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Numerical solution of time fractional partial differential equations using multiquadric quasi-interpolation scheme

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In this paper, a meshfree method is presented to solve time fractional partial differential equations. It is based on the multiquadric quasi-interpolation operator . In the present scheme, quadrature formula is used to discretise the temporal Caputo fractional derivative of order 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.
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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(xixi1),
if we are supplied with a function f:[a,b]−R, quasi-interpolation of ftakes
the form:
L(f)=
N
i=0
f(xii(x),
where each function φi(x)is a linear combination of the Hardy’s MQs basis
function (Hardy,1971),
ψi(x)=c2+(xxi)2,
and low-order polynomials and cR+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)(xx0)
2(x1x0),
ψ1(x)=ψ2(x)ψ1(x)
2(x2x1)ψ1(x)(xx0)
2(x1x0),
ψi(x)=ψi+1(x)ψi(x)
2(xi+1xi)ψi(x)ψi1(x)
2(xixi1),2iN2, (2)
ψN1(x)=(xNx)ψN1(x)
2(xNxN1)ψN1(x)ψN2(x)
2(xN1xN2),
ψN(x)=1
2+ψN1(x)(xNx)
2(xNxN1).
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(|xxkj|),
where
¯ϕ(r)=s2
(s2+r2)3/2,
and sR+is a shape parameter.
The coefficients {αj}¯
N
j=1are uniquely obtained by the interpolation condition
Sf (xki)=
¯
N
j=1
αj¯ϕ(|xkixkj|)=f(xki),1i¯
N.(3)
4M. SARBOLAND
Since, the Equation (3)issolvable(Madych & Nelson,1990), so
α=A1
X.f
X,(4)
where
X={xk1,...,xk¯
N},α=[α1,...,α¯
N]T,AX=[¯ϕ(|xkixkj|)],
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+(xxkj)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+(xxkj)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[(xkjxkj1)f(xkj+1)(xkj+1xkj1)f(xkj)+(xkj+1xkj)f(xkj1)]
(xkjxkj1)(xkj+1xkj)(xkj+1xkj1),
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+(xxi)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+βu(x,t)
x+γ2u(x,t)
x2=f(x,t),
t∈[0, T],x=[a,b],0 <α1, (9)
with the initial condition
u(x,0)=u0(x), (10)
and the boundary conditions,
u(a,t)=g1(t),t0, 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
(1α) t
0
u(x,s)
s(ts)(α)ds,0<t<T,0<α<1,
u(x,t)
t,0tT,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)unk+1(x)unk(x)
+O(t), (13)
where aα=(t)α
(2α),bα(k)=(k+1)1α(k)1αand 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)unk+1(x)unk(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)unk+1(x)unk(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,1iN1,
(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 N1 linear equations
in N1 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]N1
i,j=1,Dij =[ˇ
ψij]N1
i,j=1,¨
Dij =[¯
ψij]N1
i,j=1,
ˆ
1=[ˆ
ψ0(x1),ˆ
ψ0(x2),...,ˆ
ψ0(xN1)]T,
ˆ
2=[ˆ
ψN(x1),ˆ
ψN(x2),...,ˆ
ψN(xN1)]T,
ˇ
1=[ˇ
ψ0(x1),ˇ
ψ0(x2),...,ˇ
ψ0(xN1)]T,
ˇ
2=[ˇ
ψN(x1),ˇ
ψN(x2),...,ˇ
ψN(xN1)]T,
¯
1=[¯
ψ0(x1),¯
ψ0(x2),...,¯
ψ0(xN1)]T,
¯
2=[¯
ψN(x1),¯
ψN(x2),...,¯
ψN(xN1)]T,
and
χn+1=[χn+1(x1),χn+1(x2),...,χn+1(xN1)]T.
Subsequently, Equation (22) can be written as
un+1=M1Nun+M1, (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
1NA1un+AM1AM
1NA1(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 unthe numerical
solution of Equation (9)atthenth time level, then the error εnat the nth time
level is given by εn=unun. The error equation for discretised fractional
partial differential equation can be written as
εn+1=AM1NA1εn=Eεn,(27)
where E=AM1NA1. 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 εn0asn→∞,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
MA1εn+1=NA1εn.(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]A1;cα=(2α).
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 ηR0or(ηR0andtα
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 Land L2error norms which are defined by
L=unun=max
0jN|un(xj)un(xj)|,
L2=unun2=
h
N
j=0
(un(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)=2t2α
(2α) 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=2t2α
(2α) 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 Land 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;
L3.13118E-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);
L1.152E-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 Land L2error norms versus shape parameter cfor t=0.01
and N=50 at t=0.5 of experiment 1.
cρ(E)LL2
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.
NLOrder 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 Land 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;
L4.7807E-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);
L6.086E-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)=2t2α
(3α)+2x2, which is one-dimensional linear inhomogeneous
fractional Burgers’ equation Odibat and Momani (2009)
αu(x,t)
tα+u(x,t)
x2u(x,t)
x2=2t2α
(3α) +2x2.
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(1x),
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 Land 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;
L4.1337E-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;
L6.2292E-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;
L1.3344E-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)
tα2u(x,t)
x2=2t2α
(3α) 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)=2t2α
(3α) 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)
tα2u(x,t)
x2=ex(5+α)
24 t4tα+4ex.(35)
EUROPEAN JOURNAL OF COMPUTATIONAL MECHANICS 17
Table 9. The comparison of the Land 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;
L2.8296E-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;
L8.5878E-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;
L4.2865E-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 110 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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In this paper, we present a new method for solving a nonlinear third-order Korteweg-de Vries equation. This method is based on the multiquadric (MQ) quasi-interpolation operator and an integrated radial basis function networks scheme. In the present scheme, the second-order central divided difference of the spatial derivative is used to approximate the third-order spatial derivative, and the Taylors series expansion to discretize the temporal derivative. Then, the spatial derivative is approximated by the MQ quasi-interpolation operator . This method is applied on some test experiments and the numerical results have been compared with the exact solutions and the solutions of other numerical methods. The , and root-mean-square errors of the solutions show the efficiency and the accuracy of the method. Furthermore, the stability analysis of the method is surveyed.
We present a numerical method to solve a time-space fractional Fokker–Planck equation with a space-time dependent force field F(x, t), and diffusion d(x, t). When the problem being modelled includes time dependent coefficients, the time fractional operator, that typically appears on the right hand side of the fractional equation, should not act on those coefficients and consequently the differential equation can not be simplified using the standard technique of transferring the time fractional operator to the left hand side of the equation. We take this into account when deriving the numerical method. Discussions on the unconditional stability and accuracy of the method are presented, including results that show the order of convergence is affected by the regularity of solutions. The numerical experiments confirm that the convergence of the method is second order in time and space for sufficiently regular solutions and they also illustrate how the order of convergence can depend on the regularity of the solutions. In this case, the rate of convergence can be improved by considering a non-uniform mesh.
We present a numerical method to solve a time-space fractional Fokker-Planck equation with a space-time dependent force field F(x, t), and diffusion d(x, t). When the problem being modelled includes time dependent coefficients, the time fractional operator, that typically appears on the right hand side of the fractional equation, should not act on those coefficients and consequently the differential equation can not be simplified using the standard technique of transferring the time fractional operator to the left hand side of the equation. We take this into account when deriving the numerical method. Discussions on the unconditional stability and accuracy of the method are presented, including results that show the order of convergence is affected by the regularity of solutions. The numerical experiments confirm that the convergence of the method is second order in time and space for sufficiently regular solutions and they also illustrate how the order of convergence can depend on the regularity of the solutions. In this case, the rate of convergence can be improved by considering a non-uniform mesh.
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An inversion problem of estimating parameters for a one-dimensional time fractional thermal wave equation with fractional heat flux conditions and Caputo fractional derivatives is investigated. To begin with, the analytical solution of the direct problem is obtained. Then, based on the parameter sensitivity analysis, the least-squares method is used to estimate both the fractional order alpha and the relaxation time T simultaneously. Finally, two different heat flux distributions are given as different boundary conditions to perform the simulation experiments, respectively. By analyzing the degree of fitting curves, results show that the least-squares method performs well in parameter estimation for this fractional thermal wave equation. This study provides an effective method of estimating the parameters of fractional thermal wave equations.