Method of approximate particular solutions for constant- and variable-order fractional diffusion...

$: Method of approximate particular solutions for constant- and variable-order fractional diffusion models$
Method of approximate particular solutions forconstant- and variable-order fractional diffusion models

Zhuo-Jia Fu a,b,n, Wen Chen a,nn, Leevan Ling b

a College of Mechanics and Materials, Hohai University, Nanjing 210098, PR Chinab Department of Mathematics, Hong Kong Baptist University, Hong Kong, PR China

a r t i c l e i n f o

Article history:Received 30 April 2014Accepted 2 September 2014

Keywords:Radial basis functionCollocation methodMeshless methodFractional diffusion

a b s t r a c t

The method of approximate particular solutions (MAPS) is an alternative radial basis function (RBF)meshless method, which is defined in terms of a linear combination of the particular solutions of theinhomogeneous governing equations with traditional RBFs as the source term. In this paper, we applythe MAPS to both constant- and variable-order time fractional diffusion models. In the discretizationformulation, a finite difference scheme and the MAPS are used respectively to discretize time fractionalderivative and spatial derivative terms. Numerical investigation examples show the present meshlessscheme has highly accuracy and computationally efficiency for various fractional diffusion models.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

In recent decades, anomalous diffusion phenomena are exten-sively observed in a wide range of engineering and physics fields[1–4], such as contaminant transport, seepage, magnetic plasma,dissipation and turbulence. To describe anomalous diffusion phenom-ena, constant-order fractional diffusion equations are considered asrecent alternative models and have received fantastic success [5–7].However, various recent experimental results [8,9] show thatconstant-order fractional diffusion equations cannot fully capturesome more complicated diffusion processes, whose diffusion beha-viors depend on the time evolution, spatial variation or evenconcentration variation. To deal with these issues, variable-orderfractional diffusion equations [10,11] have been introduced, in whichthe variable-order time fractional operator can be time-dependent,spatial-dependent, and/or concentration-dependent.

Nowadays, finite difference methods (FDMs) are popular anddominant numerical techniques for temporal and spatial discretiza-tion of constant-order [12–16] and variable-order [17–20] fractionaldiffusion equations. Their convergence, accuracy, and stability haveextensively been discussed in the literatures [21–24].

For the numerical simulations of constant-order fractional diffusionequations, with traditional FDMs for temporal discretization, severalnumerical methods have been introduced to spatial discretization of

fractional derivative equations, such as the Fourier method [25],spectral method [26], finite element method [27–29], boundaryelement method [30], and radial basis function meshless collocationmethod [31–33]. In comparison with traditional FDMs for spatialdiscretization, these methods can reduce, to a certain extent, comput-ing costs for large computational domain problems. In this work, weshall extend the idea to mitigate the computing costs in the numericalsimulation of variable-order fractional diffusion equations.

We will focus on constant- and variable-order time fractionaldiffusion equations, which only have fractional derivative in time andinteger differential operator in space. We employ a finite differencemethod for temporal discretization and introduce an alternativeradial basis function (RBF) meshless method, the method of approx-imate particular solutions (MAPS) [34–37], for spatial discretization.Chen et al. [38] first proposed the method of approximate particularsolutions (MAPS) to solving partial differential equations. Then theMAPS has been successfully applied to various physical and engineer-ing problems, such as anisotropic problems [39], nonlinear Poissonproblems [40], wave problems [41], elasticity problems [42], Stokesflow problems [43], and convection-diffusion problems [44]. Incomparison with the famous RBF method, also known as the Kansamethod, the MAPS uses a newly derived RBF as interpolation basisfunction, which include some information from the consideredgoverning equation operator. And some numerical experiments[45–47] demonstrate that the MAPS outperforms the Kansa methodin terms of both the stability and accuracy, particularly in theevaluation of partial derivatives.

This paper first applies the method of approximate particularsolutions (MAPS), to 2D constant- and variable-order fractional diffu-sion problems. A brief outline of the paper is as follows. Section 2

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Engineering Analysis with Boundary Elements

http://dx.doi.org/10.1016/j.enganabound.2014.09.0030955-7997/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author at: College of Mechanics and Materials, Hohai University,Nanjing 210098, PR China.

nn Corresponding author.E-mail addresses: [email protected] (Z.-J. Fu),

[email protected] (W. Chen).

Please cite this article as: Fu Z-J, et al. Method of approximate particular solutions for constant- and variable-order fractionaldiffusion models. Eng. Anal. Boundary Elem. (2014), http://dx.doi.org/10.1016/j.enganabound.2014.09.003i

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describes the present computational formulations for fractional diffu-sion equations. In Section 3, the efficiency and accuracy of the presentapproach are examined with some benchmark examples. Finally,Section 4 concludes this paper with some remarks.

2. Methodology

2.1. Time fractional diffusion model

Without loss of generality, we consider the following variable-order time fractional diffusion equations in a bounded domain Ωwith piecewise smooth boundary ∂Ω¼ΓDþΓN (ΓD \ ΓN ¼∅)

∂α tð Þu x; tð Þ∂tα tð Þ ¼ DΔþ v!U∇�λ

� �u x; tð ÞþQ x; tð Þ;

0oα tð Þo1; xAΩ; tA 0; Tð Þ; ð1Þ

with boundary conditions

u x; tð Þ ¼ g1 x; tð Þ; xAΓD; tA 0; Tð Þ; ð2aÞ

∂u x; tð Þ∂n

¼ g2 x; tð Þ; xAΓN ; tA 0; Tð Þ; ð2bÞ

and initial condition

u x;0ð Þ ¼ u0 xð Þ; xAΩ; ð3Þ

where Q x; tð Þ, g1 x; tð Þ, g2 x; tð Þ and u0 xð Þ are known functions; D thediffusion coefficient, λ the reaction coefficient, v! the velocityvector, n the unit outward normal, T the total time to be considered,

0 0.5 1 1.5 20

0.5

1

1.5

2

x

y

Fig. 1. Schematic configuration of uniform node distribution on a square domain(boundary nodes ‘o’ and inner nodes ‘n’).

Fig. 2. Convergence rate (RMSE) of the present method with the derived RBFformulation (11(a)) by using different time steps (dt¼0.02, 0.01, 0.005, 0.002) inExample 1. (a) Full Dirichlet boundary conditions and (b) mixed boundaryconditions.

Fig. 3. Convergence rate (RMSEx) of the present method with the derived RBFformulation (11(a)) by using different time steps (dt¼0.02, 0.01, 0.005, 0.002) inExample 1. (a) Full Dirichlet boundary conditions and (b) mixed boundaryconditions.

Z.-J. Fu et al. / Engineering Analysis with Boundary Elements ∎ (∎∎∎∎) ∎∎∎–∎∎∎2





and ð∂α tð Þ=∂tα tð ÞÞ the variable-order time fractional derivative of orderα(t) with respect to t defined by

∂α tð Þu x; tð Þ∂tα tð Þ ¼

Z t

0

1Γ 1�α η

� �� ∂u x;η� �∂η

dη

t�η� �α tð Þ; 0oα tð Þo1: ð4Þ

For more details, readers are referred to Sun [11]. When α(t) is aconstant and independent with t, the above-mentioned variable-order time fractional derivative goes back to the following famousCaputo constant-order time fractional derivative [48]

∂αu x; tð Þ∂tα

¼ 1Γ 1�αð Þ

Z t

0

∂u x;η� �∂η

dηt�η� �α; 0oαo1 ð5Þ

Therefore, the constant-order time fractional derivative can be seenas a special case of the variable-order time fractional derivative.

2.2. Finite difference method for temporal discretization

In the present method, we first introduce the finite differencemethod for temporal discretization of variable-order time frac-tional derivative (4). Then we have

∂α tð Þu x;tð Þ∂tα tð Þ ¼ R t

01

Γ 1�α ηð Þð Þ∂u x;ηð Þ

∂ηdη

t�ηð Þα tð Þ

� ∑k

j ¼ 0

Z jþ1ð Þτ

jτ

∂u x; ξ� �∂ξ

1Γ 1�α tjþ1

� �� dξ

tkþ1�ξ� �α tjþ 1ð Þ

¼a0 ukþ1�uk� �þ ∑

k

j ¼ 1ajbj uk� jþ1�uk� j

� �; kZ1

a0 u1�u0� �

; k¼ 0

8>><>>: ð6Þ

where

a0 ¼τ�α t1ð Þ

Γ 2�α t1ð Þð Þ; aj ¼τ�α tjþ 1ð Þ

Γ 2�α tjþ1� �� ;

bj ¼ jþ1ð Þ1�α tjþ 1ð Þ� j1�α tjþ 1ð Þh i; j¼ 1;2;…; k:

More details can be found in Appendix A. For a constant-ordertime fractional derivative (5), we can still use formulation (6) andset α tið Þ ¼ α; i¼ 1;2;…; k; kþ1. Then the time fractional diffu-sion Eq. (1) can be rewritten as

DΔu x; tð Þþ v!U∇u x; tð Þ�λu x; tð ÞþQ x; tð Þ


k


� �; kZ1

a0 u1�u0� �

; k¼ 0

8>><>>: ð7Þ

Later, we use the θ-method to further temporal discretization ofEq. (7) and obtain

Fig. 4. Convergence rate (RMSE) of the present method with the derived RBFformulation (11(b)) by using different time steps (dt¼0.02, 0.01, 0.005, 0.002) inExample 1. (a) Full Dirichlet boundary conditions and (b) mixed boundaryconditions.

Fig. 5. Convergence rate (RMSEx) of the present method with the derived RBFformulation (11(b)) by using different time steps (dt¼0.02, 0.01, 0.005, 0.002) inExample 1. (a) Full Dirichlet boundary conditions and (b) mixed boundaryconditions.

Z.-J. Fu et al. / Engineering Analysis with Boundary Elements ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 3





where θA 0;1½ �, and θ¼0 is explicit method, θ¼1 is implicitmethod.

2.3. Method of approximate particular solutions for spatialdiscretization

The method of approximate particular solutions (MAPS) will beused for spatial discretization of Eq. (8), its computational for-mulation can be represented as

u xi; tnþ1ð Þ ¼ ∑M

j ¼ 1αnþ1j Φ ‖xi�xj‖2

� �; ð9Þ

where ‖xi�xj‖2 the Euclidean distance between the collocationnodes xi and RBF centers xj, u xi; tnþ1ð Þ is the approximate solution

at the node xi at the instant tnþ1, and αnþ1j unknown coefficients

at the instant tnþ1, M the total number of the collocation nodes,and Φ ‖xi�xj‖2

� �denotes the derived radial basis functions (RBFs)

from Eq. (8). Since Eq. (8) can be considered as the standardinhomogeneous convection-diffusion equation with the governingoperator Δþð1=DÞ v!U∇�ððλθþa0Þ=DθÞ, the present MAPS usesthe following derived RBFs as interpolation basis function

Φ rð Þ ¼ � 1μ2 ∑

p=2

i ¼ 0

Δμ2

� �i

rp ln rð Þ� pð Þ!!2μpþ2K0 μr

� �; ð10Þ

which are derived from the modified Helmholtz equationΔ�μ2� �

Φ rð Þ ¼ϕ rð Þ with the thin plate spline RBF ðϕ rð Þ ¼ rp ln rð Þ;p¼ 2;4;6;…Þ in which μ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλθþa0� �

=ðDθÞq

, K0 is the zero order

modified Bessel function of the second kind. Here we list the firstthree derived RBF formulations from Eq. (10). When p¼2, namely,

Fig. 6. Convergence rate (RMSE) of the present method with the derived RBFformulation (11(c)) by using different time steps (dt¼0.02, 0.01, 0.005, 0.002) inExample 1. (a) Full Dirichlet boundary conditions and (b) mixed boundaryconditions.

Fig. 7. Convergence rate (RMSEx) of the present method with the derived RBFformulation (11(c)) by using different time steps (dt¼0.02, 0.01, 0.005, 0.002) inExample 1. (a) Full Dirichlet boundary conditions and (b) mixed boundaryconditions.

DθΔukþ1þθ v!U∇ukþ1� λθþa0� �

ukþ1

¼�θQkþ1� 1�θ

� �DΔ ukþ v!U∇ukþQk�λuk

� ��a0ukþ ∑

k


� �; kZ1

�θQ1� 1�θ� �

DΔ u0þ v!U∇u0þQ0�λu0� �

�a0u0; k¼ 0

8>>><>>>:

ð8Þ






ϕ rð Þ ¼ r2 ln rð Þ, we have

Φ rð Þ ¼� r2 ln rð Þ

μ2 �4 ln rð Þþ4μ4 � 4

μ4K0 μr� �

; ra0;

� 4μ4þ4γ

μ4þ 4μ4 ln

μ2

� �; r¼ 0;

8<: ð11aÞ

when p¼4, namely, ϕ rð Þ ¼ r4 ln rð Þ, we have

Φ rð Þ ¼� r4 ln rð Þ

μ2 �8r2 2 ln rð Þþ1ð Þμ4 �64 ln rð Þþ96

μ6 �64K0 μrð Þμ6 ; ra0;

�96μ6þ64γ

μ6 þ64μ6 ln

μ2

� �; r¼ 0;

8><>: ð11bÞ

when p¼6, namely, ϕ rð Þ ¼ r6 ln rð Þ, we have

Φ rð Þ ¼

� r6 ln rð Þμ2 �12r4 3 ln rð Þþ1ð Þ

μ4 �96r2 6 ln rð Þþ5ð Þμ6 ;

nra0;

�2304 ln rð Þþ4224μ8 �2304K0 μrð Þ

μ8

o�4224

μ8 þ2304γμ8 þ2304

μ8 ln μ2

� �; r¼ 0;

8>>>><>>>>:

ð11cÞ

where Euler constant γ ¼ 0:57721566490153286….

3. Numerical results

In this section, the efficiency, accuracy and convergence of thepresent method are tested to constant- and variable-order fractionaldiffusion equations under a square domain Ω1 with side length 2,namely, Ω1 ¼ x1; x2ð Þj 0rx1; x2r2

. The numerical accuracy is

calculated by the relative root mean square errors (RMSE)

RMSE¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1NT

∑NT

i ¼ 1~uðxi; TÞ�uðxi; TÞð Þ2

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1NT

∑NT

i ¼ 1u2ðxi; TÞ

s;

,ð12aÞ

RMSEx¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1NT

∑NT

i ¼ 1

∂ ~uðxi; TÞ∂x1

�∂uðxi; TÞ∂x1

� �2s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1NT

∑NT

i ¼ 1

∂uðxi; TÞ∂x1

� �2s

;

,

ð12bÞwhere uðxi; TÞ and ~uðxi; TÞ represent respectively the analytical andnumerical results evaluated at xi; Tð Þ, and NT is the total number ofuniform-distributed test nodes in the computational domain. Unlessotherwise specified, NT¼441, T¼2 and θ¼1 in all the followingnumerical cases.

Example 1. Consider the 2D constant-order fractional diffusionequation in Ω1


¼Δu x; tð ÞþQ x; tð Þ; 0oαo1; xAΩ; tA 0; Tð Þ; ð13Þ

with zero initial condition u x;0ð Þ ¼ 0; xAΩ and under twodifferent types of boundary conditions:

(a) Full Dirichlet boundary conditions

u x; tð Þ ¼ t2 x1 2�x1ð Þþx2 2�x2ð Þ½ �; xAΓD ¼ ∂Ω; tA 0; Tð Þ;ð14Þ

Fig. 8. Convergence rate (RMSE) of the present method with the derived RBFformulation (11(b)) by using different time steps (dt¼0.02, 0.01, 0.005, 0.002) inExample 2. (a) Full Dirichlet boundary conditions and (b) mixed boundaryconditions.

Fig. 9. Convergence rate (RMSEx) of the present method with the derived RBFformulation (11(b)) by using different time steps (dt¼0.02, 0.01, 0.005, 0.002) inExample 2. (a) Full Dirichlet boundary conditions and (b) mixed boundaryconditions.






(b) Mixed boundary conditions

u x; tð Þ ¼ t2 x1 2�x1ð Þþx2 2�x2ð Þ½ �;xAΓD ¼ x1; x2ð Þjx1 ¼ 0;2

; tA 0; Tð Þ; ð15aÞ

∂u x; tð Þ∂n

¼ ∂ t2 x1 2�x1ð Þþx2 2�x2ð Þ½ �� ∂n

;

xAΓN ¼ x1; x2ð Þjx2 ¼ 0;2

; tA 0; Tð Þ; ð15bÞwhereQ x; tð Þ ¼ ð2t2�α=Γ 3�αð ÞÞþt2

� �x1 2�x1ð Þþx2 2�x2ð Þ½ �þ4t2.

The exact solution is

u x; tð Þ ¼ t2 x1 2�x1ð Þþx2 2�x2ð Þ½ �; xAΩ; tA 0; Tð Þ: ð16ÞFirst, we set α¼0.7 and place uniform nodes in the computationaldomain Ω1 as shown in Fig. 1. Figs. 2–7 show the convergencerate (RMSE and RMSEx) of the present method with different basisfunctions (11(a))–(11(c)) by using different time steps (dt¼0.02,0.01, 0.005, 0.002) in Example 1. Generally speaking, the numer-ical accuracy of the present method increases with the increasinginterpolation node number M. However, with a large time step(dt¼0.02, 0.01), the numerical accuracy first enhances with anincrease of M, and then any further increase of M would not gainmuch improvement in terms of accuracy. This may be caused bythe error generated from temporal discretization. From Figs. 2–7, itcan be observed that the numerical accuracy of Example 1(a) isbetter than those of Example 1(b) by using the present method

with different basis functions (11(a))–(11(c)). The present methodwith RBF formulations (11(b)) and (11(c)) can perform better thanRBF formulation (11(a)). And it cannot provide the correct numer-ical results with the partial derivative term ðð∂u x; tð Þ=∂nÞ orð∂u x; tð Þ=∂x1ÞÞ by using RBF formulation (11(a)) for 2D fractionaldiffusion equations. Therefore, we only adopt RBF formulations (11(b)) and (11(c)) as the basis functions in the MAPS to solve all thefollowing numerical cases.

Example 2. Consider the 2D variable-order fractional diffusionequation with α tð Þ ¼ 0:8þ0:2t=T in Ω1

∂α tð Þu x; tð Þ∂tα tð Þ ¼Δu x; tð ÞþQ x; tð Þ; 0oα tð Þo1; xAΩ; tA 0; Tð Þ; ð17Þ



u x; tð Þ ¼ t2 sinπx12

� �sin πx2

2

� �; xAΓD ¼ ∂Ω; tA 0;ð TÞ;



� �sin

πx22

� �;

xAΓD ¼ x1; x2ð Þjx1 ¼ 0;2


Fig. 10. Convergence rate (RMSE) of the present method with the derived RBFformulation (11(c)) by using different time steps (dt¼0.02, 0.01, 0.005, 0.002) inExample 2. (a) Full Dirichlet boundary conditions and (b) mixed boundaryconditions.

Fig. 11. Convergence rate (RMSEx) of the present method with the derived RBFformulation (11(c)) by using different time steps (dt¼0.02, 0.01, 0.005, 0.002) inExample 2. (a) Full Dirichlet boundary conditions and (b) mixed boundaryconditions.






∂u x; tð Þ∂n

¼ ∂ t2 sin ðπx1=2Þ� �

sin ðπx2=2Þ� ��

∂n;


; tA 0; Tð Þ; ð19bÞwhere Q x; tð Þ ¼ ð2t2�α tð Þ=Γ 3�α tð Þð ÞÞþðπ2t2=2Þ� �

sin ðπx1=2Þ� �


. The exact solution is


� �sin

πx22

� �; xAΩ; tA 0; Tð Þ: ð20Þ

Similar to the previous example, uniform nodes are placed inthe computational domain Ω1 as shown in Fig. 1(a). Figs. 8–11show the convergence rate (RMSE and RMSEx) of the presentmethod with different basis functions (11(b))–(11(c)) by usingdifferent time steps (dt¼0.02, 0.01, 0.005, 0.002) in Example 2.Generally speaking, it can provide the acceptable numerical results(the error is less than 1%) by using the present method with nodedensity (dh¼1/8, namely, M¼293) and dtr0:02. With appropri-ate time step (dt¼0.01), the numerical accuracy of the presentmethod increase with the increasing interpolation node numberM. However, with inappropriate time step (dt¼0.02, 0.005, 0.002),the numerical accuracy first enhances with an increase of M, andthen further increase of M may not gain much improvement inaccuracy. It reveals that the present numerical solutions ofvariable-order time fractional diffusion models are more sensitiveto the time discretization formulation and depend on the functiontype of the fractional derivative order. Therefore, coupling MAPSwith other higher order time discretization formulations [49,50]will be a sensible way to improve the numerical accuracy in the

solution of variable-order time fractional diffusion models. In thisstudy, we will not further pursue this issue.

Example 3. Consider the 2D variable-order fractional diffusionequation with α tð Þ ¼ 0:8þ0:2 sin 0:5πt=T

� �in Ω1

∂α tð Þu x; tð Þ∂tα tð Þ ¼Δu x; tð ÞþQ x; tð Þ; 0oα tð Þo1; xAΩ; tA 0; Tð Þ;

ð21Þ




� �sin

πx22

� �; xAΓD ¼ ∂Ω; tA 0; Tð Þm

ð22Þ



� �sin

πx22

� �; xAΓD ¼ x1; x2ð Þjx1 ¼ 0;2

;

tA 0; Tð Þ; ð23aÞ

∂u x; tð Þ∂n

¼ ∂ t2 sin ðπx1=2Þ� �

sin ðπx2=2Þ� ��

∂n;


; tA 0; Tð Þ; ð23bÞwhere Q x; tð Þ ¼ ð2t2�α tð Þ=Γ 3�α tð Þð ÞÞþðπ2t2=2Þ� �



. The exact solution is

Fig. 12. Convergence rate (RMSE and RMSEx) of the present method with dt¼0.02by using different RBF formulations (11(b)) and (11(c)) in Example 3. (a) FullDirichlet boundary conditions and (b) mixed boundary conditions.

Fig. 13. Convergence rate (RMSE and RMSEx) of the present method with dt¼0.01by using different RBF formulations (11(b)) and (11(c)) in Example 3. (a) FullDirichlet boundary conditions and (b) mixed boundary conditions.







� �sin

πx22

� �; xAΩ; tA 0; Tð Þ: ð24Þ

Uniformly distributed nodes in Fig. 1(a) are used. Figs. 12 and 13show the convergence rate (RMSE and RMSEx) of the presentmethod with different basis functions (11(b))–(11(c)) by using

different time steps (dt¼0.02, 0.01) in Example 3. From Figs. 12and 13, we can see that the present method with node density(dh¼1/8, namely, M¼293) provides the acceptable numericalresults (the error is less than 1%). And it has the similar conclusionwith Examples 1 and 2 that the numerical accuracy of Example 3(a) is better than those of Example 3(b) by using the presentmethod with different basis functions (11(b))–(11(c)).

Next we place irregular nodes in the computational domainΩ1

as shown in Fig. 13 to study the meshless feature of RBF-typemethods. Tables 1–4 list the present numerical errors (RMSE andRMSEx) with the RBF formulations (11(b))–(11(c)) by using bothuniform and irregular nodes with node number M¼484 and timestep dt¼0.01. From Tables 1–4, it can be seen that the present methodworks equally well with irregular nodes. And the present method withRBF formulations (11(c)) can perform better than RBF formulation (11(b)), especially for the partial derivative term ð∂uðxi; TÞÞ=∂x1.

Example 4. The last case is the following fractional diffusionmodel in Ω1


¼Δu x; tð Þ; 0oαo1; xAΩ; tA 0; Tð Þ; ð25Þ

with initial condition u x;0ð Þ ¼ f xð Þ; xAΩ and under the mixedboundary conditions

u x; tð Þ ¼ 0; xAΓD ¼ x1; x2ð Þjx1 ¼ 0;2


Table 3Numerical errors (RMSE) by using the RBF formulation (11(c)) with both uniform and irregular nodes.

Node distribution RMSE

Ex. 1(a) Ex. 1(b) Ex. 2(a) Ex. 2(b) Ex. 3(a) Ex. 3(b)

Uniform nodes 1.77E�04 2.58E�03 1.73E�04 5.61E�04 6.16E�03 1.08E�02Irregular nodes 1.89E�04 3.69E�03 1.62E�04 7.94E�04 6.17E�03 1.10E�02

Table 4Numerical errors (RMSEx) by using the RBF formulation (11(c)) with both uniform and irregular nodes.

Node distribution RMSEx



0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

t

u(-0.5,0,t)

=0.3=0.5=0.7=0.9

αααα

Fig. 14. Temporal evolution of u �0:5;0; tð Þ with different time fractional derivativeorder α by using the present method with RBF formulations (11(c)).

Table 1Numerical errors (RMSE) by using the RBF formulation (11(b)) with both uniform and irregular nodes.

Node distribution RMSE



Table 2Numerical errors (RMSEx) by using the RBF formulation (11(b)) with both uniform and irregular nodes.

Node distribution RMSEx








∂u x; tð Þ∂n

¼ 0; xAΓN ¼ x1; x2ð Þjx2 ¼ 0;2

; tA 0; Tð Þ; ð26bÞ

where

f xð Þ ¼2x1þ2; �1rx1r1=22�2x1ð Þ=3; 1=2rx1r1

(:

Finally, we set dt¼0.01 and place uniform nodes (M¼488) inthe computational domain Ω1 as shown in Fig. 1. Fig. 14 plots thetemporal evolution of u �0:5;0; tð Þ with different time fractionalderivative order α by using the present method with RBF formula-tions (11(c)). Numerical investigation shows that the concentrationu decays very fast at the beginning of diffusion process and itbecomes more and more slowly at late times. This anomalousdiffusion phenomenon becomes more and more obviously withthe smaller time fractional derivative order α.

4. Conclusions

This paper applies the method of approximate particularsolutions (MAPS) to constant- and variable-order time fractionaldiffusion models. Discretization of the time-fractional and spatialderivatives is respectively done by a finite difference scheme andthe MAPS. The present numerical experiments verify that theproposed method is a competitive meshless collocation methodfor constant- and variable-order time fractional diffusion models,and it provides satisfactory solutions and converges to the exactsolutions with the increasing interpolation node number under

both uniform and irregular distributions. Numerical investigationsshow that the present method with RBF formulation (11(c))performs better than that with RBF formulations (11(a)) and (11(b)), and it cannot provide the correct numerical results with thepartial derivative term ðð∂u x; tð Þ=∂nÞ or ð∂u x; tð Þ=∂x1ÞÞ by using RBFformulation (11(a)) for 2D fractional diffusion equations.

Moreover, it should be mentioned that the present numericalsolutions of variable-order time fractional diffusion models aremore sensitive to the time discretization formulation and dependon the function type of the fractional derivative order. Therefore,coupling MAPS with other higher order time discretization formula-tions [49,50] will be a sensible way to solve variable-order timefractional diffusion models. We leave the detailed numerical simula-tions to our future studies.

Acknowledgments

The work described in this paper was supported by the NationalScience Funds of China (Grant nos. 11302069 and 11372097), theFundamental Research Funds for the Central Universities, HohaiUniversity (Grant no. 2013B32814), the National Basic ResearchProgram of China (973 Project no. 2010CB832702), the NationalScience Funds for Distinguished Young Scholars of China (Grant no.11125208), the 111 Project (Grant no. B12032), a CERG Grant of theHong Kong Research Grant Council, and a FRG Grant of Hong KongBaptist University.

Appendix A. Time discretization of variable-order timefractional derivative term

The time discretization formulation (6) of variable-order timefractional derivative term (4) is derived as follows:

where

a0 ¼τ�α t1ð Þ

Γ 2�α t1ð Þð Þ; aj ¼τ�α tjþ 1ð Þ

Γ 2�α tjþ1� �� ;

bj ¼ jþ1ð Þ1�α tjþ 1ð Þ� j1�α tjþ 1ð Þh i; j¼ 1;2;…; k

∂α tð Þu x; tð Þ∂tα tð Þ ¼ 1

Γ 1�α tð Þð ÞZ t

0

∂u x;η� �∂η

dη

t�η� �α tð Þ

� ∑k

j ¼ 0

Z jþ1ð Þτ

jτ

∂u x; ξ� �∂ξ

1Γ 1�α tjþ1

� �� dξ


¼ ∑k

j ¼ 0

u x; tjþ1� ��u x; tj

� �τ

Z jþ1ð Þτ

jτ

1Γ 1�α tjþ1

� �� dξ


¼ ∑k

j ¼ 0

u x; tjþ1� ��u x; tj

� �τ

Z k� jþ1ð Þτ

k� jð Þτ

1Γ 1�α tk� jþ1

� �� dη

ηα tk� jþ 1ð Þ

¼ ∑k

j ¼ 0

u x; tk� jþ1� ��u x; tk� j

� �τ

Z jþ1ð Þτ

jτ

1Γ 1�α tjþ1

� �� dη

ηα tjþ 1ð Þ

¼ ∑k

j ¼ 0

u x; tk� jþ1� ��u x; tk� j

� �τ

1Γ 1�α tjþ1

� �� Z jþ1ð Þτ

jτ

dη

ηα tjþ 1ð Þ

¼ ∑k

j ¼ 0

τ�α tjþ 1ð ÞΓ 2�α tjþ1

� �� u x; tk� jþ1� ��u x; tk� j

� �� jþ1ð Þ1�α tjþ 1ð Þ� j1�α tjþ 1ð Þh i

;

¼τ � α t1ð Þ

Γ 2�α t1ð Þð Þ ukþ1�uk

� �þ ∑k

j ¼ 1

τ�α tjþ 1ð ÞΓ 2�α tjþ1

� �� u x; tk� jþ1� ��u x; tk� j

� �� jþ1ð Þ1�α tjþ 1ð Þ� j1�α tjþ 1ð Þh i

; kZ1

τ � α t1ð ÞΓ 2�α t1ð Þð Þ u

1�u0� �

; k¼ 0

8>>><>>>:


k


� �; kZ1

a0 u1�u0� �

; k¼ 0

8>><>>:






References

[1] Hilfer R. Applications of fractional calculus in physics. Singapore: WorldScientific; 2000.

[2] Metzler R, Klafter J. The random walk's guide to anomalous diffusion:a fractional dynamics approach. Phys Rep 2000;339(1):1–77.

[3] Wyss W. The fractional diffusion equation. J Math Phys 1986;27:2782–5.[4] Gorenflo R, Luchko Y, Mainardi F. Wright functions as scale-invariant solutions

of the diffusion-wave equation. J Comput Appl Math 2000;118(1–2):175–91.[5] Yang Q, Turner I, Liu F, Ilis M. Novel numerical methods for solving the time-

space fractional diffusion equation in two dimensions. SIAM J Sci Comput2011;33(3):1159–80.

[6] Gorenflo R, Mainardi F, Moretti D, Pagnini G, Paradisi P. Discrete random walkmodels for space-time fractional diffusion. Chem Phys 2007;284(1–2):521–41.

[7] Fu Z-J, Chen W, Yang H-T. Boundary particle method for Laplace transformedtime fractional diffusion equations. J Comput Phys 2013;235:52–66.

[8] Chechkin AV, Gorenflo R, Sokolov IM. Fractional diffusion in inhomogeneousmedia. J Phys A: Math Gen 2005;38(42):L679.

[9] Sun H, Chen W, Chen Y. Variable-order fractional differential operators inanomalous diffusion modeling. Phys A: Stat Mech Appl 2009;388(21):4586–92.

[10] Lorenzo CF, Hartley TT. Variable order and distributed order fractionaloperators. Nonlinear Dyn 2002;29(1–4):57–98.

[11] Sun H, Chen W, Wei H. A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems. EurPhys J Spec Top 2011;193(1):185–92.

[12] Tadjeran C, Meerschaert MM. A second-order accurate numerical methodfor the two-dimensional fractional diffusion equation. J Comput Phys2007;220(2):813–23.

[13] Murio DA. Implicit finite difference approximation for time fractional diffusionequations. Comput Math Appl 2008;56(4):1138–45.

[14] Wang H, Wang K, Sircar T. A direct O(Nlog2N) finite difference method forfractional diffusion equations. J Comput Phys 2010;229(21):8095–104.

[15] Ren J, Sun Z, Zhao X. Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions. J Comput Phys2013;232(1):456–67.

[16] Zhao X, Sun Z. A box-type scheme for fractional sub-diffusion equation withNeumann boundary conditions. J Comput Phys 2011;230(15):6061–74.

[17] Sun H, Chen W, Li C, Chen Y. Finite difference schemes for variable-order timefractional diffusion equation. Int J Bifurc Chaos 2012;22(04):1250085.

[18] Shen S, Liu F, Chen J, Turner I, Anh V. Numerical techniques for thevariable order time fractional diffusion equation. Appl Math Comput2012;218(22):10861–70.

[19] Chen CM, Liu F, Anh V, Turner I. Numerical schemes with high spatial accuracyfor a variable-order anomalous subdiffusion equation. SIAM J Sci Comput2010;32(4):1740–60.

[20] Lin R, Liu F, Anh V, Turner I. Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusionequation. Appl Math Comput 2009;212(2):435–45.

[21] Langlands TAM, Henry BI. The accuracy and stability of an implicit solutionmethod for the fractional diffusion equation. J Comput Phys 2005;205(2):719–36.

[22] Cui M. Compact finite difference method for the fractional diffusion equation.J Comput Phys 2009;228(20):7792–804.

[23] Zhuang P, Liu F. Implicit difference approximation for the time fractionaldiffusion equation. J Appl Math Comput 2006;22(3):87–99.

[24] Zhang Y, Sun Z, Zhao X. Compact alternating direction implicit scheme for thetwo-dimensional fractional diffusion-wave equation. SIAM J Numer Anal2012;50(3):1535–55.

[25] Chen C-M, Liu F, Turner I, Anh V. A Fourier method for the fractional diffusionequation describing sub-diffusion. J Comput Phys 2007;227(2):886–97.

[26] Lin Y, Xu C. Finite difference/spectral approximations for the time-fractionaldiffusion equation. J Comput Phys 2007;225(2):1533–52.

[27] Jiang Y, Ma J. High-order finite element methods for time-fractional partialdifferential equations. J Comput Appl Math 2011;235(11):3285–90.

[28] Li C, Zhao Z, Chen Y. Numerical approximation of nonlinear fractionaldifferential equations with subdiffusion and superdiffusion. Comput MathAppl 2011;62(3):855–75.

[29] Fix GJ, Roof JP. Least squares finite-element solution of a fractional order two-point boundary value problem. Comput Math Appl 2004;48(7–8):1017–33.

[30] Katsikadelis JT. The BEM for numerical solution of partial fractional differentialequations. Comput Math Appl 2011;62(3):891–901.

[31] Brunner H, Ling L, Yamamoto M. Numerical simulations of 2D fractionalsubdiffusion problems. J Comput Phys 2010;229(18):6613–22.

[32] Chen W, Ye L, Sun H. Fractional diffusion equations by the Kansa method.Comput Math Appl 2010;59(5):1614–20.

[33] Liu Q, Gu Y, Zhuang P, Liu F, Nie Y. An implicit RBF meshless approach for timefractional diffusion equations. Comput Mech 2011;48(1):1–12.

[34] Chen CS, Fan CM, Wen PH. The method of approximate particular solutions forsolving elliptic problems with variable coefficients. Int J Comput Methods2011;8(03):545–59.

[35] Ling L, Chen CS, Kwok TO. Adaptive method of particular solution For solving3d inhomogeneous elliptic equations. Int J Comput Methods 2010;07(03):499–511.

[36] Chen W, Fu ZJ, Chen CS. Recent advances on radial basis function collocationmethods. Berlin: Springer; 2013.

[37] Tsai C-C, Chen CS, Hsu T-W. The method of particular solutions for solvingaxisymmetric polyharmonic and poly-Helmholtz equations. Eng Anal BoundElem 2009;33(12):1396–402.

[38] Chen CS, Fan CM, Wen PH. The method of approximate particular solutions forsolving certain partial differential equations. Numer Methods Partial Diff Equ2012;28(2):506–22.

[39] Zhu H. The method of approximate particular solutions for solving anisotropicelliptic problems. Eng Anal Bound Elem 2014;40:123–7.

[40] Reutskiy SY. Method of particular solutions for nonlinear Poisson-typeequations in irregular domains. Eng Anal Bound Elem 2013;37(2):401–8.

[41] Kuo LH, Gu MH, Young DL, Lin CY. Domain type kernel-based meshlessmethods for solving wave equations. CMC-Comput Mater Contin 2013;33(3):213–28.

[42] Bustamante CA, Power H, Florez WF, Hang CY. The global approximateparticular solution meshless method for two-dimensional linear elasticityproblems. Int J Comput Math 2013;90(5):978–93.

[43] Bustamante CA, Power H, Sua YH, Florez WF. A global meshless collocationparticular solution method (integrated radial basis function) for two-dimensional Stokes flow problems. Appl Math Modell 2013;37(6):4538–47.

[44] Jiang T, Li M, Chen CS. The method of particular solutions for solving inverseproblems of a nonhomogeneous convection-diffusion equation with variablecoefficients. Numer Heat Transf, Part A: Appl 2012;61(5):338–52.

[45] Yao G, Siraj ul I, Sarler B. Assessment of global and local meshless methodsbased on collocation with radial basis functions for parabolic partial differ-ential equations in three dimensions. Eng Anal Bound Elem 2012;36(11):1640–8.

[46] Reutskiy SY. Method of particular solutions for solving PDEs of the second andfourth orders with variable coefficients. Eng Anal Bound Elem 2013;37(10):1305–10.

[47] Bustamante CA, Power H, Florez WF. A global meshless collocation particularsolution method for solving the two-dimensional Navier–Stokes system ofequations. Comput Math Appl 2013;65(12):1939–55.

[48] Podlubny I. Fractional differential equations. San Diego: Academic Press; 1999.[49] Cuesta E, Lubich C, Palencia C. Convolution quadrature time discretization of

fractional diffusion-wave equations. Math Comput 2006;75(254):673–96.[50] Jameson A, Schmidt W, Turkel E. Numerical solutions of the Euler equations by

finite volume methods using Runge–Kutta time-stepping schemes. AIAApaper; 1981-1259.



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