t) = 0. (43)

According to Section 4.1, the analytic solution of Eq.(41)-(43) is given by

u(x, t) =∞∑

n=1

[8

n3(−1)n+1 − 4

n3] sin(nx) exp(−[Kα(n2)

α2 + Kβ(n2)

β2 ]t). (44)

The analytical solution is compared with the numerical methods in Figure 5for t = 0.4, α = 1.8, β = 0.4, and Kα = Kβ = 0.25. It is seen that all three

23

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

4

4.5

x

u(x,

t=0.

4) a

t α=

1.8,

β=

0.4

L1/L2Standard/Shifted GrunwaldMTMAnalytical

Fig. 5. Comparison of the numerical solutions with the analytic solution at t = 0.4for the RFADE (41)-(43) with α = 1.8, β = 0.4, and Kα = Kβ = 0.25.

numerical methods perform well. Based on the matrix method from Section4.4, Figure 6 shows the solution profiles for different values of α at differenttimes t, while Figure 7 shows the solution profiles for different values of β atdifferent times t. In both figures, it can be seen that the process describedby the RFADE is again slightly more skewed to the right than that modelledby the classical ADE (α = 2, β = 1). To further demonstrate the impact of

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

4

4.5

x

u(x,

t)

α=1.2α=1.4α=1.6α=1.8α=2.0

t=0.5

t=1.5

t=2.5

t=3.5

Fig. 6. The numerical approximation of u(x, t) for the RFADE (41)-(43) with dif-ferent values of α = 1.2, 1.4, 1.6, 1.8, 2.0 when β = 0.4 and t = 0.5, 1.5, 2.5, 3.5.

24

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

4

4.5

x

u(x,

t)

β=0.1β=0.3β=0.5β=0.7β=0.9

t=0.5

t=1.5

t=2.5

t=3.5

Fig. 7. The numerical approximation of u(x, t) for the RFADE (41)-(43) with dif-ferent values of β = 0.1, 0.3, 0.5, 0.9 when α = 1.8 and t = 0.5, 1.5, 2.5, 3.5.

fractional order, another example of RFADE with a different initial conditionis now considered:

∂u(x, t)

∂t= Kα

∂αu(x, t)

∂|x|α + Kβ∂βu(x, t)

∂|x|β , 0 < x < π, t > 0, (45)

u(x, 0) = sin 4x, (46)

u(0, t) = u(π, t) = 0. (47)

Figures 8 and 9 displays the changes in the solution behaviours when α variesfrom 1 to 2 and β ranges from 0 to 1, respectively. In both figures, consis-tent with our observations for the RFDE in Section 5.1, we see that as α isdecreased from 2 to 1 or β from 1 to 0 the amplitude of the sinusoidal so-lution behaviour is increased. In Figure 10, the solution profiles of RFADEover space for 0 < t < 1 are displayed when α = 2.0, β = 1.0 (left) andα = 1.5, β = 0.7 (right), respectively. It can be observed that the fractionaladvection-dispersion process (α = 1.5, β = 0.7) is slower than the classicaladvection-dispersion process (α = 2.0, β = 1.0). Furthermore, from Figures8-10, it is seen that the solution continuously depends on the Riesz spacefractional derivatives. Finally, Figure 11 provides a comparison of the RFDE(38)-(40) and RFADE (45)-(47) with Kα = Kβ = 0.25 at time t = 0.5, whereit can be observed that there is an offset when the fractional advection termis added to the fractional diffusion process.

Example 2 We consider the following equation

25

11.2

1.41.6

1.820

12

3

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

αx

u(x,

t=0.

5) fo

r β=

0.7

Fig. 8. The numerical approximation of u(x, t) for the RFADE (45)-(47) whenβ = 0.7 and t = 0.5.

00.2

0.40.6

0.81

0

1

2

3

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

βx

u(x,

t=0.

5) fo

r α=

1.5

Fig. 9. The numerical approximation of u(x, t) for the RFADE (45)-(47) whenα = 1.5 and t = 0.5.

∂u(x, t)

∂t= Kα

∂α

∂|x|α u(x, t) + Kβ∂β

∂|x|β u(x, t) + f(x, t), 1 < x < 1, 0 < t ≤ T,(48)

with the initial and boundary conditions given by

u(x, 0) = 0, (49)

26

0

0.5

10

1

2

3

−1.5

−1

−0.5

0

0.5

1

tx

u(x,

t) fo

r α=

2.0,

β=

1.0

0

0.5

10

1

2

3

−1.5

−1

−0.5

0

0.5

1

tx

u(x,

t) fo

r α=

1.5,

β=

0.7

Fig. 10. A comparison of solutions for the RFADE (45)-(47) when α = 2.0 andβ = 1.0 (left) and α = 1.5 and β = 0.7 (right).

0 0.5 1 1.5 2 2.5 3−0.6

−0.4

−0.2

0

0.2

0.4

0.6

x

u(x,

t=0.

5)

RFADE: α=1.5,β=0.7RFDE: α=1.5

Fig. 11. Comparison of the RFDE (38)-(40) and RFADE (45)-(47) withKα = Kβ = 0.25 at time t = 0.5.

u(0, t) = u(π, t) = 0, (50)

27

where

f(x, t) =Kαtαeβt

2cos(απ/2)

{2

Γ(3− α)[x2−α + (1− x)2−α]− 12

Γ(4− α)[x3−α + (1− x)3−α]

+24

Γ(5− α)[x4−α + (1− x)4−α]

}+

Kβtαeβt

2cos(βπ/2)

{2

Γ(3− β)[x2−β + (1− x)2−β]

− 12

Γ(4− β)[x3−β + (1− x)3−β] +

24

Γ(5− β)[x4−β + (1− x)4−β]

}

+ tα−1eβt(α + βt)x2(1− x)2. (51)

The exact solution of Eq.(48)-(50) is given by

u(x, t) = tαeβtx2(1− x)2.

In this example, we take α = 1.7, β = 0.4, Kα = Kβ = 2.0, N = 100.

6 CONCLUSIONS

In this paper, three effective numerical methods for solving the RFDE andRFADE on a finite domain with homogeneous Dirichlet boundary conditionshave been described and demonstrated. They are the L1/L2-approximationmethod, the standard/shifted Grunwald method, and the matrix transformmethod. All three numerical methods perform well in approximating the an-alytic solution of the RFDE and RFADE derived using a spectral representa-tion. The methods and techniques discussed in this paper can also be appliedto solve other kinds of fractional partial differential equations, e.g., the modi-fied fractional diffusion equation where 1 < β < α ≤ 2.

Table 2Maximum errors for 3 methods

h = 1/N L2 approximation Matrix transform Shifted Grunwald

1/50 1.8144E-2 3.2649E-2 2.8191E-3

1/100 9.4917E-3 3.2901E-2 1.5093E-3

1/200 4.8584E-3 3.2961E-2 7.7821E-4

1/400 2.4586E-3 overstack 3.9459E-4

28

0 0.2 0.4 0.6 0.8 1−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x

u(x,

t=2.

0) a

t α=

1.7

L1/L2

Standard/Shifted Grunwald

MTM

Analytical

Fig. 12. A comparison of exact solution and three numerical solutions.

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