Download A class of Methods Based on Cubic Non

Spline Method for Solving the Linear Time
Fractional Diffusion Equation
Talaat S. El-Danaf
Department of Mathematics, Faculty of Science, Menoufia University, Shebeen El-Koom, Egypt.
Email: [email protected]
Abstract
In this paper, we are concerned with the problem of applying cubic parametric spline functions to
develop a numerical method for obtaining approximation for the solution for the linear time
fractional diffusion equation. The special parametric spline used in this paper is in fact
trigonometric-polynomial splines. The truncation error of the method is theoretically analyzed.
Using Von Neumann method, the proposed method is shown to be conditionally stable. A
numerical example is included to illustrate the practical implementation of the proposed method.
Keywords: Parametric spline; Time fractional diffusion; Von Neumann stability.
1 Introduction
In this article, we propose a parametric spline based method to obtain numerical solutions for
the time fractional diffusion equation of the form
 u  2u
(1)

 g ( x, t ),
a  x  b, 0    1, t  0 ,
t  x 2
subject to the conditions
(2)
u(a, t )  1 t , u(b, t )   2 t , t  0 .
and
ux, t 0   f x ,
a  x  b.
(3)
The spline functions proposed have the form [2,6,7], T3  span1, x, sin x, cos x where  is
the frequency of the trigonometric part of the spline functions which will be used to raise the
accuracy of the method. Numerical solution for the linear time fractional diffusion equation of the
form
 u  2u
(4)

 0 0  x  1, t  0
t  x 2
subject to the conditions
and
u (0, t )  0, u (1, t )  0,
ux,0   x.
(5)
.
(6)
based on implicit finite difference method [3], have been proposed. This paper is organized as
follows: In section 2, a new method depends on the use of the parametric spline is derived. In
section 3 the stability analysis is theoretically discussed. Using Von Neumann method, the proposed
method is shown to be conditionally stable. Finally, in section 4 one numerical example is included
to illustrate the practical implementation of the proposed method.
________________________________________________________________________________
Corresponding Author: Talaat S. El- Danaf: talaat11@ yahoo.com
2 Numerical Method
We set up a grid in the x , t plane with grid spacing h and k and grid points xi , t j  , where
xi  a  ih, for each i  0,1,..., N  1, and t j  jk ,
for each
j  0,1,....
j
Let Z i be an approximation to u ( xi , t j ), obtained by the segment Pi ( x, t j ) of the mixed spline
function passing through the points ( xi , Z i j ) and ( xi 1 , Z i j 1 ) . Each segment has the form [2,6,7]
Pi ( x, t j )  ai (t j ) cos ( x  xi )  bi (t j ) sin  ( x  xi )  ci (t j ) ( x  xi )  di (t j ) ,
(7)
for each i  0,1,, N . To obtain expressions for the coefficients of Eq. (7) in terms of Z i , Z i j1 ,
j
S i , and S i j 1 , we first define
j
Pi ( xi , t j )  Z i j , Pi ( xi 1 , t j )  Z i j 1 , Pi ( 2) ( xi , t j )  Si j , and Pi ( 2) ( xi 1 , t j )  Si j 1
By using Eqs. (7) and (8), we have
ai  d i  Z i j ,
ai cos   bi sin   ci h  d i  Z i j 1 ,
(8)
(9)
 ai   S i ,
2
j
 ai  2 cos   bi  2 sin   S i j 1 ,
ai  ai (t j ), bi  bi (t j ), ci  ci (t j ), d i  d i (t j ) , and   h. By solving the last four
where
equations, we obtain the following expressions:
h 2 cos  S i j  S i j 1
Z i j 1  Z i j
h S i j 1  S i j
h2 j
h2
,
(10)
,
c


d

Si  Z ij .
ai   2 S i j , bi 
i
i
2
2
2
h
 sin 






 

2.1 Spline Relations
Using the continuity condition of the first derivative at x  xi , that is Pi (1) ( xi , t j )  Pi (11) ( xi , t j ) ,
we obtain
bi   ci  ai 1 sin   bi 1 cos   ci 1 .
(11)
Using expressions in Eq. (10), Eq. (11) becomes
h 2 cos  S i j  S i j 1  Z i j 1  Z i j  hS i j 1  S i j 



h
 2 sin 
2
Z i j  Z i j 1   hS i j  S i j 1 
h 2 cos  S i j 1  S i j 
h 2 j
S
sin


cos


i 1
h
2
 2 sin 
2
After slight rearrangements, the last equation becomes
Z i j 1  2Z i j  Z i j 1  S i j 1  S i j  S i j 1 , i  1,2,, N .
(12)
h2
h2
2h 2 cos  2h 2

, 
 2 , and   h .
where  
 sin   2
 sin 

Remark 1.
1- The truncation error for Eq. (12), that is




T * i  u ij1  u ij1  2u ij   D x2 u ij1  D x2 u ij1  D x2 u ij
j
can be obtained by expanding this equation in Taylor series in terms of
derivatives as follows
2
u ( xi , t j ) and its
 h2

 h2
j
 
T * i  h 2    2  Dx2 uij  h 2     Dx4 uij  h 4 
  Dx6 uij  .
 12

 360 12 
From this expression of the local truncation error, for   2  h 2 our scheme is of Oh 2  , but


 
h2
our scheme is of O h 4 .
12
 h 2 4h 2 
,   2  h 2 , and system (12) reduces to
2- As   0, that is     0, then  ,     ,
 6 6 
ordinary cubic spline, that is
h2
Z i j 1  2Z i j  Z i j 1 
( S i j 1  4S i j  S i j 1 ), i  1,2,, N .
6
Using Eq. (1) , we can write S i j in the form
for   2  h 2 and  
 2 Z ij   Z ij
j 




g
i
 t 

x 2


Using the Caputo partial fractional derivative[3,5], we have
Sij 
  Z xi , t j 
(13)
t j Z  x , s 
1

i

t j  s  ds, t j  jk , 0    1.

1    0
t
t
Using a piecewise technique, Eq. (14) becomes
j 1
  Z xi , t j 
( q 1) k Z  x , s 
1
i
t j  s  ds, 0    1.




qk
1    q 0
t
t


(14)
(15)
Since t j  s  does not change sign on qk , q  1k  , the Weighted Mean Value Theorem for
Integrals[1], can be applied to each integration in the last summation as follows[3,4],

q 1k Z  xi , s 
Z xi , s * q 1k





t

s
ds

t

s
ds, qk  s *  q  1k .
j
j
 qk

qk
t
t
This implies that



( q 1) k
qk

Z  xi , s 
Z iq 1  Z iq

t j  s  ds 
t
k

( q 1) k
qk
t
 s  ds

j
1
1
 Z iq 1  Z iq   t j  qk   t j  qk  k 


k
1

 



 Z iq 1  Z iq    jk  qk 1   jk  qk  k 1 



k
1



1
1
1
 
Z iq 1  Z iq  j  q    j  q  1 .
k 1   
then the discrete approximation for the partial fractional derivative (15) can be written in the form
j 1
  Z xi , t j 
(16)
   j ,q Z iq 1  Z iq ,
0    1.

t
q 0
1
where  j , q   j  q 1   j  q  11 , and  
. Formula (16) allows us to express S i j in
1   1   k 
the form




3


j 1

S i j     j ,q Z iq 1  Z iq  g ij
(17)
q 0
which gives us the following useful formulas
Si1   Z i1  Z i0  g i0 , 1,0  1, j  1.
and




j 2

(18)

S i j   Z i j  Z i j 1     j ,q Z iq 1  Z iq  g ij ,  j , j 1  1, j  2
(19)
q 0
The use of Eqs. (18) and (19) in Eq. (12) respectively gives us the following systems
1   Z i11   2   Z i1  1   Z i11  Z i01  Z i0  Z i01  i1 ,
j  1, i  1,2,, N .
1
where  i  g i11  g i1  g i11 , j  1, i  1,2,, N .
and
Ai Z i j 1  Bi Z i j  Ai Z i j 1  Ai* Z i j 11  Bi* Z i j 1  Ai* Z i j 11   i j ,
(20)
(21)
i  1,2,..., N and j  2
where
Ai   ,
Ai  1   ,
*
Bi   ,
Bi  2   ,
*
and
 i     j ,q Z
j 2
j

q 0
q 1
i 1
Z
   Z
j 2
q
i 1
q 0

j ,q
q 1
i
Z
   Z
j 2
q
i
q 0

j ,q
q 1
i 1

 Z iq1   i j , j  2 .
or
j 2
j 2
q 0
q 0
 i j    j ,q Z iq11  Z iq 1  Z iq11     j ,q Z iq1  Z iq  Z iq1    i j , j  2
(22)
System (21) consists of N equations in the unknowns Z i , i  0,..., N  1. To get a solution to this
system we need 2-additional equations. These equations are obtained from conditions in (2).


Remark 2. For  ,    0, h 2 , System (21) reduces to the finite difference method[3], that is

1 j 
2 
1
Z i 1     2  Z i j  2 Z i j 1  Z i j 1   i j ,
2
h
h 
h

j  1, i  1,2,, N .
 i1   g i1 , and  i j    j ,q Z iq 1  Z iq   g ij , j  2.
j 2
where
q 0
3 Stability Analysis
The Von Neumann technique will be carried out to investigate the stability of systems (20) and
(21). The key part of Von Neumann analysis is to assume a solution of the form
(23)
Z i j   j exp Iih , I   1.
Inserting the latter expression for Z i j in Eq. (20), we obtain the characteristic equations in the form
 2    1 exp Iih   1    1 exp I i  1h  exp I i  1h 
  0 exp Iih    0 exp I i  1h   exp I i  1h ,
4
(24)
After simple calculations, Eq. (24) becomes
 1   0
   2 cos  
,   h . The quantity   2 cos  is surely
21  cos       2 cos  
positive if we choose   0, and   0 such that   2 but 1 cos   is positive or equal to
(25)
where  
zero. Then we obtain
 1    0 , and 0    1.
which implies
1   0
(26)
Substituting Eq.(23) into Eqs.(21) and (22) gives us the characteristic equation
Bi  j exp Iih   Ai  j exp I i  1h   exp I i  1h  
Bi* j 1 exp Iih   Ai* j 1 exp I i  1h   exp I i  1h    i j ,
j2
which can be simplified as
21  cos      2 cos   j     2 cos   j 1 
1
i j
exp Iih 
(27)
and
j 2
 i j     j ,q  q 1 exp Iih    exp  I   exp I  
q 0
j 2
   j ,q  q exp Iih    exp  I   exp I , j  2
(28)

q 0
which simplifies to
j 2
1
 i j     2 cos    j ,q  q 1   q  , j  2
exp Iih 
q 0
Using Eqs. (29) and (27), we obtain
(29)
j 2
 j   j 1    j ,q  q 1   q  , j  2
(30)
q 0
For j  2 , we obtain
 2   1   2,0  1   0    2  {(1   2,0 ) 1   2,0 0 }


The quantities  , 1  2,0 ,and  2,0 are positive. Then
 2  1   2,0  1   2,0  0
Inequality (26) implies
 2   1   2,0  1   2,0  0   1   2,0  0   2,0  0    0
This gives
2  0





where 0    1 for   0, and   0 such that   2 . For j  3 , Eq. (30) implies
 3   2  3,0  1   0   3,1  2   1  
 3  {1  3,1  2  3,1  3,0  1  3,0 0 }
then
 3  1  3,1   2  3,1  3,0   1  3,0   0
5
(31)



Since  , 1 3,1 ,  3,1   3,0 ,and  3, 0 are positive. Using inequalities (26) and (31), we obtain
 3  1   3,1  2   3,1   3,0  1   3,0  0





  1   3,1  0    3,1   3, 0  0   3, 0  0    0 
(32)
3  0
where 0    1 for   0, and   0 such that   2 . By the same method we can prove that
 j   0  Z i0  f i , j  1 and we have stability for   0, and   0 such that   2 .
4 Numerical Illustrations
We now obtain the numerical solutions of the linear time fractional diffusion equation (1).
Example
Consider the linear time fractional diffusion equation [3]
 u  2u

 0, 0    1, 0  x  1.
t  x 2
with boundary conditions
u (0, t )  u (1, t )  0, t  0,
and initial conditions
ux,0  x1  x,
0  x  1.
(33)
(34)
(35)
The solution will be approximated using the proposed method, with k  1 / 256, h  0.01,
  h 2 / 12 and   h 2  2 . The results are presented in Tables 1 – 2.
Table 1 : The numerical approximation by using the proposed method at
t  0.5,   h 2 / 12, and   h 2  2
Numerical Solution Numerical Solution
x
0.2
0.3
0.4
0.5
0.6
0.7
0.8
  0.5
  0.75
0.0122731
0.0168020
0.0196778
0.0206627
0.0196778
0.0168020
0.0122731
0.00860642
0.01179210
0.01381840
0.01451310
0.01381840
0.01179210
0.00860642
Table 2 : The numerical approximation by using the proposed method at
t  1.5,   h 2 / 12, and   h 2  2
Numerical Solution Numerical Solution
x
  0.75
  0.5
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.00711150
0.00973602
0.01140260
0.01197350
0.01140260
0.00973602
0.00711150
6
0.00340607
0.00466484
0.00546479
0.00573890
0.00546479
0.00466484
0.00340607
h2
12
2
and   h  2 at some different times t = 0.5 and 1.5 respectively. In Each figure the top curve is
at   0.25 , the middle curve is at   0.5 and the bottom curve is at   0.75 .
Figs.1 and 2 illustrate the behavior of the numerical solution for k  1 / 256, h  0.01,  
Fig. 2
Fig. 1
Z x, 0.5
.
Z x,1.5
0.020
.
0.015
0.015
0.010
0.010
0.005
0.005
x values
0.2
0.4
0.6
0.8
x values
1.0
0.2
Figs.1: The behavior of the numerical solution
for k  1 / 256, h  0.01,  
where t = 0.5
0.4
0.6
0.8
1.0
Figs.2: The behavior of the numerical solution
2
2
for k  1 / 256, h  0.01,   h and   h 2  2
12
where t = 1.5 and   0.25, 0.5 and 0.75
h
and   h 2  2
12
and   0.25, 0.5 and 0.75
The figures show that as  -increases the curve of the numerical solution decays.
5 Conclusion
In this paper, we considered a numerical treatment for the linear time fractional diffusion equation..
The method is shown to be conditionally stable using Von-Neumann method.
References
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2005.
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dissipative wave equation”, International Journal of Numerical methods for Heat and Fluid
Flow, 19,950-959,(2009).
[3] Diego A. Murio ”Implicit finite difference approximation for time fractional diffusion
equations”, Computers and Mathematics with applications 56,1138-1145, 2008.
[4] Jerome S. and Keith B. Oldham ” The fractional Calculus” Academic Press, Inc. 1974.
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[5] Podlubny I,. ” Fractional differential equations” Academic Press, 1999.
[6] Ramadan M. A., El Danaf T. S. and Abd Alaal F.E.I, ” Application of non-polynomial
spline approach to the solution of Burgers’ equation”, The Open Applied Mathematics
Journal, Vol.1,15-20,2007.
[7] Rashidinia, J. and Mohammadi R, ” Non-polynomial cubic spline methods for the solution of
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