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An Implicit Numerical Method for Semilinear Space-time Fractional
Diffusion Equation
D.B. Dhaigude and Gunvant A. Birajdar*
Department of Mathematics, Dr.Babasaheb Ambedkar Marathwada
University, Aurangabad. 431 004, (M.S) India.
(*Corresponding author; e-mail: [email protected])
Received: xxx, Revised: xxx, Accepted: xxx
Abstract
The aim of the study is to obtain the solution of semilinear space-time fractional diffusion equation
for first initial boundary value problem (IBVP), by applying implicit method. The main idea of the
method is to convert the problem into an algebraic system which simplify the computations. We discuss
the stability, convergence and error analysis of the implicit finite difference scheme with suitable example
using MATLAB.
Keywords: Riemann-Liouville fractional derivative, Caputo fractional derivative, Implicit finite
difference method, Stability, Convergence.
Introduction
Fractional differential equations play an important role in the study of various physical, chemical
and biological phenomena. Therefore, many researchers have attracted in the field of theory, methods and
applications of fractional differential equations. Therefore, there is a need to study reliable and efficient
technique to obtain either exact or approximate solution of fractional differential equations. The
researchers have developed few numerical techniques and obtained approximate solution of both linear
and nonlinear fractional differential equations.
Liu et al.[9] consider time fractional advection-dispersion equation and obtained its complete
solution. Eidelman and Kochubei [5] investigated the Cauchy problem for the time-fractional diffusion
equations with variable coefficients, and constructed the fundamental solution. Lin and Xu[7], Liu et
al.[8] and Yuste [16] proposed explicit finite difference scheme for the time fractional diffusion equation.
Zhuang and Liu[17], Zhuang et al.[21], Chen et al.[2] and Murio[13], developed the implicit finite
difference approximation for the time fractional diffusion equation and also discussed its stability and
convergence. Liu et al.[11] employed the implicit finite difference scheme for space-time fractional
diffusion equation. Where as Zhuang and Liu [18] developed the implicit difference scheme for the two
dimensional space-time fractional diffusion equation.
Nonlinear partial differential equations have lot of applications in various branches of
sciences[3, 12]. In fact, published papers on the numerical methods for the nonlinear fractional partial
differential equations are limited. This motivates us to consider an effective numerical method for such
problems. Choi et al.[6] and Liu et al.[10] developed the numerical technique for fractional diffusion
equation with nonlinear source term. Zhang and Liu[19] consider Riesz space fractional diffusion
equation with nonlinear source term. Also Yang et al.[15] provide the numerical solution of fractional
Fokker Plank equation with nonlinear source term and proved its stability as well as convergence.
Recently Dhaigude and Birajdar[4] developed the discrete Adomian decomposition method for nonlinear
system of fractional partial differential equations.
We consider the space-time semilinear fractional diffusion equation
 u
= a( x, t ) Dx u ( x, t )  f (u ), a( x, t ) > 0,0 <   1,1 <   2,

t
Walailak J Sci & Tech 201x; x(x): xxx-xxx.
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Implicit Finite Difference Scheme
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with initial condition
and boundary conditions
D.B. Dhaigude and G. A. Birajdar
u ( x,0) = g ( x) 0 < x < l
(2)
u (0, t ) = 0 = u (l , t ) 0 < t  T ,
(3)
it is called the first initial boundary value problem (IBVP) for space-time semilinear fractional diffusion
equation. Note that
  (.)
is Caputo fractional derivative of order  and is defined [1] as
t 
t u ( x,  )
d
 1
, 0 <  < 1;.2in


 u  (1   ) 0  (t   )
=
t   u
,
 = 1.
 t
and the Riemann-Liouville fractional derivative of order
 1
2

2
Dx u ( x, t ) =  2(2   ) x
 u ,
 x 2

t
0
 (1 <   2)
is defined as
u ( x, )d
, 1 <  < 2;.2in
(t   )  1
 = 2.
When  = 1 and  = 2 in (1), it reduces to first IBVP for reaction-diffusion equation.
Here our aim is to find the numerical solution of IBVP for space-time fractional diffusion
equation (1)-(3) by using implicit finite difference method. We replace the time derivative by Caputo
fractional derivative and space derivative by Riemann-Liouville fractional derivative respectively.
Plan of the paper is as follows. In section 2 we develop the implicit difference scheme for first
IBVP. Stability of the implicit difference scheme is proved in section 3. Section 4 shows that implicit
difference scheme is convergent. Finally test problem is given as an application of the method.
Implicit Finite Difference Scheme
Consider the first IBVP for space-time semilinear fractional diffusion equation is
 u
= a( x, t ) Dx u ( x, t )  f (u ) 0 < x < l , 0 < t  T , 0 <   1,1 <   2
t 
(4)
with initial condition
u ( x,0) = g ( x)
(5)
u (0, t ) = 0 = u (l , t )
(6)
and boundary conditions
Our aim is to find the discrete solution of fractional IBVP (4)-(6). We divide the whole domain into
equal parts of rectangles. Define t k = k , k = 0,1,2,..., n , xi = ih, i = 0,1,2,..., m where
=
T
l
&h =
are the temporal and spatial steps respectively.
m
m
k
Let u i be the numerical approximation to u ( xi , tk ) . First, we approximate the time derivative
as follows.
Walailak J Sci & Tech 201x; x(x): xxx-xxx.
Implicit Finite Difference Scheme
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D.B. Dhaigude and G. A. Birajdar
t u ( x,  )
 u
1
d
=


0
t
(1   )
 (t   )
k
( j 1) u ( x,  )
1
d
=


(1   ) j =0 ( j )
 (tk 1   )
k u( x , t
1
d
i j 1 )  u ( xi , t j ) ( j 1)
;


( j )
(1   ) j =0

(t k 1   )
k u( x , t
1
i j 1 )  u ( xi , t j ) ( k  j 1) d
=

(k  j )  
(1   ) j =0

k u( x , t
1
i k 1 j )  u ( xi , t k  j ) ( j 1) d
=

( j )  
(1   ) j =0

 u
 
 
=
[
u
(
x
,
t
)

u
(
x
,
t
)]

i k 1
i k
t  (2   )
(2   )
k
[u( x , t
i
k 1 j
)  u ( xi , tk  j )][( j  1)1  j1 ]
j =1
where
1
b j = ( j  1)
 j1 , i = 0,1,2,..., m; j = 0,1,2,..., k , we have
  u ( xi , t k 1 )
t 
=
 
(2   )
[u ( xi , t k 1  u ( xi , t k )] 
 
(7)
k
(2   )
[u ( x , t
i
k 1 j
)  u ( xi , t k  j )]b j  O (
1
)
j =1
For every  (0  n  1 < n) the Riemann-Liouville derivative exists and coincides with the
Grunwald-Letnikow derivative. The relationship between the Riemann-Liouville and Grunwald-Letnikov
definitions also has another consequence which is important for the numerical approximations of
fractional order differential equations. This allows the use of the Riemann-Liouville definitions during
problem formulation and then the Grunwald-Letnikov definitions for obtaining the numerical solution.

For Dx u ( x, t ) , we adopt the shifted Grunwald formula at all time levels for approximating the second
order space derivative.
Dx u ( xi , tk 1 ) =
where the Grunwald weights are defined by
g 0 = 1, g j = (1) j
1
h
i 1
g u( x  ( j 1)h, t
j
i
k 1
)  O(h)
(8)
j =0
(  )(   1)(   2)...(   j  1)
, j = 1,2,3,...
j!
(9)
and the nonlinear function approximate as
f ( xi , tk , u( xi , tk 1 )) = f ( xi , tk , u( xi , tk ))  O( )
We rewrite the equation (7) as follows
k
k
k
j =1
j =1
j =1
[u( xi , tk 1 j )  u( xi , tk  j )]b j = u( xi , tk 1 j )b j  u( xi , tk  j )b j
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Implicit Finite Difference Scheme
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D.B. Dhaigude and G. A. Birajdar
k 1
k
= b1u( xi , tk )  u( xi , tk 1 j )b j  u( xi , tk  j )b j  bk u( xi , t0 )
j =2
j =1
k 1
k 1
j =1
j =1
= b1u( xi , tk )  u( xi , tk  j )b j 1  u( xi , tk  j )b j  bk u( xi , t0 )
k 1
= b1u( xi , tk )  (b j 1  b j )u( xi , tk  j )  bk u( xi , t0 )
(11)
j =1
Now using equation (7)-(??)
i 1
u( xi , tk 1 ) = u( xi , tk )  r g j u( xi  j 1 , tk 1 )  b1u( xi , tk ) 
j =0
k 1
(b
 b j 1 )u( xi , tk  j )  bk u( xi , tk )  r1 f (u( xi , tk ), xi , tk )  Rik 1
j
(12)
j =1
where r = r (i, k ) =
Let u
k
i
aik  (2   )
, r1 =   (2   )

h
| Rik 1 | c1  ( 1  h    )
be the numerical approximation of u ( xi , tk ) and let f i
k
(13)
be the numerical approximation of
f ( xi , tk , u ( xi , tk )) . Therefore the complete discrete form of first IBVP (4)-(6) is
(1   r )ui1  r
i 1

g j ui11 j
j = 0, j 1
i 1
(1   r )uik 1  r

g j uik11 j
j = 0, j 1
= ui0  r1 f i 0 , k = 0
k 1
= (1  b1 )uik  r1 f i k  (b j  b j 1 )uik  j  bk ui0 , k > 1
j =1
(14)
initialcon dition ui0 = g i i = 0,1,2,..., m  1
andtheboun darycondit ions u0k = umk k = 0,1,2,..., n
For k = 0, i = 1,2,..., m  1 in equation (14) we get the set of ( m  1) equations. The matrix equation
(15)
(16)
is
AU 1 = U 0  r1F 0
1
1
1
1
T
0
0
0
0
T
0
0
0
0
T
where U = [u1 , u2 ,..., um 1 ] ;U = [u1 , u2 ,..., um 1 ] ; F = [ f1 , f 2 ,..., f m 1 ] and A is a square
matrix of order (m  1)  (m  1) such that
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Implicit Finite Difference Scheme
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 1  r

 rg 2
 rg3

.
A=



 rg
m 1


D.B. Dhaigude and G. A. Birajdar
rg 0
1  r
 rg 2
rg 0
1   r  rg 0
.
.
.
.
.
 rg m  2
.
.
.
 rg 2









1  r 


This can be written as
0,

Ai , j = 1  r ,
 rg
i  j 1 ,

Also for
when j > i  1;
when j = i;
otherwise.
k = 1, i = 1,2,..., m  1 the matrix equation is
AU 2 = (1  b1 )U 1  b1U 0  r1F 1
In general,
k  1, i = 1,2,..., m  1 we can write as
k 1
AU k 1 = (1  b1 )U k  (b j  b j 1 )U ik  j  bkU 0  r1F k , k > 0
(18)
j =1
k
k
k
k
T
where F = [ f (u1 ), f (u2 ),..., f (um 1 )] ; U
k 1
= [u1k 1 , u2k 1 ,..., umk 11 ]T
Stability
Lemma 3.1 In equation (14), the coefficients bk (k = 0,1,..., ) and g j ( j = 0,1,2,...) satisfy
1. b j > b j 1 , j=0,1,2,...;
2. b0 = 1 , b j > 0 , j=0,1,...;
3.
g1 =   , g j  0( j  1),  j =0g j = 0 ;

4. For any positive integer n, we have

n
gj < 0.
j =0
k
Let u i be the approximate solution of the implicit finite difference scheme (4)-(6), and let f i
k
be the approximations of f i . Setting
 ik = uik  uik ,
we obtain the roundoff error equation.
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Implicit Finite Difference Scheme
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(1   r ) i1  r
i 1

j = 0, j 1
i 1
(1   r ) ik 1  r
for
g j  i11 j

D.B. Dhaigude and G. A. Birajdar
=  i0  r1 ( f i 0  f i 0 ), k = 0
k 1
= (1  b1 ) ik  r1 ( f i k  f i k )  (b j  b j 1 ) ik  j  bk  i0 , k > 1
g j  ik11 j
j = 0, j 1
j =1
i = 1,2,..., m  1, k = 0,1,2,..., n . Assuming || E || = max  ik .
k
1i  m 1
We now analyze the stability via method of induction. When
|  |= max{|  |, |  |,..., | 
1
l
1
1
1
2
1
m 1
k = 1, assume that
|} i.e.
|  l1 | (1  r ) |  l1 | r
 (1  r ) |  l1 | r
| (1  r ) l1  r
i 1
g
j = 0, j 1
i 1
g
j = 0, j 1
j
|  l1 |
j
|  l1 j 1 |
i 1
g
1
j l  j 1
j = 0, j 1
|
=|  l11  r1 ( f i 0  f i 0 ) |
(19)
|  l0 |  r1 L |  l0 |
 (1  r1 L) |  l0 |
|| E1 ||  C ||  0 || ( C = 1  r1L)
(20)
|| E k 1 || =|  lk 1 |= max{| 1k 1 |,|  2k 1 |,...,|  mk 11 |}
Let
and
assume
that
|| E ||  C ||  || , j = 1, 2,...k , we have
j
0
|  lk 1 | (1  r ) |  lk 1 | r
| 1  r ) lk 1  r
i 1
i 1
g
j = 0, j 1
g
j = 0, j 1
k 1
j l
j
|  lk 1 |
|
k 1
=| (1  b1 ) ik  r1 ( f i k  f i k )  (b j  b j 1 ) ik  j  bk  i0 |
j =1
k 1
 (1  b1 ) |  ik | r1 | ( f i k  f i k ) | (b j  b j 1 ) |  ik  j | bk |  i0 |
j =1
 (1  b1 ) |  |  r1 L |  | (b1  bk ) |  lk | bk |  i0 |
k
l
k
l
 (1  r1 L) |  l0 |
|| E k 1 ||  C0 ||  0 || , ( C0 = C  r1L)
Hence, the following theorem is obtained.
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Implicit Finite Difference Scheme
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Theorem 3.1 Suppose that
D.B. Dhaigude and G. A. Birajdar
 ik (i = 1,2,..., m  1, k = 1,2,..., n)
is the solution of the roundoff
error equation (4)-(6) and the nonlinear source term satisfies Lipschitz condition, then there is a positive
constant C0 such that ||  ||  C0 ||  || , k = 1, 2,..., n .
k
0
4 Convergence
In this section, we analyze the convergence of the implicit finite difference scheme (4)-(6). Let
u( xi , tk ) be the exact solution of the IBVP (4)-(6) at mesh point ( xi , tk ) and let u ik be the numerical
solution of the (4)-(6)computed using implicit finite difference scheme.
Define ei = u ( xi , t k )  ui and E = (e1 , e2 ,..., em 1 )
k
k
(1   r )ei1  r
k
k
k
i 1

i 1

T
= ei0  r1 ( f i 0  f i 0 )  Ri1 , k = 0
g j ei11 j
j =0, j 1
(1   r )eik 1  r
k
g j eik11 j
= (1  b1 )eik  r1 ( f i k  f i k ) 
j =0, j 1
k 1
(b
j
 b j 1 )eik  j  bk ei0   Rik 1 , k > 1
j =1
where
i = 1,2,..., m  1, k = 0,1,2,..., n .
| Rlk 1 | c1  ( 1  h    ) for i = 1, 2,..., m  1, k = 0,1, 2,..., n
Using mathematical induction and Lemma 1, we give the convergence analysis as follows. For
k = 1, assuming that || e1 || =| el1 |= max ei1
1i  M 1
| el1 | (1  r ) | el1 | r
 (1  r ) | el1 | r
| (1  r )el1  r
i 1
g
j = 0, j 1
| el1 |
j
i 1
g
j = 0, j 1
j
| el1 j 1 |
i 1
ge
j = 0, j 1
1
j l  j 1
|
=| el1  r1 ( f i 0  f i 0 )  Rl1 |
(22)
| el0 |  r1 L | el0 |  | Rl1 |
|| E1 || | Rl1 |
Using
e 0 = 0 and | Rl1 | c1  ( 1  h    ) , we obtain
|| e1 ||  b01c1  ( 1  h    )
Suppose that it holds for
k 1
and | el
j , || e j ||  bj 11c1  ( 1  h   ), j =1, 2,..., k
|= max{| e1k 1 |, | e2k 1 |,..., | emk 11 |}. Note that bj 1  bk1
| elk 1 | (1  r ) | elk 1 | r
i 1
g
j = 0, j 1
j
| elk 1 |
Walailak J Sci & Tech 201x; x(x): xxx-xxx.
Implicit Finite Difference Scheme
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D.B. Dhaigude and G. A. Birajdar
| (1  r ) lk 1  r
i 1
 g
j = 0, j 1
k 1
j l  j 1
|
k 1
=| (1  b1 )eik  r1 ( f i k  f i k )  (b j  b j 1 )eik  j  bk ei0  Rlk 1 |
j =1
k 1
 (1  b1 ) | eik | r1 | ( f i k  f i k ) | (b j  b j 1 ) | eik  j |  | Rlk 1 |
j =1
 (1  b1 ) || e ||  r1 L || e || (b1  bk ) || e k ||  | Rlk 1 |
k
k
 bk1{(1  b1 )  r1 L  (b1  bk )  bk }c1  ( 1  h    )
 bk1{1  r1 L}c1  ( 1  h    )
|| ek 1 ||  C0 k   ( 1  h    ), ( (1  r1L)c1 = C0 )
If
(23)
k  T is finite then, the following theorem is obtained.
k
Theorem 4.1 Let u i be the approximate value of u ( xi , tk ) computed by using implicit finite
difference scheme and source term satisfy the Lipschitz condition. Then there is a positive constant C0
such that | ui  u ( xi , t k ) | C0 (  h)
k
Test Problem
Example 4.1 Consider the space-time semilinear fractional diffusion equation
with initial condition,
and the boundary conditions
where
 0.9u 1.8u
= 1.8  u 2  f , 0 < x <  , 0 < t  T
0.9
t
x
u ( x,0) = sinx ;
u (0, t ) = 0 = u ( , t );
f = t 0.1sinxE1,1.1 (t )  et sin( x  0.9 )  sin 2 xe2t and exact solution is u( x, t ) = et sinx
In the following table we compare the exact solution and obtained solution at t = 0.01
I.F.D.M.
Exact Solution
Absolute Error
Relative Error
% Error
0.5064
0.5050
0.0014
0.0028
0.2772

0.8751
0.8747
0.0004
4.573  10
4
0.04578

1.0103
1.0101
0.00024
2.376  10
4
0.0238
0.8754
0.8747
0.0007
8.0027  10
0.5064
0.5050
0.0014
u ( x, t )

u ( ,0.01)
6
u ( ,0.01)
3
u ( ,0.01)
2
2
u ( ,0.01)
3
5
u ( ,0.01)
6
Walailak J Sci & Tech 201x; x(x): xxx-xxx.
0.0028
4
0.08
0.2772
Implicit Finite Difference Scheme
http://wjst.wu.ac.th
D.B. Dhaigude and G. A. Birajdar
Comparison between the exact solution and the numerical solution when t=0.02 & t=0.05
Example 4.2 Consider the space-time semilinear fractional diffusion equation
 u
u
=
x
 sinu , 0 < x < 1, 0 < t  T , 0 <   1,1 <   2
t 
x 
u ( x,0) = x(1  x);
with initial conditions,
and boundary conditions
u (0, t ) = 0 = u (1, t );
Numerical solution of u ( x, t ) at different time steps when  = 0.9 and  = 1.8
Walailak J Sci & Tech 201x; x(x): xxx-xxx.
Implicit Finite Difference Scheme
D.B. Dhaigude and G. A. Birajdar
http://wjst.wu.ac.th
Acknowledgements
The second author is thankful to UGC New Delhi, India for financial support under the scheme Research
Fellowship in Science for Meritorious Students " vide letter No.F.4-3/2006(BSR) /11-78/ 2008(BSR).
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