Preconditioned Gauss-Seidel Method for the Solution of Time-fractional
Diffusion Equations
A. Sunarto
1
and J. Sulaiman
2
1
Department Tadris Matematika, IAIN Bengkulu, Indonesia
2
Department Matemathics with economics, Universiti Malaysia Sabah, Malaysia
Keywords:
Linear System, Preconditioned Gauss-Seidel (PGS) method.
Abstract:
In this paper, we deal with the application of an unconditionally implicit finite difference approximation equa-
tion of the one-dimensional linear time fractional diffusion equations via the Caputo’s time fractional deriva-
tive. Based on this implicit approximation equation, the corresponding linear system can be generated in which
its coefficient matrix is large scale and sparse. To speed up the convergence rate in solving the linear system
iteratively, we construct the corresponding preconditioned linear system. Then we formulate and implement
the Preconditioned Gauss-Seidel (PGS) iterative method for solving the generated linear system. One example
of the problem is presented to illustrate the effectiveness of PGS method. The numerical results of this study
show that the proposed iterative method is superior to the basic GS iterative method.
1 INTRODUCTION
Based on previous studies in (Mainardi, 1997; Di-
ethelm and Freed, 1999; Meerschaert and Tadjeran,
2004; Zhang, 2009)many successful mathematical
models, which are based on fractional partial deriva-
tive equations (FPDEs), have been developed. Fol-
lowing to that, there are several methods used to
solve these models. For instance, we have transform
method (Chaves, 1998), which is used to obtain ana-
lytical and/or numerical solutions of the fractional dif-
fusion equations (FDE). Other than this method, other
researchers have proposed finite difference methods
such as explicit and implicit (Agrawal, 2002; Yuste
and Acedo, 2005; Yuste, 2006). Also it is pointed
out that the explicit methods are conditionally stable.
Therefore, we discretize the time-fractional diffusion
equation via the implicit finite difference discretiza-
tion scheme and Caputo’s fractional partial derivative
of order α in order to derive a Caputo’s implicit finite
difference approximation equation. This approxima-
tion equation leads a tridiagonal linear system. Due
to the properties of the coefficient matrix of the lin-
ear system which is sparse and large scale, iterative
methods are the alternative option for efficient solu-
tions.As far as iterative methods are concerned, it can
be observed that many researchers such as Ghuang-
hui (Ghuang-hui et al., 2006), Young (Young, 2014),
Hackbusch (Hackbush, 1994) and Saad (Saad, 1996)
have proposed and discussed several families of itera-
tive methods. In addition to that, the concept of block
iteration has also been introduced by Evans (Evans,
1985), Ibrahim and Abdullah (Ibrahim and Abdullah,
1995), Evans and Yousif (Evans, 1985) to demon-
strate the efficiency of its computation cost. Among
the existing iterative methods, the preconditioned iter-
ative methods (Ghuang-hui et al., 2006), Zhao (Zhao
et al., 2000), Hoang-hao (Hhonghao et al., 2009), Gu-
nawardena (Gunawardena et al., 1991), Saad (Saad,
1996) have been widely accepted to be one of the ef-
ficient methods for solving linear systems.
Because of the advantages of these iterative meth-
ods, the aim of this paper is to construct and inves-
tigate the effectiveness of the Preconditioned Gauss-
Seidel (PGS) iterative method for solving time frac-
tional parabolic partial differential equations (TP-
PDE’s) based on the Caputo’s implicit finite differ-
ence approximation equation. To investigate the ef-
fectiveness of the PGS method, we also implement
the Gauss Seidel (GS) iterative methods being used a
control method.
To demonstrate the effectiveness of PGS method,
let time fractional parabolic partial differential equa-
tion (TPPDE’s) be defined as
268
Sunarto, A. and Sulaiman, J.
Preconditioned Gauss-Seidel Method for the Solution of Time-fractional Diffusion Equations.
DOI: 10.5220/0009882502680273
In Proceedings of the 2nd International Conference on Applied Science, Engineering and Social Sciences (ICASESS 2019), pages 268-273
ISBN: 978-989-758-452-7
Copyright
c
 2020 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

∂
α
U(x,t)
∂
α
=α(x)
∂
2
U(x,t)
∂x
2
+
b(x)
∂U(x,t)
∂x
+ c(x)U(x,t)
(1)
where α(x), b(x) and c(x) are known functions or
constants, whereas α is a parameter which refers to
the fractional order of time derivative.
The outline of this paper is organized as follows:
In Section 2 and 3, an approximate the formula of
the Caputo’s fractional derivative operator and nu-
merical procedure for solving time fractional diffu-
sion equation (1) by means of the implicit finite dif-
ference method are given. In Section 4, formulation
of the PGS iterative method is introduced. In Section
5 shows numerical example and its results and con-
clusion is given in Section 6.
2 PRELIMINARIES
Before constructing the Caputo’s implicit finite dif-
ference approximation equation of Problem (1), the
following are some basic definitions for fractional
derivative theory which are used in this paper.
Definition 1. (Young, 2014) The Riemann-
Liouville fractional integral operator, J
α
of order−α
is defined as
J
α
f (x) =
1
r(α)
Z
x
v
(x − t)
α
f (t)dt,
α > 0x > 0
(2)
Definition 2. (Young, 2014) The Caputo’s frac-
tional partial derivative operator, D
α
of order−α is
defined as
D
α
f (x) =
1
r(m − α)
Z
x
0
f
m
(t)
(x − t)
α−m+1
dt
,α > 0
(3)
with m − 1 < α ≤ m, m ∈ N,x > 0
To obtain the numerical solution of Problem (1)
with Dirichlet boundary conditions, firstly we derive
an implicit finite difference approximation equation
based on the Caputo’s derivative definition and the
non-local fractional derivative operator. This implicit
approximation equation can be categoried as uncondi-
tionally stable scheme. To facilitate us in getting this
approximation equation of Problem (1), let the solu-
tion domain of the problem be restricted to the finite
space domain 0 ≤ x ≤ y , with 0 < α < 1, whereas
the parameter α refers to the fractional order of time
derivative. In addition to that, consider boundary con-
ditions of Problem (1) be given as
U(0,t) = g
0
(t),U (l,t) = g
1
(t),
and the initial condition
U(x,0) = f (x),
where g
0
(t),g
1
(t), and f(x) are given functions. A
discretize approximation to the time fractional deriva-
tive in Eq. (1) by using Caputo’s fractional partial
derivative of order α , is defined as (Young, 2014;
Hackbush, 1994).
∂u(x,t)
∂t
a
=
1
r(n − 1)
Z
∞
0
∂u(x − s)
∂t
(t − s)
−α
ds,t > 0, 0 < α < 1
(4)
3 APPROXIMATION FOR
FRACTIONAL DIFFUSION
EQUATION
According to Eq. (4), the formulation of Caputo’s
fractional partial derivative of the first order approxi-
mation method is given as
D
α
t
U
i,n
∼
=
σ
α,k
n
∑
j=1
ω
(α)
j
(U
i,n
− j + 1 − U
i,n− j
)
(5)
and we have the following expressions
σ
α,k
=
1
r(1 − α)(1 − α)k
α
and
ω
(α)
j
= j
1−α
Before discretizing Problem (1), let the solution
domain of the problem be partitioned uniformly. To
do this, we consider some positive integers m and
n in which the grid sizes in space and time direc-
tions for the finite difference algorithm are defined as
h = ∆x =
γ−0
m
and k = ∆t =
T
n
respectively. Based
on these grid sizes, we construct the uniformly grid
network of the solution domain where the grid points
in the space interval [0,γ] are indicated as the num-
bers x
i
= ih,i = 0, 1,2, ...,m and the grid points in the
time interval [0,T ] are labeled t
j
= jk, j = 0, 1,2, ...,n.
Then the values of the function U(x,t) at the grid
point are denoted as U
i, j
= U(x
i
,t
j
).
Preconditioned Gauss-Seidel Method for the Solution of Time-fractional Diffusion Equations
269

By using Eq. (5) and the implicit finite difference
discretization scheme, the Caputo’s implicit finite dif-
ference approximation equation of Problem (1) to the
grid point centered at (x
i
,t
j
) = (ih,nk) is given as
σ
α,k
Σ
n
j=1
ω
(α)
j
(U
i,n− j+1
−U
(
i,n − j)) =
a
1
1
h
2
(U
i−1,n
− 2U
i,n
+U
i+1,n
)
+ b
i
1
2h
(U
i+1,n
−U
i−1,n
) + c
i
U
i,n
,)
(6)
for i = 1,2..., m − 1.
Based on Eq. (6), this approximation equation is
known as the fully implicit finite difference approxi-
mation equation which is consistent first order accu-
racy in time and second order in space. Basically, the
approximation equation (6) can be rewritten based on
the specified time level. For instance, we have for
n ≥ 2:
σ
/al pha,k
Σ
n
j=1
ω
(/al pha)
(U
i,n= j+1
− U
i,n− j
) =
(
a
i
h
2
−
b
i
2h
)U
i−1,n
+ (c
i
−
2a
i
h
2
)U
i,n
+ (
a
i
h
2
−
b
i
2h
)U
i+1,n
(7a)
∴ σ
/al pha,k
Σ
n
j=1
ω
(/al pha)
(U
i,n= j+1
− U
i,n− j
) =
p
i
U
i−1,n
+ q
i
U
i,n
+ r
i
U
i+1,n
,
where
p
i
=
a
i
h
2
−
b
i
2h
,q
i
= c
i
−
2a
i
h
2
,r
i
=
a
i
h
2
−
b
i
2h
Also, we get for n = 1,
−p
i
U
i−1,1
+ q
i
∗U
i,1
− r
i
U
i+1,1
= f
i,1
,i = 1,2,...,m −
1
(7b)
where
ω
(/al pha)
= 1,q
i
∗ = σ
/al pha,k
− q
i
, f
i,1
= σ
/al pha,k
U
i,1
Based on Eq. (7b), it can be seen that the tridiag-
onal linear system can be constructed in matrix form
as
AU
v
= f
v
(8)
where
A =








q
∗
1
−r
1
−p
2
q
∗
2
−r
2
−p
3
q
∗
3
−r
3
.
.
.
.
.
.
.
.
.
−p
m−2
q
∗
m−2
−r
m−2
p
m−1
−p
∗
m−1








(m-1)x(m-1)
U
v
=

U11 U
21
U
31
· · · U
m−2,1
U
m−1,1

T
f
v
=

U
11
+ p
1
U
01
U
21
U
31
· · ·
U
m−2,1
U
m−1,1+p
m−1
U
m,1

T
4 FORMULATION OF
PRECONDITIONED
GAUSS-SEIDEL ITERATIVE
METHOD
In relation to the tridiagonal linear system in Eq. (8),
it is clear that the characteristics of its coefficient ma-
trix are large scale and sparse. As mentioned in Sec-
tion 1, many researchers have discussed various itera-
tive methods such as Ghuang-Hui (Ghuang-hui et al.,
2006), Zhao (Zhao et al., 2000), Hoang-hao (Hhong-
hao et al., 2009), Gunawardena (Gunawardena et al.,
1991), Young (Young, 2014), Hackbusch (Hackbush,
1994), Saad (Saad, 1996), Yousif and Evans (Evans
and Yousif, 1986). To obtain numerical solutions
of the tridiagonal linear system (8), we consider the
Preconditioned Gauss-Seidel (PGS) iterative method
(Ghuang-hui et al., 2006; Zhao et al., 2000; Hhong-
hao et al., 2009; Gunawardena et al., 1991), which
is the most known and widely using for solving any
linear systems.
Before applying the PGS iterative method, we
need to transform the original linear system (8) into
the preconditioned linear system
A
∗
x
v
= f
∗
v
(9)
where
A
∗
= PAP
T
,
f
∗
v
= P f
v
,U
v
= P
T
x
v
Actually, the matrix P is called a preconditioned ma-
trix and defined as (Kohno et al., 1997).
P = I + S
ICASESS 2019 - International Conference on Applied Science, Engineering and Social Science
270

where
S =








0 −r
1
0 0 0 0
0 0 −r
2
0 0 0
0 0 0 −r
3
0 0
0 0
.
.
.
.
.
.
.
.
.
0
0 0 0 0 0 −r
m−1
0 0 0 0 0 0








(m-1)x(m-1)
and the matrix I is an identical matrix. To formu-
late PGS method, let the coefficient matrix in (8) be
expressed as summation of the three matrices
A
∗
= D − L − V
(10)
where D, L and V are diagonal, lower triangular
and upper triangular matrices respectively. By us-
ing Eq. (9) and (10) , the formulation of PGS itera-
tive method can be defined generally as (Ghuang-hui
et al., 2006; Zhao et al., 2000; Hhonghao et al., 2009;
Gunawardena et al., 1991; Kohno et al., 1997).
x
(k+1)
v
= (D − L)
−1
V x
(k)
v
+ (D − L)
−1
f
∗
v
(11)
where x
(k+1)
v
represents an unknown vector at (k +
1)
th
iteration. The implementation of the PGS itera-
tive method can be described in Algorithm 1.
Algorithm 1: PGS method
i Initialize U
v
← 0 and ε ← 10
−10
.
ii For j = 1,2,· · · ,nImplement
For i = 1, 2,· · · , m − 1calculate
x
(k+1)
v
= (D − L)
−1
V x
(k)
v
+ (D − L)
−1
f
∗
v
U
(k+1)
v
= P
T
x
(k+1)
v
Convergence test. If the convergence criterion i.e



U
(k+1)
v
−U
(k)
v



≤ ε = 10
−10
is satisfied, go to
Step (iii). Otherwise go back to Step (a).
iii Display approximate solutions.
5 NUMERICAL EXAMPLE
By using approximation Eq.(7), we consider one ex-
ample of the time fractional diffusion equation to test
the effectiveness of the Gauss-Seidel (GS), and Pre-
conditioned Gauss-Seidel (PGS) iterative methods. In
order to compare the effectiveness of these two pro-
posed iterative methods, three criteria have been con-
sidered such as number of iterations, execution time
(in seconds) and maximum absolute error at three dif-
ferent values of α = 0.25,α = 0.50 and α = 0.75. For
implementation of both iterative schemes, the con-
vergence test considered the tolerance error, which is
fixed as ε = 10
−10
.
Let us consider the time fractional initial boundary
value problem be (Ali et al., 2013).
∂
/al pha
U(x,t)
∂t
/al pha
=
∂
2
U(x,t)
∂x
2
,0 < /al pha ≤ 1,0 ≤ x ≤
y,t > 0
(12)
where the boundary conditions are stated in fractional
terms.
U
(0,t)
=
2kt
/al pha
r(/al pha + 1)
,U (l,t) = l
2
+
2kt
/al pha
r(/al pha + 1)
(13)
and the initial condition
U(x,0) = x
2
(14)
From Problem (12), as taking /al pha = 1, it can
be seen that Eq. (12) can be reduced to the standard
diffusion equation
∂U (x,t)
∂t
=
∂
2
U(x,t)
∂x
2
,0 ≤ x ≤ γ,t > 0
(15)
subjected to the initial condition
U(x,0) = x
2
,
and boundary conditions
U(0,t) = 2kt,U(l,t) = l
2
+ 2kt,
Then the analytical solution of Problem (15) is ob-
tained as follows
U(x,t) = x
2
+ 2kt.
Now by applying the series
U(x,t) = Σ
m−1
n=0
∂
n
U(x,0)
∂t
n
t
n
n!
+
Σ
∞
n−1
Σ
m−1
i=0
∂
mn+U
U(x,0)
∂t
mn+i
t
n/al pha+i
r(n/al pha+i+1)
to U(x,t) for 0 < /al pha ≤ 1 it can be shown that the
analytical solution of Problem (12) is given as
U(x,t) = x
2
+ 2k
t
/al pha
r(/al pha+1)
.
All results of numerical experiments for Problem
(12), obtained from implementation of GS and PGS
iterative methods are recorded in Table 1 at different
values of mesh sizes, m = 128, 256, 512, 1024, and
2048.
Preconditioned Gauss-Seidel Method for the Solution of Time-fractional Diffusion Equations
271

6 CONCLUSION
In order to get the numerical solution of the time
fractional diffusion problems, the paper presents the
derivation of the Caputo’s implicit finite difference
approximation equations in which this approxima-
tion equation leads a linear system. From observa-
tion of all experimental results by imposing the GS
and PGS iterative methods, it is obvious at /alpha =
0.25 that number of iterations have declined approx-
imately by 64.87-99.82% corresponds to the PGS it-
erative method compared with the GS method. Again
in terms of execution time, implementations of PGS
method are much faster about 4.96-93.03% than the
GS method. It means that the PGS method requires
the least amount for number of iterations and com-
putational time at /al pha = 0.25 as compared with
GS iterative methods. Based on the accuracy of both
iterative methods, it can be concluded that their nu-
merical solutions are in good agreement.
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APPENDIX
Table 1: Comparison of number iterations, the execution
time (seconds) and maximum errors for the iterative meth-
ods using example at α = 0.25,0.50,0.75.
M Method
128
GS
PGS
256
GS
PGS
512
GS
PGS
1024
GS
PGS
2048
GS
PGS
ICASESS 2019 - International Conference on Applied Science, Engineering and Social Science
272

Table 2: Comparison of number iterations, the execution
time (seconds) and maximum errors for the iterative meth-
ods using example at α = 0.25,0.50,0.75. (extension)
α = 0.25
K Time Max Error
21017 37.73 9.97e-05
7292 35.86 9.96e-05
77231 343.63 1.00e-04
26884 261.56 9.98e-05
281598 2747.34 1.02e-04
98422 1916.28 1.00e-04
1017140 68285.36 1.09e-04
357258 14064.44 1.04e-04
3631638 158914.30 1.38e-04
21156 4104.17 1.36e-04
Table 3: Comparison of number iterations, the execution
time (seconds) and maximum errors for the iterative meth-
ods using example at α = 0.25,0.50,0.75 (extension).
α = 0.25
K Time Max Error
13601 5.92 9.86e-05
4715 2.23 9.84e-05
50095 42.17 9.90e-05
17417 16.68 9.87e-05
183181 339.85 1.01e-04
63298 123.01 9.96e-05
663971 2454.53 1.08e-05
232784 1007.47 1.03e-05
2380946 17795.25 1.38e-04
19153.0 3239.84 134e-05
Table 4: Comparison of number iterations, the execution
time (seconds) and maximum errors for the iterative meth-
ods using example at α = 0.25,0.50,0.75 (extension).
α = 0.75
K Time Max Error
6695 2.94 1.30e-04
2319 1.93 1.30e-04
24732 20.70 1.30e-04
8585 12.37 1.30e-04
90783 166.75 1.32e0-4
31619 62.78 1.31e-04
330622 1209.39 1.40e-04
115617 820.93 1.35e-04
1192528 8794.26 1.71e-04
12899 1305.5 1.35e-04
Preconditioned Gauss-Seidel Method for the Solution of Time-fractional Diffusion Equations
273