A Tau Approach for Solving Fractional Diffusion Equations using Legendre-Chebyshev Polynomial Method

IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 1 Ver. VI (Jan - Feb. 2015), PP 42-48
www.iosrjournals.org
DOI: 10.9790/5728-11164248 www.iosrjournals.org 42 | Page
A Tau Approach for Solving Fractional Diffusion Equations using
Legendre-Chebyshev Polynomial Method
Osama H. Mohammed1
, Radhi A. Zaboon2
, Abbas R. Mohammed1
1
Department of Mathematics and Computer Applications, College of Science, Al-Nahrain University, Baghdad -
Iraq.
2
Department of Mathematics, College of Science, Al-Mustansiriyah University, Baghdad - Iraq.
Abstract: In this paper, a modified numerical algorithm for solving the fractional diffusion equation is
proposed. Based on Tau idea where the shifted Legendre polynomials in time and the shifted Chebyshev
polynomials in space are utilized respectively.
The problem is reduced to the solution of a system of linear algebraic equations. From the computational point
of view, the solution obtained by this approach is tested and the efficiency of the proposed method is confirmed.
Keywords: Tau method, Shifted Chebyshev polynomial, Shifted Legendre polynomial, Fractional diffusion
equation.
I. Introduction
Fractional calculus is a generalization of classical calculus which provides an excellent tool to describe
memory and hereditary properties of various materials and processes. The field of the fractional differential
equations draws special interest of researchers in several areas including chemistry, physics, engineering,
finance and social sciences [1].
There are many advantages of fractional derivatives. One of them is that it can be seen as a set of
ordinary derivatives that give the fractional derivatives the ability to describe what integer-order derivatives
cannot [2].
Fractional diffusion equations [3] are the generalization of classical diffusion equations. These
equations play important roles in molding anomalous diffusion systems and sub-diffusion systems. In recent
years, they have been applied in various areas of engineering, science, finance, applied mathematics, bio-
engineering etc, [4].
Most fractional diffusion equations do not have closed form solutions, so approximation and numerical
techniques such as He's variational iteration method [5,6], Adomian's decomposition technique [7], the
homotopy analysis method [8], finite difference scheme [9], and other methods [10], [11], [12] and [13]. [14]
Described sinc-Legendre collocation method for the fractional diffusion equations and [15] investigating the
numerical solution for a class of fractional diffusion-wave equations with a variable coefficients using sinc-
Chebyshev collocation method.
In this paper we develop a numerical algorithm to solve fractional diffusion equation based on Tau
method using shifted Legendre polynomials-shifted Chebyshev polynomials. The required approximate solution
is expanded as a series with the elements of shifted Legendre polynomials in time and shifted Chebyshev
polynomials in space with unknown coefficients. The proposed approach will reduce the problem to the solution
to a system of linear algebraic equations.
This paper is organized as follows:
In the next section, we introduce the definitions of the fractional derivatives and integration which are
necessary for the late of this paper. In sections 3 and 4 we present the shifted Legendre and shifted Chebyshev
polynomials respectively. Section 5 is devoted to constructing and analyzing the numerical algorithm. As a
result, a system of linear algebraic equations is formed and the solution of the considered problem is obtained.
In section 6, numerical examples are given to demonstrate the effectiveness and convergence of the proposed
method. Finally a brief conclusion is given in section 7.
II. Fractional derivatives and integration [1]
In this section we shall give some definitions which are needed in this paper later on.
Definition (1): The Riemann-Liouville fractional integral operator of order 0  is defined as:
t
1
t 0
1
I f(x) (x t) f(t)dt, 0, x 0
( )
 
    
   . … (1)

A Tau Approach for Solving Fractional Diffusion Equations using Legendre-Chebyshev ….
0
tI f(x) f(t)
Definition (2): The Riemann-Liouville fractional derivative operator of order 0  is defined as:
n
x
n 1
n 0
1 d
D f(x) (x t) f(t)dt
(n ) dx
 
 
   … (2)
Where n is an integer and n-1< α ≤ n.
Definition (3): The fractional derivative of f(x) in the Caputo sense is defined as
(n)
x
c
1 n0
1 f (t)
D f(x) dt, n 1 n, n
(n ) (x t)

 
     
    . … (3)
Where α> 0 is the order of the derivative and n is the smallest integer greater than or equal to  . For the Caputo
derivative, we have
D C 0
 , (C is a constant) … (4)
0
0
0, for and .
D x ( 1)
x , for and or and .
( 1 )
 

      

  
               

 
… (5)
Note: We use the ceiling function    to denote the smallest integer greater than or equal to α and the function
   to denote the largest integer less than or equal to α.
III. The shifted Legendre polynomials [16]
The well-known Legendre polynomials are defined on the interval [-1,1] and can be determined with
the aid of the following recurrence formula
i 1 i i 1
2i 1 i
L (z) zL (z) L (z)
i 1 i 1
 

 
 
, i=1, 2,… .
Where 0L (z) 1 and 1L (z) z . In order to use these polynomials on the interval x [0,h] we define the so-
called shifted Legendre polynomials by introducing the change of variable z (2x h) h  . The shifted Legendre
polynomials in x are then obtained as:
h h h
i 1 i i 1
(2i 1)(2x h) i
L (x) L (x) L (x), i 1,2,...
(i 1)h i 1
 
 
  
 
. … (6)
Where h
0L (x) 1 and h
1L (x) (2x h) h  .
The analytic form of the shifted Legendre polynomial h
iL (x) of degree i is given by
ki
h i k
i 2 k
k 0
(i k)! x
L (x) ( 1)
(i k)! (k!) h



 

 … (7)
The orthogonality condition is
h
h h
i j0
h
, fori j.
L (x)L (x)dx 2i 1
0, fori j.


 
 

A function f(x) defined over [0,1] may be approximated as:
n m T
sr srs 1 r 0
ˆf(x) c (x) C (x) f(x) 
     
Where C and (x) are n(m+1)×1 matrices given by:
10 11 1m 20 21 2m nmC [c ,c ,...,c ,c ,c ,...c ,...,c ]
And
h h h T
m,h 0 1 m(x) [L (x),L (x),...,L (x)]  … (8)
IV. The shifted Chebyshev polynomials [17]
The well-known Chebyshev polynomials are defined on the interval [-1, 1] and can be determined with
the aid of the following recurrence formulae
i 1 i i 1T (t) 2tT (t) T (t), i 1,2,...    . … (9)

Where h
0T (t) 1 and h
1T (t) t . In a shifted Chebyshev polynomial, the domain is transformed to values
between 0 and h by introducing the change of variables t (2x h) 1  . Let the shifted Chebyshev polynomials
i
2x
T 1
h
 
 
 
be denoted by h
iT (x) , then the h
iT (x) can be generated by the following recurrence relation
h h h
i 1 i i 1
2x
T (x) 2 1 T (x) T (x)
h
 
 
   
 
… (10)
Where h
0T (x) 1 and h
1T (x) (2x h) 1  . The analytic form of the shifted Chebyshev polynomials h
iT (x) of
degree i is given by
2ki
h i k k
i k
k 0
(i k 1)!2
T (x) i ( 1) x
(i k)!(2k)!h


 
 

 … (11)
The orthogonality condition is
h
h h
i j h ij j0
T (x)T (x)W (x)dx h  … (12)
Where h
2
1
W (x)
hx x


,
j
jh
2

  , with 0 2  and j 1  , j 1 .
A function f(x) defined over [0,1] may be approximated as:
n m T
sr srs 1 r 0
ˆf(x) c (x) C (x) f(x) 
     
Where C and (x) are n(m+1)×1 matrices given by:
10 11 1m 20 21 2m nmC [c ,c ,...,c ,c ,c ,...c ,...,c ]
And
h h h T
m,h 0 1 m(x) [T (x),T (x),...,T (x)]  … (13)
Lemma (1), [17]: Let h
iT (x) be a shifted Chebyshev polynomial, then:
Dα h
iT (x) 0, i 0,1,..., 1.     
Proof: The lemma can be easily proved by making use of the relations (5) and (11).
Theorem (1), [17]: Fractional derivative of order α>0 for the Chebyshev polynomials is given by:
Dα ( )
m,h m,h(x) D (x)
  … (14)
Where ( )
D 
is the (m+1) × (m+1) operational matrix of fractional derivative of order α in Caputo sense and is
defined as follows: (Note that in ( )
D 
, the first    rows are all zero).
,0,k ,1,k ,m,k
k k k
( )
i i i
i,0,k i,1,k i,m,k
k k k
m m m
m,0,k m,1,k m,m,k
k k k
0 0 0
0 0 0
D
            
            
               

               
               






  



   




  

  
  
  


  

  

  
















 
 

… (15)
Where i, j,k is given by
i k
i, j,k
j
1
( 1) 2i(i k 1)! (k 1 )
2
1
h (k )(i k)! (k j 1) (k j 1)
2


     
 
             

V. The Approach
In this section we will find an approximate solution for the one-dimensional space fractional diffusion
equation of the form:
u(x,t) u(x,t)
c(x) q(x,t),
t x


 
 
 
0 x   , 0 t   , 1 2,   … (16)
with initial condition
u(x,0) f(x), 0 x ,    … (17)
And boundary conditions
0u(0,t) g (t), 0 t ,    … (18)
1u( ,t) g (t), 0 t ,    … (19)
Based on Tau idea where the shifted Legendre polynomial in time and the shifted Chebyshev polynomial in
space are utilized respectively.
A function u(x,t) of two independent variables defined for 0<x<1, 0<t< may be expanded in terms of
the shifted Legendre and shifted Chebyshev polynomials as
n m
T
n,m ij i j n, m,
i 0 j 0
u (x,t) a L (t)T (x) (t)A (x)

 
    
 … (20)
Where the shifted Chebyshev vector m, (x)  is defined as eq. (13) and the shifted Legendre vector n, (t) is
defined as eq. (8).
The shifted Chebyshev and the shifted Legendre coefficients matrix A is given by
00 0m
n0 nm
a a
A
a a
 
 
  
 
 

 

Where
ij i j0 0
j
2i 1 2
a u(x,t)L (t)T (x)W (x)dtdx, i 0,1,...,n, j 0,1,...,m


  
        
 


 … (21)
The integration of n, (t) from 0 to t can be expressed as:
t
n, n,0
(t )dt P (t) 
    … (22)
Where P is an (n+1)×(n+1) operational matrix of integration given by
0 0
1 1
n 1 n 1
n
0
P
0
0
 
  
 
  
  
 
  
  
   … (23)
With k
1
2(2k 1)
 

To solve problem (16) – (19) we approximate f(x) by (m+1) terms of the shifted Legendre-shifted
Chebyshev polynomial.
m m
h T
i j 0 i j n, m,
j 0 j 0
f(x) f T (x) T (t). f T (x) (t)F (x) f(x;t),
 
     
  … (24)
Where F is known (n+1)×(m+1) matrix and can be shown by
0 1 m 1 mf f f f
0 0 0 0
F
0 0 0 0
 
 
 
 
 
 


   

… (25)
In addition, we approximate c(x), u(x,t) and q(x,t) by the shifted Legendre-shifted Chebyshev polynomials as
T
m m,c (x) C (x),   … (26)
T
n,m n, m,u (x,t) (t)A (x),    … (27)
T
n,m n, m,q (x,t) (t)Q (x),    … (28)

Where the vector T
0 1 mC [c ,c ,...,c ] and the matrix Q are known, but A is (n+1) × (m+1) unknown matrix. Now
integrating eq. (16) from 0 to t and using eq. (17), we can get
t t
0 0
u(x,t)
u(x,t) f(x) c(x) dt q(x,t)dt.
x



  
  … (29)
Using eqs. (14), (22), (26) and (27), then we obtain
 t t
T T ( ) T T T ( )
m, n, m, m, n, m,o 0
T T ( ) T
n, m, m,
u(x,t)
c(x) dt (C (x)) (t)dt A(D (x)) (C (x))( (t)P AD (x))
x
(t)P AD (x) (x)C.

 
 



      

   
    
 

…(30)
Let
T T
m, m, m,(x) (x)C H (x),     … (31)
Where H is a (m+1) × (m+1) matrix. In order to illustrate H, eq. (31) can be written as
m m
k k j kj k
k 0 k 0
c T (x)T (x) H T (x), j 0,1,...,m.
 
    
Multiplying both sides of the above equation by iT (x)W (x), i 0,1,...,m
 and integrating the result from 0 to 
, we obtain
m
k i j k ij i i0 0
k 0
c T (x)T (x)T (x)W (x)dx H T (x)T (x)W (x)dx, i, j 0,1,...,m.

   
 
    
  … (32)
Now, we suppose
i,j,k i j k0
w T (x)T (x)T (x)W (x)dx, 

  
 … (33)
By eqs. (12), (32) and (33), we have
m
ij k i,j,k
k 0i
2
H c w , i, j 0,1,...,m.

 
 
 … (34)
So eq. (30) can be written as
t
T T ( ) T
n, m,0
u(x,t)
c(x) dt (t)P AD H (x).
x




 
  … (35)
In addition, by eqs. (22) and (28), we get
t
T T
n,m n, m,0
q (x,t)dt (t)P Q (x)    … (36)
Finally, applying eqs. (24), (27), (35) and (36), the residual n,mR (x,t) for eq. (29) can be written as
T T ( ) T T T
n,m n, m, n, m,R (x,t) (t)(A F P AD H P Q) (x) (t)E (x),
          
Where
T ( ) T T
E A F P AD H P Q
   
According to the typical Tau method (see [18]), we generate (n+1)×(m+1) linear algebraic equations as follows
ijE 0, i 0,1,...,n, j 0,1,...,m 2.    … (37)
Also, substituting eq. (27) in eqs. (18) and (19), we obtain
T
n, m, 0(t)A (0) g (t),   … (38)
T
n, m, 1(t)A ( ) g (t),    … (39)
Respectively, eqs. (38) and (39) are collocated at (n+1) points. In addition, by comparing the coefficients eqs.
(38) and (39) then we can obtain (n+1)×(m+1) coefficients of A. Consequently, the n,mu (x,t) given in eq. (27)
can be calculated.
VI. Illustrative examples
To validate the effectiveness of the proposed method for problem (16) – (19) we consider the following
two examples. Our results are compared with the exact solution or with the previous works on the same
problem.
Example (1): Consider the following space fractional differential equation in
1.8
1.8
u(x,t) u(x,t)
c(x) q(x,t), 0 x 1, t 0,
t x
 
    
 
With the coefficient function

1.8
c(x) (1.2)x 
The source function
3 2 t
q(x,t) (6x 3x )e
 
The initial condition
2 3
u(x,0) x x , 0 x 1,   
And the boundary conditions
u(0,t) u(1,t) 0. 
The exact solution of this problem is given in [17] as:
2 3 t
u(x,t) (x x )e .
 
Following table (1) represent a comparison of the absolute errors between the proposed method and methods
[16] and [17] for u(x,2) of example (1).
Table (1) comparison of the absolute errors with methods [16] and [17] for u(x,2) of example (1).
x
n=m=3 n=m=5
Method[16] Method[17]
Proposed
method
Method[16] Method[17]
Proposed
method
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.1 2.89e-05 5.845e-06 6.379e-06 4.47e-06 1.261e-05 1.252e-05
0.2 1.09e-04 2.653e-05 7.168e-05 2.78e-07 1.634e-06 9.609e-07
0.3 2.20e-04 8.329e-05 1.732e-04 5.81e-06 1.425e-05 1.565e-05
0.4 3.40e-04 1.505e-04 2.884e-04 1.02e-05 2.513e-05 2.719e-05
0.5 4.45e-04 2.145e-04 3.946e-04 1.17e-05 2.774e-05 3.024e-05
0.6 5.15e-04 2.613e-04 4.693e-04 1.08e-05 2.332e-05 2.599e-05
0.7 5.27e-04 2.771e-04 4.898e-04 8.54e-06 1.553e-05 1.809e-05
0.8 4.60e-04 2.480e-04 4.335e-04 6.06e-06 8.317e-06 1.043e-05
0.9 2.91e-04 1.603e-04 2.777e-04 3.67e-06 3.758e-06 5.062e-06
1.0 0.0 0.0 0.0 0.0 0.0 0.0
Example (2): Consider the space fractional differential equation
1.5
1.5
u(x,t) u(x,t)
c(x) q(x,t), 0 x 1, t 0,
t x
 
    
 
With the coefficient function
0.5
c(x) (1.5)x , 
The source function
2
q(x,t) (x 1)cos(t 1) 2xsin(t 1),    
The initial condition
2
u(x,0) (x 1)sin(1), 
And the boundary conditions
u(0,t) sin(t 1), u(1,t) 2sin(t 1).   
The exact solution of this problem is given in [17] as:
2
u(x,t) (x 1)sin(t 1).  
Following table (2) represent a comparison between the solutions obtained by the proposed method for u(x,1)
and the exact solution of example (2).
Table (2) comparison between the solution obtained by the proposed method and the exact solution for
u(x,1) of example (2).
x
n=m=2
The Approach
n=m=3
The Approach
n=m=6
The Approach
n=m=7
The Approach
exact
0.1 0.1140 0.1341 0.1426 0.1426 0.1425
0.2 0.1217 0.1377 0.1468 0.1468 0.1468
0.3 0.1302 0.1441 0.1539 0.1538 0.1538
0.4 0.1397 0.1534 0.1637 0.1637 0.1637
0.5 0.1500 0.1654 0.1764 0.1764 0.1764
0.6 0.1611 0.1803 0.1919 0.1919 0.1919
0.7 0.1731 0.1978 0.2103 0.2103 0.2103
0.8 0.1859 0.2181 0.2314 0.2314 0.2314
0.9 0.1997 0.2411 0.2554 0.2554 0.2554

VII.Conclusion
In this paper we develop and analyze the numerical algorithm for the fractional diffusion equation.
Based on Tau method the shifted Legendre polynomial and the shifted Chebyshev polynomial are used to
reduce the problem to the solution of a system of linear algebraic equations. The solution obtained by this
method is in good agreement with the exact solution and other previous works that are used to solve fractional
diffusion equation. The effectiveness of the presented method is tested and confirmed through out the numerical
results.
References
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3334.
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A Tau Approach for Solving Fractional Diffusion Equations using Legendre-Chebyshev Polynomial Method

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