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European Journal of Computational Mechanics

ISSN: 1779-7179 (Print) 1958-5829 (Online) Journal homepage: http://www.tandfonline.com/loi/tecm20

Numerical solution of time fractional partial

differential equations using multiquadric quasi-

interpolation scheme

M. Sarboland

To cite this article: M. Sarboland (2018): Numerical solution of time fractional partial differential

equations using multiquadric quasi-interpolation scheme, European Journal of Computational

Mechanics, DOI: 10.1080/17797179.2018.1469833

To link to this article: https://doi.org/10.1080/17797179.2018.1469833

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EUROPEAN JOURNAL OF COMPUTATIONAL MECHANICS, 2018

https://doi.org/10.1080/17797179.2018.1469833

Numerical solution of time fractional partial diﬀerential

equations using multiquadric quasi-interpolation scheme

M. Sarbolanda

aDepartment of Mathematics, Saveh Branch, Islamic Azad University, Saveh, Iran

ABSTRACT

In this paper, a meshfree method is presented to solve time

fractional partial diﬀerential equations. It is based on the

multiquadric quasi-interpolation operator LW2. In the present

scheme, quadrature formula is used to discretise the temporal

Caputo fractional derivative of order α∈(0, 1]and the quasi-

interpolation is used to approximate the solution function and

its spatial derivatives. Our numerical results are compared with

the exact solutions as well as the results obtained from the

other numerical schemes. It can be easily seen that the proposed

method is a reliable and eﬀective method to solve fractional

partial diﬀerential equation. Furthermore, the stability analysis

of the method is surveyed.

ARTICLE HISTORY

Received 26 October

2017

Accepted 17 April 2018

KEYWORDS

Time fractional partial

differential equation;

multiquadric

quasi-interpolation

scheme

1. Introduction

In the last decade, fractional order partial diﬀerential equations are increasingly

used to model problems in mathematical physics (Fan & Jiang,2014;Qi & Jiang,

2011;Ray,2015;Zhuang, Liu, Turner, & Gu,2014), mathematics (Chen, Liu,

Zhuang, & Anh,2009;Feng, Zhuang, Liu, & Turner,2015;Zhuang, Liu, Anh,

&Turner,2008), coloured noise (Sun, Abdelvahab, & Onaral,1984), ﬂuid and

continuum mechanics (Carpinteri & Mainardi,1997), ﬁnance (Sabatelli, Keating,

Dudley, & Richmond,2002;Song & Wang,2013;Wyss,2000) and biological

processes and systems (Magin, Ingo, Colon-Perez, Triplett, & Mareci,2013).

Some of fractional partial diﬀerential equations (FPDEs) have been studied and

solved, such as the time fractional nonlinear Sine-Gordon and Klein-Gordon

equations (Dehghan, Abbaszadeh, & Mohebbi,2015), the space fractional wave

equation (Odibat and Momani,2006), the time–space fractional telegraph equa-

tion (Momani,2005;Orssingher & Beghin,2004;Orssingher & Zhao,2003;

Zhao & Li,2012), the fractional Fokker-Planck equation (Aminataei & Karimi

Vanani,2013;Chen, Liu, Zhuang, & Anh,2009;Pinto & Sousa,2017), the time–

space fractional diﬀusion wave equation (Povstenko,2010) and the fractional

kdv equation (Debnath & Bhatta,2004). Since most FPDEs do not have exact

CONTACT M. Sarboland m.sarboland@gmail.com

© 2018 Informa UK Limited, trading as Taylor & Francis Group

2M. SARBOLAND

analytic solutions, so approximation and numerical methods are used exten-

sively. Recently, the variational iterative method (Momani & Odibat,2007), the

Adomian decomposition method (El-Sayed & Gaber,2006;Momani & Odibat,

2006) and radial basis function (RBF) meshless method (Uddin & Haq,2011;

Vanani & Aminataei,2012) have been applied to solve such problems.

In the present work, we present a meshless approach for solving time FPDEs

based on the multiquadric (MQ) quasi-interpolation operator LW2.

MQ quasi-interpolation is a linear combination of MQ-RBF and the approxi-

mated function. In 1992, (Beatson & Powell,1992) proposed three univariate MQ

quasi-interpolations named as LA,LBand LC.(Wu & Schaback,1994)presented

the MQ quasi-interpolation LD. In recent years, (Jiang, Wang, Zhu, & Xu,

2011) have introduced a new MQ quasi-interpolation scheme. This approach is

based on inverse multiquadric (IMQ) RBF interpolation, and Wu and Schaback’s

operator LDthat have the advantages of high approximation order. Up to now,

MQ quasi-interpolation is applied for solving diﬀerent types of PDEs, see (Jiang

&Wang,2012;Sarboland & Aminataei,2014,2015b,2015a).

The outline of the present paper is as follows. A brief description of the MQ

quasi-interpolation scheme is given in Section 2. In Section 3, we apply our

numerical method for the time FPDEs. The stability analysis of the method is

discussed in Section 4. The results of several numerical experiments are explained

in Section 5. In Section 6, we conclude our results.

2. The MQ quasi-interpolation scheme

In this section, we describe three MQ quasi-interpolation schemes named as

LD,LWand LW2. More details can be seen in Beatson and Powell (1992), Jiang

et al. (2011)andWu and Schaback (1994).

For a given area =[a,b]and a ﬁnite set of diﬀerent points

a=x0<x

1<... <x

N=b,h=max

1iN(xi−xi−1),

if we are supplied with a function f:[a,b]−→R, quasi-interpolation of ftakes

the form:

L(f)=

N

i=0

f(xi)φi(x),

where each function φi(x)is a linear combination of the Hardy’s MQs basis

function (Hardy,1971),

ψi(x)=c2+(x−xi)2,

and low-order polynomials and c∈R+is a shape parameter. In Wu and

Schaback (1994), Wu and Schaback introduced the MQ quasi-interpolation

EUROPEAN JOURNAL OF COMPUTATIONAL MECHANICS 3

operator LDthat is deﬁned as

LDf(x)=

N

i=0

f(xi)

ψi(x),(1)

where

ψ0(x)=1

2+ψ1(x)−(x−x0)

2(x1−x0),

ψ1(x)=ψ2(x)−ψ1(x)

2(x2−x1)−ψ1(x)−(x−x0)

2(x1−x0),

ψi(x)=ψi+1(x)−ψi(x)

2(xi+1−xi)−ψi(x)−ψi−1(x)

2(xi−xi−1),2iN−2, (2)

ψN−1(x)=(xN−x)−ψN−1(x)

2(xN−xN−1)−ψN−1(x)−ψN−2(x)

2(xN−1−xN−2),

ψN(x)=1

2+ψN−1(x)−(xN−x)

2(xN−xN−1).

Now, suppose that {xki}¯

N

i=1is a smaller set of the given points {xi}N

i=0where ¯

N

is a positive integer satisfying ¯

N<Nand 0 =k0<k

1<... <k

¯

N+1=N.

Using the IMQ-RBF, the second derivative of f(x)can be approximated by RBF

interpolant Sf as

Sf =

¯

N

j=1

αj¯ϕ(|x−xkj|),

where

¯ϕ(r)=s2

(s2+r2)3/2,

and s∈R+is a shape parameter.

The coeﬃcients {αj}¯

N

j=1are uniquely obtained by the interpolation condition

Sf (xki)=

¯

N

j=1

αj¯ϕ(|xki−xkj|)=f(xki),1i¯

N.(3)

4M. SARBOLAND

Since, the Equation (3)issolvable(Madych & Nelson,1990), so

α=A−1

X.f

X,(4)

where

X={xk1,...,xk¯

N},α=[α1,...,α¯

N]T,AX=[¯ϕ(|xki−xkj|)],

f

X=[f(xk1),...,f (xk¯

N)]T.

Using the fand the coeﬃcient αdeﬁned in Equation (4), a function e(x)is

constructed in the form

e(x)=f(x)−

¯

N

j=1

αjs2+(x−xkj)2.(5)

Now, using LDoperator deﬁned by Equation (1)onthedata{(xi,e(xi))}N

i=0with

the shape parameter c, the MQ quasi-interpolation operator LWis deﬁned as

follows:

LWf(x)=

¯

N

j=1

αjs2+(x−xkj)2+LDe(x). (6)

The shape parameters cand sshould not be the same constant in Equation (6).

In Equation (3), the value of f

xkjcan be replaced by

f

xkj=2[(xkj−xkj−1)f(xkj+1)−(xkj+1−xkj−1)f(xkj)+(xkj+1−xkj)f(xkj−1)]

(xkj−xkj−1)(xkj+1−xkj)(xkj+1−xkj−1),

when the data’s {(xki,f(xki))}¯

N

i=1are given, and {xi}¯

N

i=1aren’t equally spaced

points. So, if f

Xin Equation (4) is replaced by

F

X=[f

xk1,...,f

xk¯

N

]T,(7)

the quasi-interpolation operator deﬁned by Equations (5)and(6) is denoted by

LW2. The linear reproducing property and the high convergence rate of LW2

were also studied in Jiang et al. (2011).

The operator LW2can be written in the compact form

LW2f(x)=

N

i=0

f(xi)

ψi(x),(8)

where the basis functions

ψi(x)are a linear combination of functions

ψi(x)and

¯

φi(x)=s2+(x−xi)2.SeeSarboland and Aminataei (2014) for details of

compactness approach.

EUROPEAN JOURNAL OF COMPUTATIONAL MECHANICS 5

3. The numerical method

In this paper, we concentrate on the numerical solution of time fractional order

partial diﬀerential equation of the type:

∂αu(x,t)

∂tα+β∂u(x,t)

∂x+γ∂2u(x,t)

∂x2=f(x,t),

t∈[0, T],x∈=[a,b],0 <α≤1, (9)

with the initial condition

u(x,0)=u0(x), (10)

and the boundary conditions,

u(a,t)=g1(t),t≥0, u(b,t)=g2(t), (11)

where u0(x),g1(t),g2(t)and f(x,t)are known functions, βand γare real

parameters, αis the order time derivative and ∂αu(x,t)

∂tαis the Caputo fractional

partial derivative that deﬁned as follows:

∂αu(x,t)

∂tα=1

(1−α) t

0

∂u(x,s)

∂s(t−s)(−α)ds,0<t<T,0<α<1,

∂u(x,t)

∂t,0≤t≤T,α=1, (12)

where (.) is the Gamma function.

Now, we present the numerical scheme for solving the Equation (9)usingthe

MQ quasi-interpolation LW2. In this approach, the quadrature formula is ﬁrst

employed for discretisation of the time fractional derivative similar to the work

that has been done in Murio (2008). Then, the solution function is approximated

by Equation (8), and their spatial derivatives are then obtained by symbolic

diﬀerentiation. In the end, the collocation method is applied.

3.1. The discretisation of time fractional derivative

According to the simple quadrature formula see Murio (2008), the term

∂αu(x,tn+1)

∂tα,tn=ntwith step size tcan be arranged as

∂αu(x,tn+1)

∂tα=aαun+1(x)−un(x)+aα

n

k=1

bα(k)un−k+1(x)−un−k(x)

+O(t), (13)

where aα=(t)−α

(2−α),bα(k)=(k+1)1−α−(k)1−αand un(x)=u(x,tn).See

Murio (2008) for more details.

6M. SARBOLAND

Substituting Equation (13) into Equation (9) yields the following time discre-

tised form of the fractional diﬀerential equation:

aαun+1(x)−un(x)+aα

n

k=1

bα(k)un−k+1(x)−un−k(x)

+βun+1

x(x)+γun+1

xx (x)=fn+1(x),(14)

where fn+1(x)=f(x,tn+1).

Equation (14) can be written in the following form by some rearrangement of

terms:

aαun+1(x)+βun+1

x(x)+γun+1

xx (x)=aαun(x)+χn+1(x), (15)

where

χn+1(x)=−aα

n

k=1

bα(k)un−k+1(x)−un−k(x)+fn+1(x).

Now, the unknown function un(x)is approximated using MQ quasi-

interpolation scheme, and its spatial derivatives un

x(x)and un

xx(x)are calculated

by diﬀerentiating such closed form quasi approximation as follows:

un(x)=

N

i=0

un

iˆ

ψi(x), (16)

un

x(x)=

N

i=0

un

i

∂ˆ

ψi

∂x(x)=

N

i=0

un

iˇ

ψi(x),(17)

un

xx(x)=

N

i=0

un

i

∂2ˆ

ψi

∂x2(x)=

N

i=0

un

i¯

ψi(x), (18)

where ∂ˆ

ψi

∂x=ˇ

ψiand ∂2ˆ

ψi

∂x2=¯

ψi.

At the end, replacing (16)-(18)into(15) and applying collocation method yields

N

k=0

un+1

kaαˆ

ψik +βˇ

ψik +γ¯

ψik=aα

N

k=0

un

kˆ

ψik +χn+1

i,1iN−1,

(19)

where χn+1

i=χn+1(xi),ˆ

ψik =ˆ

ψk(xi),ˇ

ψik =ˇ

ψk(xi),and ¯

ψik =¯

ψk(xi), whereas

according to (9):

un

0=u(x0,tn)=u(a,tn)=g1(tn), (20)

EUROPEAN JOURNAL OF COMPUTATIONAL MECHANICS 7

un

N=u(xN,tn)=u(b,tn)=g2(tn). (21)

Substituting (20)and(21)into(19) generates a system of N−1 linear equations

in N−1 unknown parameters un+1

i.

Now, Equation (19) can be written in the matrix form

[aαA+βD+γ¨

D]un+1=aαAun+χn+1+aα

1

k=0

2

j=1

(−1)kgn+k

jˆ

j

+

2

j=1

gn+1

j[βˇ

j+γ¯

j], (22)

where

Aij =[ˆ

ψij]N−1

i,j=1,Dij =[ˇ

ψij]N−1

i,j=1,¨

Dij =[¯

ψij]N−1

i,j=1,

ˆ

1=[ˆ

ψ0(x1),ˆ

ψ0(x2),...,ˆ

ψ0(xN−1)]T,

ˆ

2=[ˆ

ψN(x1),ˆ

ψN(x2),...,ˆ

ψN(xN−1)]T,

ˇ

1=[ˇ

ψ0(x1),ˇ

ψ0(x2),...,ˇ

ψ0(xN−1)]T,

ˇ

2=[ˇ

ψN(x1),ˇ

ψN(x2),...,ˇ

ψN(xN−1)]T,

¯

1=[¯

ψ0(x1),¯

ψ0(x2),...,¯

ψ0(xN−1)]T,

¯

2=[¯

ψN(x1),¯

ψN(x2),...,¯

ψN(xN−1)]T,

and

χn+1=[χn+1(x1),χn+1(x2),...,χn+1(xN−1)]T.

Subsequently, Equation (22) can be written as

un+1=M−1Nun+M−1, (23)

where

M=aαA+βD+γ¨

D,N=aαA,

=χn+1+aα

1

k=0

2

j=1

(−1)kgn+k

jˆ

j+

2

j=1

gn+1

j[βˇ

j+γ¯

j].(24)

In order to make the reduction in error, the obtained uifrom (23) is substituted

by Equation (16) that can be written as follows:

un=Aun+gn

1ˆ

1+gn

2ˆ

2, (25)

8M. SARBOLAND

Figure 1. The space–time graph of estimated solution by MQQI for x∈[0, 1],t∈[0, 2]and

α=0.6 of experiment 1.

and the obtained value is considered as ui. Therefore, it’s taken from (23)and

(25)that

un+1=AM

−1NA−1un+AM−1−AM

−1NA−1(gn

1ˆ

1+gn

2ˆ

2)

+gn+1

1ˆ

1+gn+1

2ˆ

2.(26)

Hence, the unknown parameters uiare speciﬁed from (26) instead of (23).

4. The stability analysis

In this section, the stability analysis of our numerical scheme is presented using

spectral radius of the ampliﬁcation matrix similar to the work that Islam et al. did

in ul-Islam, Haq, and Uddin (2009). Let unbe the exact and u∗nthe numerical

solution of Equation (9)atthenth time level, then the error εnat the nth time

level is given by εn=un−u∗n. The error equation for discretised fractional

partial diﬀerential equation can be written as

εn+1=AM−1NA−1εn=Eεn,(27)

where E=AM−1NA−1. It is noteworthy that this error includes both time and

spatial errors at every time level. For the stability of the numerical scheme, we

must have εn→0asn→∞,i.e.ρ(E)1, which is the necessary and suﬃcient

condition for the numerical scheme to be stable, where ρ(E)denotes the spectral

radius of the ampliﬁcation matrix E. Equation (27) is equivalent with

MA−1εn+1=NA−1εn.(28)

EUROPEAN JOURNAL OF COMPUTATIONAL MECHANICS 9

Figure 2. Absolute error and exact and estimated solutions at t=2 with t=0.01, N=121

and α=0.6 of experiment 1.

Now, Equation (28) can be written into following form using the values of M

and Ndeﬁned in Equation (24):

[I+(t)αR]εn+1=Iεn,

where

R=cα[βD+γ¨

D]A−1;cα=(2−α).

The condition of stability will be satisﬁed if maximum eigenvalue of the matrix

E=[I+(t)αR]−1is less than or equal to unity, i.e.

1

1+(t)αηR1, (29)

where ηRdenotes the eigenvalue of the matrix R. The above inequality holds true

if either ηR≥0or(ηR≤0andt≥α

−2

ηR). It is clear that the condition number

and magnitude of the eigenvalue of the matrix Rdepend on the shape parameter

and the number of collocation points. Hence, the condition number and the

spectral radius of the matrix Eare dependent to the shape parameter and the

number of collocation points. Since it is not possible to ﬁnd explicit relationship

among the spectral radius of the matrix and the shape parameter, this dependency

is approximated numerically by keeping the number of collocation points ﬁxed.

5. The numerical experiments

In this section, the proposed method is applied for ﬁve experiments. The nu-

merical results of the FPDEs using this scheme are compared with the analytical

solutions and solutions in Uddin and Haq (2011) (MQ scheme). Our numerical

10 M. SARBOLAND

scheme is denoted by MQQI. The L∞and L2error norms which are deﬁned by

L∞=u∗n−un∞=max

0jN|u∗n(xj)−un(xj)|,

L2=u∗n−un2=

h

N

j=0

(u∗n(xj)−un(xj))2,

are used to measure the accuracy. Also, the stability analysis of the methods

is considered for ﬁrst experiment. In all experiments, the shape parameter sis

considered twice the shape parameter c.

The computations associated with the experiments discussed above were

performed in Maple 16 on a PC with a CPU of 2.4 GHZ.

Experiment 1

Consider Equation (9) with the parameters β=1, γ=0 and the inhomo-

geneous term f(x,t)=2t2−α

(2−α) sin (x)+tcos (x)that α=0.6. In this case, the

FPDE (9) has the following form

∂αu(x,t)

∂tα+∂u(x,t)

∂x=2t2−α

(2−α) sin (x)+tcos (x), (30)

that it’s a one-dimensional linear inhomogeneous fractional wave equation.

The exact solution of (30)isgivenbyu(x,t)=tsin (x)Odibat and Momani

(2009). The initial condition in (10) and the boundary conditions in (11)canbe

obtained from the exact solution.

The results are presented for α=0.6 and compared with the exact solutions

and the results of the MQ scheme Uddin and Haq (2011)inTable1.Inthis

case, we use the shape parameter c=0.09, t=0.01 and N=121. It can be

seen from Table 1that the obtained results are in good agreement with the exact

solutions and the results of the MQ scheme. We also compare the numerical

solutions with the exact solution for various values of xat diﬀerent times in

Table 2.

Table 3shows the relation between the spectral radius of the matrix Eand the

diﬀerent values of the parameter cwith ﬁxed number collocation points N=50

at t=0.5 . It is visible that if the values of shape parameter care greater than the

critical value c=0.18, then ρ(E)≥1 and hence the MQQI method becomes

unstable. Therefore, the interval stability of proposed scheme is (0, 0.18)which

is a small interval.

Moreover, the spatial rate of convergence obtained using our scheme are

presented with t=0.01 and diﬀerent values of Nat t=1 in Table 4.It

can be seen from Table 4that the convergence rate increases with the smaller

spatial step size and the error norms decrease. The time rate of convergence with

N=40 and diﬀerent values of tis shown in Table 5. It is observable that the

convergence rate decrease with a smaller time step size.

EUROPEAN JOURNAL OF COMPUTATIONAL MECHANICS 11

Figure 3. The space–time graph of estimated solution by MQQI for x∈[0, 1],t∈[0, 2]and

α=0.5 of experiment 2.

Figure 4. Absolute error and exact and estimated solutions at t=2 with t=0.01, N=51 and

α=0.5 of experiment 2.

The space–time plot of the estimated function is shown in Figure 1.Further,

the graphs of absolute error and the estimated solution using our scheme are

given at t=2inFigure2.

12 M. SARBOLAND

Table 1. Comparison of the L∞and L2errors between the numerical results of our scheme and

the results of Uddin and Haq (2011) with α=0.6, t=0.01 and N=121 at different times of

experiment 1.

Time0.10.511.52

MQQI;

L∞3.13118E-06 1.80871E-05 4.23211E-05 6.93573E-05 9.78477E-05

L21.11664E-06 9.68487E-06 2.67175E-05 4.86397E-05 7.47262E-05

MQ Uddin and Haq (2011);

L∞1.152E-06 8.673E-06 1.894E-05 2.984E-05 4.070E-05

L29.299E-07 1.730E-05 5.717E-05 1.142E-04 1.854E-04

Table 2. Comparison of results with the exact solution with α=0.6, t=0.01 and N=121 for

different values of xand tof experiment 1.

x0.5 1 1.5 2 2.5

t=0.5

Exact 0.23971 0.42074 0.49875 0.45465 0.29924

MQQI 0.23971 0.42074 0.49875 0.45465 0.29924

t=1

Exact 0.47943 0.84147 0.99750 0.90930 0.59847

MQQI 0.47943 0.84147 0.99749 0.90930 0.59847

t=1.5

Exact 0.71914 1.26221 1.49624 1.36395 0.89771

MQQI 0.71914 1.26221 1.49624 1.36395 0.89771

t=2

Exact 0.95885 1.68294 1.99499 1.81860 1.19694

MQQI 0.95885 1.68294 1.99499 1.81860 1.19694

Table 3. The spectral radius and L∞and L2error norms versus shape parameter cfor t=0.01

and N=50 at t=0.5 of experiment 1.

cρ(E)L∞L2

0.00001 0.99998 7.72317E-04 8.31526E-04

0.00010 0.99994 7.69874E-04 8.29521E-04

0.00100 0.99991 7.45230E-04 8.09243E-04

0.01000 0.99998 4.90979E-04 5.95721E-04

0.05000 1.00000 3.60981E-04 2.67189E-04

0.10000 1.00000 4.98564E-04 2.43980E-04

0.15000 1.00000 3.27169E-05 1.34562E-04

0.17000 0.99999 3.73084E-05 1.71903E-05

0.18000 2.18403 1.18919E+01 6.36746E-00

Table 4. The spatial rate of convergence at t=1 with t=0.01 of experiment 1.

NL∞Order L2Order

40 1.38814E-03 — 8.13572E-04 —

80 4.20181E-04 1.72406 2.23448E-04 1.86432

120 2.03546E-04 1.78755 1.04041E-04 1.88522

160 1.19899E-04 1.83965 6.00625E-05 1.90974

200 7.91060E-05 1.86368 3.90995E-05 1.92375

EUROPEAN JOURNAL OF COMPUTATIONAL MECHANICS 13

Table 5. The time rate of convergence at t=1 with N=40 of experiment 1.

tL

∞Order L2Order

0.100 1.38864E-03 — 8.15129E-04 —

0.050 1.38829E-03 0.00026 8.14134E-04 0.00176

0.010 1.38814E-03 0.00007 8.13572E-04 0.00043

0.005 1.38813E-03 0.00002 8.13529E-04 0.00008

0.001 1.38812E-04 0.00001 8.13506E-04 0.00002

Table 6. The comparison of the L∞and L2errors of our method with the results of Uddin and

Haq (2011) with α=0.5, t=0.01 and N=51 at different times of experiment 2.

Time 0.1 0.5 1 1.5 2

MQQI;

L∞4.7807E-04 5.4079E-04 5.5630E-04 5.6368E-04 5.6897E-04

L23.8201E-04 4.2643E-04 4.3724E-04 4.4260E-04 4.4654E-04

MQ Uddin and Haq (2011);

L∞6.086E-02 2.958E-02 2.114E-02 1.732E-02 1.503E-02

L22.613E-01 1.277E-01 9.134E-02 7.485E-01 6.494E-01

Experiment 2

In this experiment, we consider Equation (9) with β=1, γ=−1, [a,b]=[0,

1] and f(x,t)=2t2−α

(3−α)+2x−2, which is one-dimensional linear inhomogeneous

fractional Burgers’ equation Odibat and Momani (2009)

∂αu(x,t)

∂tα+∂u(x,t)

∂x−∂2u(x,t)

∂x2=2t2−α

(3−α) +2x−2.

The exact solution of this experiment is

u(x,t)=x2+t2.(31)

The initial condition of the problem is obtained from (31)att=0andthe

boundary conditions in (11) can be obtained from the exact solution.

Numerical results are listed for α=0.5 with t=0.01 and compared with

the results of Uddin and Haq (2011) in Table 6. Also, the numerical solutions

are compared with the exact solutions for diﬀerent values of xand tin Table

7. We use the shape parameter c=0.016 with N=51. Moreover, the space–

time graph of the estimated solution is presented in Figure 3. Also, the graphs of

absolute error and the estimated and analytical functions at t=2 are shown in

Figure 4.

Experiment 3

In this experiment, the Equation (9) is considered as the following form

∂αu(x,t)

∂tα=∂2u(x,t)

∂x2, (32)

with the initial condition

u(x,0)=4x(1−x),

14 M. SARBOLAND

Table 7. Comparison of the results with the exact solutions with α=0.5, t=0.01 and N=51

for different values of xand tof experiment 2.

x0.12 0.24 0.36 0.48 0.60 0.72 0.84 0.94

t=0.5

Exact 0.2644 0.3076 0.3796 0.4804 0.6100 0.7684 0.9556 1.1716

MQQI 0.2647 0.3080 0.3801 0.4809 0.6105 0.7689 0.9560 1.1718

t=1.0

Exact 1.0144 1.0576 1.1296 1.2304 1.3600 1.5184 1.7056 1.9216

MQQI 1.0147 1.0580 1.1301 1.2309 1.3605 1.5190 1.7060 1.9218

t=1.5

Exact 2.2644 2.3076 2.3796 2.4804 2.6100 2.7684 2.9556 3.1716

MQQI 2.2647 2.3080 2.3801 2.4810 2.6106 2.7689 2.9560 3.1718

t=2.0

Exact 4.0142 4.0576 4.1296 4.2304 4.3600 4.5184 4.7056 4.9216

MQQI 4.0147 4.0580 4.1301 4.2310 4.3606 4.5189 4.7060 4.9218

Table 8. The comparison of the L∞and L2errors of our method with t=0.01, c=0.01 and

N=64 at different times of experiment 4.

Time 0.2 0.4 0.6 0.8 1

α=0.2;

L∞4.1337E-06 1.9913E-05 4.6348E-05 8.3447E-05 1.3125E-04

L22.8311E-06 1.3469E-05 3.1316E-05 5.6361E-05 8.8643E-05

α=0.5;

L∞6.2292E-06 1.5261E-05 3.5725E-05 7.2620E-05 1.2012E-04

L24.8696E-05 7.4800E-06 2.4576E-05 4.9477E-05 8.1482E-05

α=0.9;

L∞1.3344E-04 1.1983E-04 9.5632E-05 7.0402E-05 1.0225E-04

L29.4744E-05 8.6462E-04 7.1672E-05 5.3182E-05 4.0570E-05

and the boundary conditions

u(0, t)=u(1, t)=0.

The exact solution of Equation (32) is not known (Podlubny, Chechkin, Skovranek,

Chen, & Jara,2009). We have solved this problem by MQQI scheme with N=20

and t=0.01 for α=1, 0.7, and 0.5. The results are shown in Figure 5. This

experiment is also solved by radial basis functions method (Uddin & Haq,2011)

and matrix approach (Podlubny et al.,2009). It should be observed that our

results are very much identical with the results obtained in Podlubny et al. (2009)

and Uddin and Haq (2011).

Experiment 4

In this experiment, we consider the time fractional diﬀusion equation

∂αu(x,t)

∂tα−∂2u(x,t)

∂x2=2t2−α

(3−α) sin (2πx)+4π2t2sin (2πx), (33)

with the initial condition

u(x,0)=0,

EUROPEAN JOURNAL OF COMPUTATIONAL MECHANICS 15

Figure 5. The space–time graph of estimated solution by MQQI for x∈[0, 1],t∈[0, 0.06]and

α=1, 0.7 and 0.5 of experiment 3.

and the boundary conditions

u(0, t)=u(1, t)=0,

wherein β=0, γ=−1, [a,b]= [0, 1] and f(x,t)=2t2−α

(3−α) sin (2πx)+

4π2t2sin (2πx).

The exact solution of this experiment is Li, Liang, and Yan (2017)

u(x,t)=t2sin (2πx). (34)

Numerical results of Equation (33) are obtained for c=0.016 and N=64 with

t=0.01. We have compared the exact solution and numerical solutions for

our problem using values of α=0.2, α=0.5andα=0.9 and tabulated in

16 M. SARBOLAND

Figure 6. Absolute error and exact and estimated solutions with t=0.01, N=64 and α=0.5

of experiment 4.

Figure 7. Absolute error and exact and estimated solutions with t=0.01, N=64 and α=0.5

of experiment 5.

Table 8. We can obviously see in this table that the exact and numerical solutions

acquired by the scheme are in harmony with respect to each other.

Also, the graphs of approximate solutions acquired for α=0.5, t=0.01

and N=64 at various times have been illustrated in Figure 6.

Experiment 5

As a ﬁfth experiment, we solve Equation (9) with β=0, γ=−1and[a,b]=

[0, 1] which is one-dimensional time fractional diﬀusion equation

∂αu(x,t)

∂tα−∂2u(x,t)

∂x2=ex(5+α)

24 t4−tα+4ex.(35)

EUROPEAN JOURNAL OF COMPUTATIONAL MECHANICS 17

Table 9. The comparison of the L∞and L2errors of our method with t=0.01, c=0.01 and

N=64 at different times of experiment 5.

Time 0.2 0.4 0.6 0.8 1

α=0.3;

L∞2.8296E-06 1.7079E-05 5.2127E-05 1.2224E-04 2.4844E-04

L22.0883E-06 1.2818E-05 4.0039E-05 9.6229E-05 1.9984E-04

α=0.5;

L∞8.5878E-06 5.7022E-05 1.7262E-04 3.8371E-04 7.2382E-04

L26.3229E-06 4.2069E-05 1.3404E-04 2.8677E-04 5.4620E-04

α=0.9;

L∞4.2865E-05 4.5074E-04 1.6991E-03 4.2690E-03 8.6414E-03

L23.1956E-05 3.3319E-04 1.2517E-03 3.1406E-03 6.3535E-03

Table 10. Comparison of the results with the exact solutions with α=0.5, t=0.01 and

N=64 for different values of xand tof experiment 5.

x0.125 0.250 0.375 0.500 0.625 0.750 0.875

t=0.2

Exact 0.0008 0.0009 0.0010 0.0012 0.0013 0.0015 0.0017

MQQI 0.0008 0.0009 0.0010 0.0012 0.0013 0.0015 0.0017

t=0.4

Exact 0.0183 0.0208 0.0236 0.0267 0.0302 0.0343 0.0388

MQQI 0.0184 0.0208 0.0236 0.0268 0.0303 0.0343 0.0389

t=0.6

Exact 0.1138 0.1289 0.1461 0.1656 0.1875 0.2125 0.2408

MQQI 0.1138 0.1290 0.1462 0.1657 0.1877 0.2127 0.2409

t=0.8

Exact 0.4151 0.4704 0.5330 0.6040 0.6844 0.7756 0.8788

MQQI 0.4153 0.4707 0.5334 0.6044 0.6848 0.7759 0.8791

t=1.0

Exact 1.3315 1.2840 1.4550 1.6487 1.8682 2.1170 2.3989

MQQI 1.3315 1.2845 1.4556 1.6494 1.8689 2.1176 2.3993

The exact solution of this experiment is Li et al. (2017)

u(x,t)=extα+4.(36)

The initial condition in (10) and the boundary conditions in (11) can be obtained

from the exact solution.

A comparison of the analytical and the obtained numerical solutions for values

α=0.3, α=0.5andα=0.9 has been given in Table 9. Also, the numerical

solutions are compared with the exact solutions for diﬀerent values of xand t

in Table 10. It is clear in these tables that the approximate results are consistent

with the theoretical results.

The graphs of absolute error and approximate solutions acquired for α=0.5,

t=0.01 and N=64 at diﬀerent times have been shown in Figure 7.

6. Conclusion

In this paper, a numerical scheme based on high accuracy MQ quasi-interpolation

scheme and RBFs approximation scheme has been given for solving the time

18 M. SARBOLAND

fractional partial diﬀerential equations. The accuracy of the method can be

improved by selecting the appropriate shape parameter.

The numerical results which are presented in the previous section demonstrate

that the performance of the method is in excellent agreement with the exact

solutions. Tables 1–10 show that the MQQI scheme is more accurate than MQ

scheme in more cases.

It should be noted that in this paper, we used the univariate MQ quasi-

interpolation scheme to solve the one-dimensional FPDEs, but this scheme can

be extended and implemented for two-dimensional FPDEs similar work that we

did in Sarboland and Aminataei (2015a). Besides, we use uniformly points in our

numerical experiments, but our schemes can be used for the scattered points.

Disclosure statement

No potential conﬂict of interest was reported by the authors.

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