PSEUDO−SPECTRAL OPERATIONAL MATRIX FOR NUMERICAL SOLUTION OF SINGLE AND MULTI−TERM TIME FRACTIONAL DIFFUSION EQUATION

Abstract

This paper presents a new numerical approach to solve single and multi−term time fractional diffusion equation. In this work, the space dimension is discretized to the Gauss−Lobatto points. We use the normalized Grunwald approximation for the time dimension and pseudo−spectral successive integration matrix for the space dimension. This approach shows that with less number of points, we can approximate the solution with more accuracy. Some examples with numerical results in tables and figures displayed.

Full text

Turk J Math, V., N., ..., pp.xx-xx
PSEUDO−SPECTRAL OPERATIONAL MATRIX FOR NUMERICAL
SOLUTION OF SINGLE AND MULTI−TERM TIME FRACTIONAL
DIFFUSION EQUATION
SAEID GHOLAMI1, ESMAIL BABOLIAN1,2, MOHAMMAD JAVIDI3
Abstract. This paper presents a new numerical approach to solve single and multi−term
time fractional diffusion equation. In this work, the space dimension is discretized to the
Gauss−Lobatto points. We use the normalized Grunwald approximation for the time
dimension and pseudo−spectral successive integration matrix for the space dimension.
This approach shows that with less number of points, we can approximate the solu-
tion with more accuracy. Some examples with numerical results in tables and figures
displayed.
Keywords: Pseudo−Spectral Integration Matrix, normalized Grunwald approximation,
Gauss−Lobatto Points, Multi−Term Fractional Diffusion Equation.
AMS Subject Classification: 35R11,65M70
1. Introduction
In the recent years, due to the accuracy of the fractional differential equations in de-
scribing a variety of engineering and physics fields, such as, kinetic [39, 35, 23, 34, 24, 27],
solid mechanics [32], quantum systems [38], magnetic plasma [25] and economics [3], many
researchers are interested in fractional calculus. In [39] the concepts of fractional kinetic
such as, particle dynamics in different potentials, particle advection in fluids, plasma
physics and fusion devices, quantum optics and etc were discussed. The fractional ki-
netic of the diffusion, diffusion−advection and Fokker−Planck type were presented which
derived of the generalization of the master and the Langevin equations were presented
[23].
However, because of the complex structure of the fractional kinetic equations, the an-
alytical solutions of these equations are very rare. Hence, the study on the numerical
methods to solve these equations is increasing. The time fractional diffusion equation,
is one of these equations which we will focus on the new numerical solution for it. In
this equation the first−order time derivative replaced by a fractional derivative of order
0< α ≤1. Some numerical methods for the single time fractional diffusion equations
present as follows.
1Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran,
e-mail: saeid gholami@ymail.com
.
2Faculty of Mathematical Sciences and Computer, Kharazmy University, Tehran, Iran,
e-mail: babolian@tmu.ac.ir
.
3Department of Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran,
e-mail: mo javidi@yahoo.com
Manuscript received xx.
1

2 TURK J MATH, V., N., ...
Valko and Abate [36] proposed numerical inversion of 2−D Laplace transform to solve
the time fractional diffusion equation on a semi−infinite domain. Li and Xu [15] developed
the numerical solution for time fractional diffusion equation based on the spectral meth-
ods for both time and space dimensions, also in [17] presented a numerical approach based
on FDM in time and Legendre spectral method in space. Igor Podlubny and co−workers
in [31] presented a general method based on the matrix form representation of discretized
fractional operator in [30]. Diego A. Murio [24] developed an implicit unconditionally sta-
ble numerical approach to solve time fractional diffusion equation on a finite slab. Scherer
et al. [33] for numerical solution of time fractional diffusion equation with non−zero initial
condition presented a modification of the Grunwald−Letnikov approximation for Caputo
time derivative. The authors of [12] proposed a numerical method to solve FPDEs based
on high−order finite element method for space and FDM for time. A numerical method
for solution of time fractional diffusion equation in one−and two−dimensional cases pre-
sented in [5] which applied FDM in time and the Kansa method in space dimension. Dou
and Hon [7] proposed a numerical computation for backward time fractional diffusion
equations and also some examples in one−and two−dimensional cases were considered.
Wei and Zhang [37] considered a Cauchy problem of 1−D time fractional diffusion equa-
tion. Finally, Sinc−Harr collocation method [28], new difference scheme [1] to solve time
fractional diffusion equation is presented.
In this paper, we consider the multi−term time fractional diffusion equation
∂αu(x, t)
∂tα+
m
X
i=1
bi
∂βiu(x, t)
∂tβi
−∂2u(x, t)
∂x2=f(x, t),(x, t)∈[−1,1] ×[0, T ] (1)
with initial and boundary conditions
u(x, 0) = v(x), u(−1, t) = g1(t), u(1, t) = g2(t),(x, t)∈[−1,1] ×[0, T ]
where 0 ≤βi≤1 and ∂αu
∂tαis the Caputo fractional derivative of order 0 ≤α≤1. Unlike
the single−term fractional diffusion equation, the mathematical studies on the numerical
solution for the multi−term fractional diffusion equation are very rare. some studies on
the multi−term fractional diffusion equation are presented in [13,16,19,21].
Now, we present a new numerical approach to solve time fractional diffusion equation
which the space dimension is discretized to the Gauss−Lobatto points, then pseudo−spectral
integration matrix is applied. Hence, we review briefly the history of the pseudo−spectral
integration matrix which is the main method in this paper. El−Gendi [9] present an
operational matrix based on Clenshaw−Curtis quadrature scheme [6] to solve some lin-
ear integral equations of Fredholm and Volterra types, then he extended this method
for solution of the linear integro−differential and ODEs. El−Gendi with co−workers
[10] present a new matrix for successive integration of a function which was generaliza-
tion of the El−Gendi operational matrix [9]. Elsayed M. E. Elbarbary [8] using some
properties of integrals and derivatives of Chebyshev polynomials and modified El−Gendi
successive integration matrix [10] to derive an operational matrix for n−fold integrations
(Pseudo−Spectral Integration Matrix) of a function. This matrix has more accurate re-
sults. Gholami [11] for the first time, applied this matrix with FDM to solve a PDE, then
in [2] with co−authors used this matrix to solve a PDE alone. Now, we apply the pseudo-
spectral successive integration matrix for the space dimension and normalized Grunwald
approximation for the time dimension to solve single and multi−term time fractional
diffusion equations.

S. GHOLAMI, E. BABOLIAN, M. JAVIDI : PSEUDO−SPECTRAL INTEGRATION MATRIX ... 3
2. Preliminaries
2.1. Concepts of Fractional Derivatives. In this subsection we present the most im-
portant definitions for the fractional derivatives.
Definition 2.1.1. The Riemann−Liouville fractional derivative of order m−1< α < m
is
aDα
xf(x) = 1
Γ(m−α)
dm
dξmZξ
a
(ξ−η)m−α−1f(η)dηξ=x
,(2)
xDα
bf(x) = 1
Γ(m−α)
dm
dξmZb
ξ
(η−ξ)m−α−1f(η)dηξ=x
.(3)
Definition 2.1.2. The Caputo fractional derivative of order m−1< α < m is
C
aDα
xf(x) = 1
Γ(m−α)Zx
a
(x−η)m−α−1f(m)(η)dη, (4)
C
xDα
bf(x) = 1
Γ(m−α)Zb
x
(η−x)m−α−1f(m)(η)dη. (5)
Definition 2.1.3. [20] The Grunwald−Letnikov fractional derivative of order m−1<
α < m is
Dα
a+f(x) = lim
h→0,nh=x−ah−α
n
X
j=0
(−1)jα
jf(x−jh),(6)
Dα
b−f(x) = lim
h→0,nh=b−xh−α
n
X
j=0
(−1)jα
jf(x+jh).(7)
From [29] we can write
Dα
a+f(x) =
m−1
X
j=0
f(j)(a)(x−a)j−α
Γ(j−α+ 1) +1
Γ(m−α)Zx
a
(x−η)m−α−1f(m)(η)dη, (8)
Dα
b−f(x) =
m−1
X
j=0
(−1)jf(j)(b)(b−x)j−α
Γ(j−α+ 1) +(−1)m
Γ(m−α)Zb
x
(η−x)m−α−1f(m)(η)dη, (9)
for m−1< α < m. Using repeated integration by parts then differentiation of Riemann−Liouville
fractional derivative we have
1
Γ(m−α)
dm
dξmZξ
a
(ξ−η)m−α−1f(η)dη =
m−1
X
j=0
f(j)(a)(ξ−a)j−α
Γ(j−α+ 1)
+1
Γ(m−α)Zξ
a
(ξ−η)m−α−1f(m)(η)dη. (10)
Similarly
1
Γ(m−α)
dm
dξmZb
ξ
(η−ξ)m−α−1f(η)dη =
m−1
X
j=0
(−1)jf(j)(b)(b−ξ)j−α
Γ(j−α+ 1)
+(−1)m
Γ(m−α)Zb
ξ
(η−ξ)m−α−1f(m)(η)dη, (11)

4 TURK J MATH, V., N., ...
These equations show that
aDα
xf(x) = Dα
a+f(x),bDα
xf(x) = Dα
b−f(x).(12)
Indeed, the Grunwald−Letnikov derivative and the Riemann−Liouville derivative are
equivalent if the function f(x) has m−1 continuous derivatives and f(m)(x) is integrable on
closed interval [a, b]. Using this fact [18], by the relationship between Riemann−Liouville
fractional derivative and Grunwald−Letnikov fractional derivative we will derive a numer-
ical solution such that we use the Riemann−Liouville definition during problem formu-
lation and then the Grunwald−Letnikov definition for achieving the numerical solution.
From the standard Grunwald definition we have
Definition 2.1.4. [40] The standard Grunwald formula for u(x, t) which a≤x≤bis
Dα
a+u(x, t) = lim
M1→∞ h−α
1
M1
X
j=0
(−1)jα
ju(x−jh1, t),(13)
Dα
b−u(x, t) = lim
M2→∞ h−α
2
M2
X
j=0
(−1)jα
ju(x+jh2, t),(14)
where M1, M2∈N, h1=x−a
M1, h2=b−x
M2and g(j)
αare the normalized Grunwald weights
functions defined as
g(j)
α=−α−j+ 1
jg(j−1)
α, j = 1,2,3, ... (15)
with g(0)
α= 1.
Let Ω = [a, b]×[0, T ],(x, t)∈Ω, tk=kτ, k = 0(1)n, xi=a+ih, i = 0(1)m,with
τ=T
nand h=b−a
mbe time and space steps, respectively. From [22], for u(x, t)∈
L1(Ω), Dα
a+u(x, t)∈`(Ω) and Dα
b−u(x, t)∈`(Ω),we obtain
Dα
a+u(xi, tk) = h−α
i
X
j=0
(−1)jα
ju(xi−j, tk) + O(h),(16)
Dα
b−u(xi, tk) = h−α
m−i
X
j=0
(−1)jα
ju(xi+j, tk) + O(h).(17)
2.2. Pseudo−spectral integration matrix. We assume that (PNf)(x) is the Nth order
Chebyshev interpolating polynomial of the function f(x) at the points (xk, f (xk)) where
(PNf)(x) =
N
X
j=0
fjϕj(x),(18)
with
ϕj(x) = 2αj
N
N
X
r=0
αrTr(x)Tr(xj),(19)
where ϕj(xk) = δj,k, (δj,k is the Kronecker delta) and α0=αN= 1/2 , αj= 1 for
j= 1(1)N−1. Since (PNf)(x) is a unique interpolating polynomial of order N, it can

S. GHOLAMI, E. BABOLIAN, M. JAVIDI : PSEUDO−SPECTRAL INTEGRATION MATRIX ... 5
be expressed in terms of a series expansion of the classical Chebyshev polynomials, hence
we have
(PNf)(x) =
N
X
r=0
arTr(x),(20)
where
ar=2αr
N
N
X
j=0
αjf(xj)Tr(xj).(21)
The successive integration of f(x) in the interval [−1, xk] can be estimated by successive
integration of (PNf)(x). Thus we have
In(f)'
N
X
r=0
arZx
−1Ztn−1
−1Ztn−2
−1
... Zt2
−1Zt1
−1
Tr(t0)dt0dt1... dtn−2dtn−1.(22)
Theorem 1. [14] The exact relation between Chebyshev functions and its derivatives is
expressed as
Tr(x) =
n
X
m=0
(−1)mn
m
2nχm
T(n)
r+n−2m, r > n,
where
χm=
n
Y
j=0
j6=n−m
(r+n−m−j).
Theorem 2. [8] The successive integration of Chebyshev polynomials is expressed in
terms of Chebyshev polynomials as
Zx
−1Ztn−1
−1Ztn−2
−1
... Zt2
−1Zt1
−1
Tr(t0)dt0dt1... dtn−2dtn−1=
n−γr
X
m=0
βr
(−1)mn
m
2nχm
ξn,m,r(x),
where
ξn,m,r(x) = Tr+n−2m(x)−
n−1
X
i=0
ηiT(i)
r+n−2m(−1),
ηi=
i
X
j=0
xj
(i−j)!j!, χm=
n
Y
j=0
j6=n−m
(r+n−m−j),
βi=2i= 0,
1i > 0,γi=


n i = 0,
n−i+ 1 1 ≤i≤n,
0i > n.
Thus, from Theorem 2 and relations (21) and (22),we have
In(f)'
N
X
j=0 2αj
N
N
X
r=0
αrTr(xj)
n−γr
X
m=0
βr
(−1)mn
m
2nχm
ξn,m,r(x)f(xj).

6 TURK J MATH, V., N., ...
The matrix form of the successive integration of the function f(x) at the Gauss-Lobatto
points xkis
[In(f)] = "N
X
j=0 2αj
N
N
X
r=0
αrTr(xj)
n−γr
X
m=0
βr
(−1)mn
m
2nχm
ξn,m,r(x)f(xj)#= Θ(n)[f].(23)
The elements of the matrix Θ(n)are
ϑ(n)
k,j =2αj
N
N
X
r=0
αrTr(xj)
n−γr
X
m=0
βr
(−1)mn
m
2nχm
ξn,m,r(xk).(24)
The matrix Θ(n)in (23), presented in [8], is called the pseudo−spectral integration matrix.
3. Single and Multi−Term Fractional Diffusion Equations
3.1. Time Fractional Diffusion Equation. We consider the time fractional diffusion
equation
∂αu(x, t)
∂tα=k(t)∂2u(x, t)
∂x2+q(t)u(x, t) + f(x, t),(x, t)∈[−1,1] ×[0, T ],(25)
with initial condition
u(x, 0) = v(x),−1≤x≤1,
and boundary conditions
u(−1, t) = g1(t), u(1, t) = g2(t),0≤t≤T,
where ∂αu(x,t)
∂tαis the Caputo fractional derivative of order 0 ≤α≤1, also v(x), g1and g2
are known functions. In the equation (12) illustrated that the Grunwald−Letnikov deriva-
tive and Riemann−Liouville derivative are equivalent under discussed conditions. Hence,
we use this fact to derive a numerical approach [18] for solution of fractional differential
equations such that in these equations we use the Riemann−Liouville definition during
problem formulation and then the Grunwald−Letnikov definition for deriving the nu-
merical solution. The relationship between Caputo derivative ∂α
∂tαand Riemann−liouville
derivative 0Dα
tis as [4]
∂αu(x, t)
∂tα=0Dα
tu(x, t)−u(x, 0)
tαΓ(1 −α),0≤α≤1,
Hence, we can write the equation (25) for 0 ≤α≤1 as
0Dα
tu(x, t)−v(x)
tαΓ(1 −α)=k(t)∂2u(x, t)
∂x2+q(t)u(x, t)+f(x, t),(x, t)∈[−1,1]×(0, T ].
(26)
Now, we apply the pseudo−spectral integration matrix to descretization of the space
dimension to the Gauss−Lobatto points xi=−cos iπ
Nfor N∈N. Assume that
∂2u(x, t)
∂x2


xi
=ϕ(xi, t),(27)
∂u(x, t)
∂x 


xi
=
N
X
j=0
ϑ(1)
i,j ϕ(xj, t) + c1,(28)

S. GHOLAMI, E. BABOLIAN, M. JAVIDI : PSEUDO−SPECTRAL INTEGRATION MATRIX ... 7
u(xi, t) =
N
X
j=0
ϑ(2)
i,j ϕ(xj, t) + c1(xi+ 1) + c2,(29)
for i= 0(1)N. We can find the constants c1and c2to satisfy the boundary conditions.
From these conditions we obtained
c1=−1
2N
X
j=0
ϑ(2)
N,j ϕ(xj, t) + g1(t)−g2(t), c2=g1(t).
By substituting c1and c2into (29), we have
u(xi, t) =
N
X
j=0
ϑ(2)
i,j ϕ(xj, t)−1
2(xi+ 1)
N
X
j=0
ϑ(2)
N,j ϕ(xj, t) + pi(t),(30)
which
Pi(t) = −1
2(xi+ 1)g1(t)−g2(t)+g1(t).(31)
Now, we substitute (27) and (30) into main equation (26) to obtain
N
X
j=0
ϑ(2)
i,j 0Dα
tϕ(xj, t)−1
2(xi+ 1)
N
X
j=0
ϑ(2)
N,j 0Dα
tϕ(xj, t)
=q(t)N
X
j=0
ϑ(2)
i,j ϕ(xj, t)−1
2(xi+ 1)
N
X
j=0
ϑ(2)
N,j ϕ(xj, t) + pi(t)
+k(t)ϕ(xi, t)−0Dα
tpi(t) + v(xi)
tαΓ(1 −α)+f(xi, t), i = 0(1)N. (32)
Let
tk=kτ, k = 0(1)m, τ =T
m
and use the Grunwald−Letnikov approximation instance the Riemann−Liouville deriva-
tive in time dimension to obtain the numerical formula as
0Dα
tkϕ(xj, t) = τ−α
k
X
r=0
g(r)
αϕ(xj, tk−r), k = 0(1)m, (33)
where g(r)
αare normalized Grunwald weights functions. Hence, we insert (33) into (32) to
obtain
τ−α
k
X
r=0
g(r)
αN
X
j=0
ϑ(2)
i,j ϕ(xj, tk−r)−1
2(xi+ 1)
N
X
j=0
ϑ(2)
N,j ϕ(xj, tk−r)
=q(tk)N
X
j=0
ϑ(2)
i,j ϕ(xj, tk)−1
2(xi+ 1)
N
X
j=0
ϑ(2)
N,j ϕ(xj, tk) + pi(tk)+k(tk)ϕ(xi, tk)
+v(xi)
tα
kΓ(1 −α)−0Dtα
kpi(tk) + f(xi, tk), i = 0(1)N, k = 1(1)m. (34)
We recall from [2] to summarize
Ai=ϑ(2)
i,0, ϑ(2)
i,1, ..., ϑ(2)
i,N −1
2(Xi+ 1)ϑ(2)
N,0, ϑ(2)
N,1, ..., ϑ(2)
N,N ,(35)
Φk=ϕ0,k, ϕ1,k , ..., ϕN,kt,(36)

8 TURK J MATH, V., N., ...
then apply g(0)
α= 1 to obtain
τ−α−q(tk)AiΦk−k(tk)ϕ(xi, tk) = v(xi)
tα
kΓ(1 −α)−0Dtα
kpi(tk)
+pi(tk)q(tk) + f(xi, tk)−τ−α
k
X
r=1
g(r)
αAiΦk−r,(37)
for i= 0(1)Nand k= 1(1)m. indeed, (37) is the following system
AkΦk=Bk.(38)
for i= 0(1)Nand k= 1(1)msince Φ0=ϕ(xi,0) = V00(xi) from the initial condition.
All unknowns ϕ(xi, tk) obtained by solving this system, finally we can approximate the
solutions from the equation (30).
3.2. The Multi−Term Time Fractional Diffusion Equation. We consider the multi−term
time fractional diffusion equation (1) for m= 1 and b1= 1 as
∂αu(x, t)
∂tα+∂βu(x, t)
∂tβ=∂2u(x, t)
∂x2+f(x, t),(x, t)∈[−1,1] ×[0, T ],(39)
with initial condition
u(x, 0) = v(x),−1≤x≤1,
and boundary conditions
u(−1, t) = g1(t), u(1, t) = g2(t),0≤t≤T,
with 0 ≤α, β ≤1. Similar to pervious subsection, we apply Pseudo−Spectral integration
matrix for xi=−cos(iπ
N) and we have same procedure to (27) −(31), exactly. Now, from
the relationship between Caputo and Riemman−Liouville fractional derivatives, we can
write the equation (39) as
0Dα
tu(x, t) + 0Dβ
tu(x, t) = ∂2u(x, t)
∂x2+F(x, t),(x, t)∈[−1,1] ×[0, T ],(40)
which
F(x, t) = f(x, t) + v(x)t−α
Γ(1 −α)+t−β
Γ(1 −β).
Substituting (27) and (30) into main equation (40), give us
N
X
j=0
ϑ(2)
i,j 0Dα
tϕ(xj, t)−1
2(xi+ 1)
N
X
j=0
ϑ(2)
N,j 0Dα
tϕ(xj, t)
+
N
X
j=0
ϑ(2)
i,j 0Dβ
tϕ(xj, t)−1
2(xi+ 1)
N
X
j=0
ϑ(2)
N,j 0Dβ
tϕ(xj, t)
=ϕ(xi, t) + Hα,β(xi, t), i = 0(1)N, (41)
which
Hα,β(xi, t) = F(xi, t)−0Dα
tPi(t)−0Dβ
tPi(t).
Let
tk=kτ, k = 0(1)m, τ =T
m

S. GHOLAMI, E. BABOLIAN, M. JAVIDI : PSEUDO−SPECTRAL INTEGRATION MATRIX ... 9
and use the normalized Grunwald−Letnikov approximation instance the Riemann−Liouville
derivative as
0Dα
tkϕ(xj, t) = τ−α
k
X
r=0
g(r)
αϕ(xj, tk−r), k = 0(1)m, (42)
0Dβ
tkϕ(xj, t) = τ−β
k
X
r=0
g(r)
αϕ(xj, tk−r), k = 0(1)m, (43)
by using the notations in [2], equations (42) and (43) we can write the equation (41) as
τ−α
k
X
r=0
g(r)
α+τ−β
k
X
r=0
g(r)
βAiΦk−r−ϕ(xi, tk) = Hα,β(xi, tk),(44)
for i= 0(1)Nand k= 1(1)m. Finally, because g(0)
α= 1 for any α, we have
τ−α+τ−βAiΦk−ϕ(xi, tk) = Hα,β(xi, tk)−Ai
k
X
r=1 τ−αg(r)
α+τ−βg(r)
βΦk−r,(45)
for i= 0(1)Nand k= 1(1)m. indeed, (45) is the following system
A Φk= Bk, i = 0(1)N, k = 1(1)N. (46)
for k= 0 from the initial condition we have Φ0=ϕ(xi,0) = V00(xi). All unknowns
ϕ(xi, tk) obtained by solving this system, finally we can approximate the solutions from
the equation (30).
4. Numerical results
Example 1. Consider the time fractional diffusion equation in [1] by translating 0 ≤
x≤1 to −1≤X≤1 as
∂αw(X, t)
∂tα= 4k(t)∂2w(X, t)
∂X 2−q(t)w(X, t) + f(X, t),(47)
where 0 ≤t≤1 and 0 ≤α≤1 with initial condition
w(x, 0) = 0,−1≤X≤1,
and boundary conditions
w(−1, t) = w(1, t)=0,0≤t≤1,
which k(t) = et,q(t) = 1 −sin (2t) and
f(X, t) = π2t2et+t2(1 −sin (2t)) + 2t2−α
Γ(3 −α)sin π(X+ 1)
2.
The exact solution of this equation is w(X, t) = t2sin π(X+1)
2. The numerical results of
this problem are presented in the Tables 1 −3 and Figures 1 −4.
Example 2. We consider the one−dimensional multi−term time fractional diffusion
equation in [13] by translating 0 ≤x≤1 to −1≤X≤1 as
∂αw(X, t)
∂tα+∂βw(X, t)
∂tβ−4∂2w(X, t)
∂X 2=f(X, t),(X, t)∈[−1,1]×[0,1],0≤α, β ≤1,
(48)

10 TURK J MATH, V., N., ...
with initial and boundary conditions
w(X, 0) = 1−X2
4, w(−1, t) = w(1, t) = 0,
and
f(X, t) = t2−α
Γ(3 −α)+t2−β
Γ(3 −β)1−X2
2+ 2(1 + t2),
the exact solution of (48) is
w(X, t) = (1 + t2)1−X2
4.
The numerical results of this problem are presented in the Tables 4 −5 and Figures 5 −7.
Example 3. Consider the time fractional diffusion equation in [28] by translating
0≤x≤1 to −1≤X≤1 as
∂αw(X, t)
∂tα−4∂2w(X, t)
∂X 2=f(X, t),(X, t)∈[−1,1] ×[0,1],0≤α≤1,(49)
with initial and boundary conditions
w(X, 0) = 0, w(−1, t) = w(1, t) = 0,
and
f(X, t) = Γ(3)
Γ(3 −α)t2−α+ 4π2t2sin π(X+ 1),
the exact solution of (49) is
w(X, t) = t2sin π(X+ 1).
The numerical results of this problem are presented in the Tables 6−8 and Figures 8−10.
5. Conclusion
In this paper, a new numerical approach for solution of single and multi−term time frac-
tional diffusion equation presented which pseudo−spectral operational matrix has critical
role in it. For the first attempt, Two numerical methods, pseudo−spectral integration
matrix and normalized Grunwald approximation applied simultaneously. The significance
of this work is the presentation a new discretization of the space dimension based on the
Gauss−Lobatto points. This method shows that with less number of points we can approx-
imate the solutions with enough accuracy. Finally, we hope to use the pseudo−spectral
operational matrix for solution of fractional partial differential equations alone.
References
[1] Alikhanov AA. A new difference scheme for the time fractional diffusion equation. J Comput Phys
2015; 280: 424-438.
[2] Babolian E, Gholami S, Javidi M. A Numerical Solution for One-dimensional Parabolic Equation
Based On Pseudo-Spectral Integration Matrix. Appl Comput Math 2014; 13: 306-315.
[3] Baillie RT. Long memory processes and fractional integration in econometrics. J Econom 1996; 73:
5-59.

S. GHOLAMI, E. BABOLIAN, M. JAVIDI : PSEUDO−SPECTRAL INTEGRATION MATRIX ... 11
[4] Chen S, Liu F, Zhuang P, Anh V. Finite difference approximations for the fractional Fokker-Planck
equation. Appl Math Model 2009; 33: 256-273.
[5] Chen W, Ye L, Sun H. Fractional diffusion equations by the Kansa method. Comp Math Appl 2010;
59: 1614-1620.
[6] Clenshaw CW. The numerical solution of linear differential equations in Chebushev series. Proceed-
ings of the Cambridge Philosophical Society 1957; 53: 134-149.
[7] Dou FF, Hon YC. Numerical computation for backward time-fractional diffusion equation. Eng Anal
with Bound Elem 2014; 40: 138-146.
[8] Elbarbary Elsayed M E. Pseudo-spectral integration matrix and boundary value problems. Int J
Comp Math 2007; 84: 1851-1861.
[9] El-Gendi SE. Chebyshev solution of differential, integral, and integro-differential equations. Comp J
1969; 12: 282-287.
[10] El-Gendi SE, Nasr H, El-Hawary HM. Numerical solution of Poisson’s equation by expansion in
Chebyshev polynomials. CMS 1992; 84: 443-449.
[11] Gholami S. A Numerical Solution for One-dimensional Parabolic Equation Using Pseudo-spectral
Integration Matrix and FDM. R J Appl Sci Eng Tech 2014; 7: 801-806. pp.801-806.
[12] Jiang Y, Ma J. High-order finite element methods for time-fractional partial differential equations.
J Comput Appl Math 2011; 235: 3285-3290.
[13] Jin B, Lazarov R, Liu Y, Zhou Z. The Galerkin finite element method for a multi-term time-fractional
diffusion equation. J Comput Phys 2015; 281: 825-843.
[14] Khalifa AK, Elbarbary EME, Abd-Elrazek MA. Chebyshev expansion method for solving second
and fourth order elliptic equations. Appl Math and Comput 2003; 135: 307-318.
[15] Li X, Xu C. A space-time spectral method for the time fractional diffusion equation. SIAM J Numer
Anal 2009; 47: 21082131.
[16] Li Z, Liu Y, Yamamoto M. Initial-boundary value problems for multi-term time-fractional diffusion
equations with positive constant coefficients. Appl Math Comput 2013; 257: 381-397.
[17] Lin Y, Xu C. Finite difference/spectral approximations for the time fractional diffusion equation. J
Comput Phys 2007; 225: 1533-1552.
[18] Liu F, Anh V, Turner I. Numerical solution of the space fractional Fokker-Planck equation. J Comput
Appl Math 2004; 166: 209-219.
[19] Liu F, Meerschaert MM, McGough RJ, Zhuang P, Liu X. Numerical methods for solving the multi-
term time-fractional wave-diffusion equation. Frac Cal Appl Anal 2013; 16: 9-25.
[20] Lorenzo CF, Hartley TT. Initialization, Conceptualization, and Application in the Generalized Frac-
tional Calculus. Lewis , USA: Lewis Research Center, 1998.
[21] Luchko Y. Initial-boundary-value problems for the generalized multi-term time-fractional diffusion
equations. J Math Anal Appl 2011; 374: 538-548.
[22] Meerschaert MM, Tadjeran C. Finite difference approximations for fractional advection-dispersion
equations. J Comput Appl Math 2004; 172: 65-77.
[23] Metzler R, Klafter J. The random walk’s guide to anomalous diffusion: A fractional dynamics
approach. Phys Reports 2000; 339: 1-77.
[24] Murio AD. Implicit finite difference approximation for time fractional diffusion equations, Comp
Math Appl 2008; 56: 1138-1145.
[25] Negrete DDC, Carreras BA, Lynch VE. Front dynamics in reaction-diffusion systems with Levy
flights: A fractional diffusion approach. Phys Rev Lett 2003; 91: 018301-018304.
[26] Pagnini G. Short note on the emergence of fractional kinetics. Pysica A 2014; 409: 29-34.
[27] Philippa B, Robson RE, White RD. Generalized phase-space kinetic and diffusion equations for
classical and dispersive trasnport. New J Phys 2014; 16: 073040.
[28] Pirkhedri A, Javadi HHS. Solving the time fractional diffusion equation via Sinc-Haar collocation
method. Appl Math Comput 2015; 257: 317-326.
[29] Podlubny I. Fractional Differential Equations. San Diego, USA: Academic Press, 1999.
[30] Podlubny I. Matrix approach to discrete fractional calculus. Fractional Calculus and Applied Analysis
2000; 3: 359-386.
[31] Podlubny I, Chechkin A, Skovranek T, Chen Y, Jara VB. Matrix approach to discrete fractional
calculus II: partial fractional differential equations, J Comput Phys 2009; 228: 3137-3153.

12 TURK J MATH, V., N., ...
tα= 0.1α= 0.75 α= 0.85 α= 0.95
0.25 4.98E−4 8.47E−3 1.07E−2 1.32E−2
0.5 8.64E−4 9.74E−3 1.18E−2 1.40E−2
0.75 1.13E−3 9.12E−3 1.06E−2 1.20E−2
1 1.35E−3 8.02E−3 8.95E−3 9.77E−3
Table 1. Max error for example 1 when N= 4 and m= 4.
tα= 0.1α= 0.75 α= 0.85 α= 0.95
0.1 9.56E−5 2.49E−3 3.26E−3 4.14E−3
0.2 1.84E−4 3.50E−3 4.54E−3 5.77E−3
0.3 2.65E−4 3.94E−3 5.00E−3 6.25E−3
0.4 3.43E−4 4.11E−3 5.08E−3 6.20E−3
0.5 4.18E−4 4.13E−3 4.98E−3 5.93E−3
0.6 4.92E−4 4.06E−3 4.79E−3 5.57E−3
0.7 5.67E−4 3.94E−3 4.57E−3 5.19E−3
0.8 6.44E−4 3.80E−3 4.32E−3 4.82E−3
0.9 7.24E−4 3.65E−3 4.09E−3 4.48E−3
1 8.07E−4 3.51E−3 3.87E−3 4.18E−3
Table 2. Max errors for example 1 when N= 4 and m= 10.
[32] Rossikhin YA, Shitikova MV. Applications of fractional calculus to dynamic problems of linear and
nonlinear hereditary mechanics of solids. Appl Mech Rev 1997; 50: 15-67.
[33] Scherer R, Kalla SL, Boyadjiev L, Al-Saqabi B. Numerical treatment of fractional heat equations.
Appl Numer Math 2008; 58: 1212-1223.
[34] Sibatov RT, Uchaikin VV. Fractional differential kinetics of charge transport in unordered semicon-
ductors. Semiconductors 2007; 41: 335-340.
[35] Sokolov IM, Klafter J, Blumen A. Fractional kinetics. Phys Today 2002; 55: 48-54.
[36] Valko PP, Abate J. Numerical inversion of 2-d Laplace transforms applied to fractional diffusion
equation. Appl Numer Math 2005; 53: 73-88.
[37] Wei T, Zhang ZQ. Stable numerical solution to a Cauchy problem for a time fractional diffusion
equation. Engineering Analysis with Boundary Elements 2014; 40: 128-137.
[38] Wu JN, Huang HC, Cheng SC, Hsieh WF. Fractional Langevin equation in quantum systems with
memory effect. Appl Math 2014; 5: 1741-1749.
[39] Zaslavsky G. Chaos, fractional kinetics and anomalous transport. Phys Rep 2002; 371: 461-580.
[40] Zhuang P, Liu F, Anh V, Turner I. Numerical methods for the variable-order fractional advection-
diffusion equation with a nonlinear source term. SIAM J NUMER ANAL 2009; 47: 1760-1781.

S. GHOLAMI, E. BABOLIAN, M. JAVIDI : PSEUDO−SPECTRAL INTEGRATION MATRIX ... 13
tα= 0.1α= 0.75 α= 0.85 α= 0.95
0.05 2.62E−5 8.73E−4 1.15E−3 1.45E−3
0.1 5.18E−5 1.35E−3 1.80E−3 2.31E−3
0.15 7.69E−5 1.65E−3 2.18E−3 2.81E−3
0.2 1.02E−4 1.84E−3 2.40E−3 3.09E−3
0.25 1.27E−4 1.96E-3 2.53E−3 3.22E−3
0.3 1.53E−4 2.04E-3 2.59E−3 3.26E−3
0.35 1.79E−4 2.09E−3 2.62E−3 3.24E−3
0.4 2.06E-4 2.12E−3 2.62E−3 3.19E−3
0.45 2.34E−4 2.13E−3 2.60E−3 3.12E−3
0.5 2.63E−4 2.14E−3 2.57E−3 3.04E−3
0.55 2.93E−4 2.13E−3 2.53E−3 2.96E−3
0.6 3.25E−4 2.12E−3 2.49E−3 2.87E−3
0.65 3.57E−4 2.10E−3 2.44E−3 2.78E−3
0.7 3.91E−4 2.08E−3 2.39E−3 2.70E−3
0.75 4.26E−4 2.06E−3 2.35E−3 2.62E−3
0.8 4.62E−4 2.04E−3 2.30E−3 2.54E−3
0.85 5.00E−4 2.02E−3 2.26E−3 2.48E−3
0.9 5.40E−4 2.01E−3 2.22E−3 2.41E−3
0.95 5.80E−4 1.99E−3 2.19E−3 2.35E−3
1 6.23E−4 1.97E−3 2.15E−3 2.30E−3
Table 3. Max errors for example 1 when N= 4,8 and m= 20.
tα= 0.25 α= 0.5α= 0.95
0.25 4.25E−3 5.55E−3 5.54E−3
0.5 3.19E−3 4.59E−3 6.59E−3
0.75 3.16E−3 4.65E−3 6.99E−3
1 3.39E−3 5.00E−3 7.28E−3
Table 4. Max error for example 2 when N=m= 4 and β= 0.2.
tα= 0.25 α= 0.5α= 0.95
0.1 3.76E−3 4.92E−3 2.17E−3
0.2 2.12E−3 2.97E−3 2.52E−3
0.3 1.57E−3 2.23E−3 2.56E−3
0.4 1.33E−3 1.92E−3 2.55E−3
0.5 1.22E−3 1.79E−3 2.53E−3
0.6 1.18E−3 1.74E−3 2.53E−3
0.7 1.17E−3 1.73E−3 2.54E−3
0.8 1.18E−3 1.75E−3 2.57E−3
0.9 1.21E−3 1.79E−3 2.59E−3
1 1.24E−3 1.84E−3 2.61E−3
Table 5. Max error for example 2 when N= 5, m = 10 and β= 0.2.

14 TURK J MATH, V., N., ...
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
x
w(x,0.25)
exact
alpha=0.1
alpha=0.75
alpha=0.85
alpha=0.95
Figure 1. Comparison of numerical solutions of the example 1 at t=0.25
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
x
w(x,0.75)
exact
alpha=0.1
alpha=0.75
alpha=0.85
alpha=0.95
Figure 2. Comparison of numerical solutions of the example 1 at t=0.75

S. GHOLAMI, E. BABOLIAN, M. JAVIDI : PSEUDO−SPECTRAL INTEGRATION MATRIX ... 15
−1 −0.8 −0.6 −0.4 −0.2 00.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
x
numerical solution for N=m=8, alpha=0.75
t
w(x,t)
Figure 3. The approximation solution of example 1 when α= 0.75.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
x
w(x,t)
numerical solution for N=m=8, alpha=0.1
Figure 4. The approximation solution of example 1 when α= 0.1.

16 TURK J MATH, V., N., ...
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
x
w(x,6/9)
Figure 5. Comparison of numerical and exact solutions of the example 2
for α= 0.95, β = 0.2 at t=6
9.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
x
w(x,t)
Figure 6. Comparison of numerical and exact solutions of the example 2
for α= 0.95, β = 0.2.

S. GHOLAMI, E. BABOLIAN, M. JAVIDI : PSEUDO−SPECTRAL INTEGRATION MATRIX ... 17
−1 −0.8 −0.6 −0.4 −0.2 00.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
x
numerical solution for N=m=9, alpha=.95 and beta=.2
t
w(x,t)
Figure 7. The approximation solution of example 2 when α= 0.95 and
β= 0.2.
−1 −0.8 −0.6 −0.4 −0.2 00.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x
numerical solution for N=8, m=10, alpha=0.1
t
w(x,t)
Figure 8. The approximation solution of example 3 for N=m= 8 when
α= 0.1.

18 TURK J MATH, V., N., ...
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
x
w(x,0.3)
exact
alpha=0.1
alpha=0.2
alpha=0.5
alpha=0.99
Figure 9. Comparison of numerical solutions of the example 3 at t= 0.3.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x
w(x,t)
t=1
t=0.7
t=0.5
t=0.3 t=0
−−−o−−− exact
−−−*−−− alpha=0.1
Figure 10. Comparison of numerical solutions of the example 3 at some
values of t

S. GHOLAMI, E. BABOLIAN, M. JAVIDI : PSEUDO−SPECTRAL INTEGRATION MATRIX ... 19
tα= 0.1α= 0.5α= 0.99
0.25 4.50E−3 3.43E−3 2.58E−4
0.5 1.82E−2 1.66E−2 1.33E−2
0.75 4.13E−2 3.91E−2 3.61E−2
1 7.36E−2 7.11E−2 6.83E−2
Table 6. Max error for example 3 when N= 4 and m= 4.
tα= 0.1α= 0.5α= 0.99
0.25 1.55E−4 1.42E−3 5.25E−3
0.5 3.00E−4 2.15E−3 5.77E−3
0.75 4.36E−4 2.70E−3 5.84E−3
1 5.64E−4 3.15E−3 5.85E−3
Table 7. Max error for example 3 when N= 8 and m= 4.
tα= 0.1α= 0.5α= 0.99
0.1 2.70E−5 3.49E−4 1.84E−3
0.2 5.25E−5 5.35E−4 2.21E−3
0.3 7.64E−5 6.73E−4 2.30E−3
0.4 9.90E−5 7.88E−4 2.32E−3
0.5 1.21E−4 8.89E−4 2.32E−3
0.6 1.41E−4 9.78E−4 2.32E−3
0.7 1.61E−4 1.06E−3 2.31E−3
0.8 1.81E−4 1.13E−3 2.30E−3
0.9 2.00E−4 1.20E−3 2.27E−3
1 2.17E−4 1.27E−3 2.09E−3
Table 8. Max errors for example 3 when N= 8 and m= 10.