On the Solution of Modified Fractional Diffusion Equations

On the Solution of Modified Fractional Diffusion Equations
BADR S. ALKAHTANI
Mathematics Department, College of Science, King Saud University, P.O.Box
1142, Riyadh 11989, Saudi Arabia
Email: alhaghog@gmail.com
ADEM KILICMAN
Department of Mathematics and Institute for Mathematical Research,
Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia,
Email: akilic@upm.edu.my
Abstract
In this paper, we applied the Homotopy Analysis method (HAM) to obtain the analytical
solutions of the general space-time fractional diffusion equation. The explicit solutions of the
equations have been presented in the closed form by using initial conditions. Further
examples are also discussed to explain this method.
MSC 2010: 26A33, 34A08, 34K37, 35R11
Keywords: Homotopy Analysis Method , Analytical solution, Fractional diffusion equation,
Jumarie‟s fractional derivative.
1. Introduction
In recent years, fractional derivatives are fruitfully applied in many mathematical
problems occur in the area of mathematical physics, chemistry, engineering etc. The solutions
of fractional differential equations have great importance in the field of science and
engineering. Many nonlinear and linear equations are appeared in various fields. There is no
analytical technique or method to solve all differential equations. Only approximate solutions
can be derived using the linearization or perturbation methods (see [1]-[6]). The Homotopy
analysis method provides a powerful approach to find analytical approximate solution to
linear and nonlinear problems, and it is powerful tool for scientists, engineers, and applied
mathematicians, because it provides immediate analytic and numerical solution for both
linear and nonlinear fractional differential equations without linearization or discretization.
2. Jumarie’s Fractional Derivative
Recently, a new modified Riemann-Liouville left derivative is proposed by G. Jumarie [78] (see also [9] and [10]). Comparing with the classical Caputo derivative, the definition of
the fractional derivative is not required to satisfy higher integer-order derivative than  .v
Secondly  th derivative of a constant is zero. For these merits, Jumarie modified derivative
successfully applied in the probability calculus [9], fractional Laplace problem [10].
According
to
the
definition
of
Jumarie‟s
Fractional
Derivative,
define
f : R  R , x  f ( x) denotes a continuous (but not necessarily differentiable) function and let
the partition h  0 in the interval [0, 1]. Through the fractional Riemann Liouville integral

0 I x f ( x) 
x
1
( x   ) 1 f ( )d , 0    1
 0
)1)
The Modified Fractional Derivative is defined as

0 D x f ( x) 
x
1
dn
( x   ) n  ( f ( )  f (0)) d ,
n 
(n   ) dx 0
(2)
where x  [0,1], n  1    n and n  1.
G. Jumarie [8] (seel also in [9] and [10]) has defined the fractional difference in following
ways

 
  ( FW  1) f ( x)   (1) k   f [ x  (  k )h],
0
k
(3)
where FWf ( x)  f ( x  h) . Then the fractional derivative is defined as the following limit,
( )
f
 lim h0
 f ( x)
.
h
(4)
The proposed modified Riemann–Liouville derivative as shown in Eq. (2) is strictly
equivalent to Eq. (4). Further, Jumarie has introduced various properties of the modified
Riemann –Liouville derivative as given below (see [9] for further properties).
(a) Fractional Leibniz product law
0
Dx (uv)  u ( ) v  uv ( ) ,
(5)
(b) Fractional Leibniz formulation
0
I x 0 Dx f ( x)  f ( x)  f (0), 0   1,
(6)
Therefore, the integration by part can be used during the fractional calculus
a
I b u ( ) v  (uv) / ba  a I b uv ( ) ,
(c) Integration with respect to (d )  .
(7)
Assume f (x) denotes a continuous R  R function, we use the following equality for
the integral with respect to (d ) 
x
1
( x   )  1 f ( ) d , 0    1
0 I x f ( x) 

 0

(8)
x
1

f ( )(d ) .

(1   ) 0
3. Space-Time Fractional Differential Equation
If we consider the following modified one-dimensional non-homogenous space-time
fractional diffusion equation considered which is a modification of Fractional Diffusion
Equations used in [14]
u ( x, t )
 d ( x) D2 u ( x, t )  q( x, t ),
t
(9)
on a finite domain x L  x  x R with 0    1 . Note that D 2 is Jumarie‟s Fractional
Derivatives of order 2 . It is assume that the diffusion coefficient (or diffusivity) d ( x)  0
and initial condition u( x, t  0)  s( x) for x L  x  x R along with Dirichlet boundary
conditions of the form u( x L , t )  0 and u( x R , t )  bR (t ).
Next, we consider two-dimensional space-time fractional diffusion equation
u ( x, y, t )
 d ( x, y) D2 u ( x, y, t )  e( x, t ) D2  u( x, y, t )  q( x, y, t ),
t
(10)
on finite rectangular domain x L  x  x H and y L  y  y H , where fractional orders 0    1
and 0    1 , and the diffusion coefficients d ( x)  0 and e( x, y)  0 . The „forcing‟ function
q( x, y, t ) can be used to represent sources and sinks. We will assume that the fractional
diffusion equation has a unique and sufficiently smooth solution under the following initial
and boundary conditions. Assume the initial condition
u( x, y, t  0)  f ( x, y)
for
x L  x  x H , y L  y  y H and Dirichlet boundary condition u( x, y, t )  B( x, y, t ) on the
boundary (perimeter) of the rectangular region x L  x  x H , y L  y  y H , with the additional
restriction that B( x L , y, t )  B( x, y L , t )  0 . The classical dispersion equation in twodimensions is given by  and   1 . The values of 0    1 , or 0    1 model a super
diffusive process in that coordinate Eq. (10) also uses Jumarie‟s fractional derivatives of order
 and  .
4. Homotopy Analysis Method (HAM)
For the following nonlinear differential equation
FD u  x, y, t   0,
(11)
where FD is a nonlinear operator for this problem, x,y and t denote an independent
variables, u  x, y, t  is unknown function. To apply Homotopy Analysis Method (HAM),
we need to construct the following deformation:
1  q  L U  x, y, t; q   u0  x, y, t   q
H  x, y, t  FD U  x, y, t ; q   ,
(12)
where q  0, 1 is the embedding parameter,   0 is an auxiliary parameter, H  x, y, t   0
is an auxiliary function, L is an auxiliary linear operator, u0  x, y, t  is an initial guess of
u  x, y, t  and U  x, y, t; q  is an unknown function of the independent variables x, t and q.
Obviously, when q  0 and q  1, it holds
U  x, y, t;0   u0  x, y, t  , U  x, y, t;1  u  x, y, t  ,
(13)
Respectively. Using the parameter q, we expand U x, t; q  in Taylor series as follows:

U  x, y, t; q   u0  x, y, t    um  x, y, t  q m ,
(14)
m 1
where
m
1  U  x, y , t ; q 
um 
q  0,
m!
mq
Assume that the auxiliary linear operator, the initial guess, the auxiliary parameter  and the
auxiliary function H x, t  are selected such that the series (12) is convergent at q  1 , then
due to (12) we have

u x, t   u 0 x, t    u m x, t .
(15)
m 1
Let us define the vector

u n x, t   u0 x, t , u1 x, t ,..., u n x, t ,
Differentiating equations (10) m times with respect to the embedding parameter q, then setting
q  0 and finally dividing them by m!, we have the so-called mth-order deformation equation

Lu m x, t    m u m1 x, t   H x, t Rm u m1 ,
(16)
where
1 
Rm  um1  
 m  1!
and
m 1
FD U  x, t ; q  
 m1q
q0
,
0 m  1,
1 m  1.
m  
Finally, for the purpose of computation, we will approximate the HAM solution of Eq. (9) by
the following truncated series:
m 1
m  x, t    uk  x, t  .
k 0
The Homotopy Analysis Method
(HAM) contains the auxiliary parameter . which
provides us with a simple way to adjust and control the convergence region of solution series
for large value of t ([11, 12], see also [12]) .
In this paper, we will solve space-time diffusion equation by Homotopy Analysis Method .
The derivatives are understood in the Jumarie‟s Fractional Derivative sense. By the present
method, numerical results can be obtained with using a few iterations. Now we will obtain the
solution of Space-Time (One and Two Dimensional) Fractional Differential Equations.
5. Numerical Applications
In this section, we apply the proposed algorithm of Homotopy Analysis Method
(HAM) using Jumarie‟s approach for fractional order diffusion equation:
Example 5.1 We consider a one-dimensional fractional diffusion equation for the Eq. (1), as
taken [14,16]
u ( x, t )
1.8u ( x, t )
 d ( x)
 q( x, t ),
t
x1.8
(17)
on a finite domain 0  x  1 , with the diffusion coefficient
d ( x)  (2.2) x 2.8 6  0.183634 x 2.8 ,
(18)
the source/sink function
q( x, t )  (1  x)e t x 3 , 0  x  1,
(19)
with the initial conditions
u( x, 0)  x 3 ,
and the boundary conditions
u(0, t )  0, u(1, t )  e t , t  0.
(20)
According to Eq. (12), the zeroth-order deformation can be given by

1  q LU x, t; q   u0 x, t   qH x, t  u( x, t )  d ( x) 

We choose the auxiliary linear operator
t

u ( x, t )
 q( x, t ) .
1.8
x

1.8
(21)
LUx, t; q   D t Ux, t; q ,
with the property LC   0,
where C is an integral constant. We also choose the auxiliary function to be Hx, t   1.
Hence, the mth-order deformation can be given by

Lum x, t   mum 1 x, t   H x, t Rm um 1 ,
where
u m 1 ( x, t )
1.8 u m 1 ( x, t )

Rm u m1  
 d ( x)
 q( x, t ).
t
x1.8
Now the solution of the mth-order deformation equation (14) for m  1 become

u m x, t    m u m1 x, t   L1 Rm u m1 .
(22)
(23)
Consequently, the first few terms of the HAM series solution for   1 are as follows:
u0  e t x 3  e t x 4  x 4 ,
4( e  t  1  t ) x 5
u1 x, t   ( e  1) x 
,
2.2
t
u 2  x, t  
t
4
4(e  1  t ) x

2.2
5
t
 1  t)x5
2!
,
3.2  2.2 2
80(e t 

It obvious that the “noise” terms appear between the components u0 and u1 , and it is
cancelled. The closed form solution is
u( x, t )  et x3.
Fig. 5.1 The surface shows the solution u ( x, t ) for Eq (17). .
Example 5.2 Now, we consider a two-dimensional fractional diffusion equation for the Eq.
(2), considered in [15,16]
u ( x, y, t )
1.8u ( x, y, t )
1.6u ( x, y, t )
 d ( x, y)

e
(
x
,
y)
 q( x, y, t ),
t
x1.8
y1.6
(24)
on a finite rectangular domain 0  x  1 , 0  y  1 , for 0  t  Tend with the diffusion
coefficients
d ( x, y)  (2.2) x 2.8 y 6,
(25)
e( x, y)  2 xy 2.6 (4.6) ,
(26)
and the forcing function
q( x, y, t )  (1  2 xy )e t x 3 y 3.6 ,
(27)
with the initial condition
u( x, y,0)  x 3 y 3.6 ,
(28)
and Dirichlet boundary conditions on the rectangle in the form
u( x,0, t )  u(0, y, t )  0, u( x,1, t )  e t x 3 ,
(29)
u(1, y, t )  e t y 3.6 ,
(30)
and
for all t  0.
According to Eq. (12), the zeroth-order deformation can be given by
1  q  L U  x, y, t; q   u0  x, y, t  
 u ( x, y, t )

1.8u ( x, y, t )
1.6u ( x, y, t )
 q H  x, y , t  
 d ( x, y )

e
(
x
,
y
)
 q ( x, y , t )  ,
1.8
1.6
t
x
y


(31)
We choose the auxiliary linear operator
L U  x, y, t; q    Dt U  x, y, t; q  ,
with the property LC   0,
where C is an integral constant. We also choose the auxiliary function to be Hx, t   1.
Hence, the mth-order deformation can be given by
L um  x, y, t   mum1  x, y, t   H  x, y, t  Rm  um1  ,
where
Rm  um1  
um 1 ( x, y, t )
t
 d ( x, y)
1.8um 1 ( x, y, t )
x1.8
 e( x, t )
1.6um 1 ( x, y, t )
y1.6
 q( x, y, t )
(32)
Now the solution of the mth-order deformation Eq. (14) for m  1 become
um  x, y, t   mum1  x, y, t   L1  Rm  um1  .
(33)
Consequently, the first few terms of the HAM series solution for   1 are as follows
u0  x 3 y 3.6 e t  2 x 4 y 4.6 e t  2 x 4 y 4.6 ,
2  4.6  5 5.6
 8
t
u1 x, t   x 4 y 4.6 ( e t  1)  

 x y ( e  1  t ),
2
.
2
3


u 2  x, t  
1106 5 5.6 t
9101827 6 6.6 t
t2
x y (e  1  t ) 
x y (e  1  t  ,
165
272250
2! ,

It obvious that the “noise” terms appear between the components u0 and u1 , and it is
cancelled. The closed form solution is
u( x, y, t )  x 3 y 3.6 e t .
Fig. 5.2 The surface shows the solution u ( x, t ) for Eq. (24).
6. Conclusions
In this paper, the application of Homotopy Analysis Method (HAM) was extended to
obtain explicit and numerical solutions of linear and non-homogeneous space-time fraction
diffusion equations. The obtained results demonstrate the reliability of the algorithm and its
wider applicability to fractional differential equations. The advantage of HAM is the auxiliary
parameter which provides a convenient way of controlling the convergence region of series
solutions. Also it is clear that the solutions agree with the exact solutions appear in [14,16].
7. Acknowledgements
This project was supported by Deanship of Scientific Research, College of Science Research
Center, King Saud University. Authors are sincerely thankful to referee for his constructive
comments to improve this paper.
References
[1]
S. Momani, An explicit and numerical solutions of the fractional KdV equation,
Math. Comp. Simul. 70 (2), (2005), 110-118.
[2]
S. Momani and R. Ibrahim, Analytical solutions of a fractional oscillator by the
decomposition method, Int. J. Pure Appl. Math. 37 (1), (2007), 119-132.
[3]
S. Momani, Analytic and approximate solutions of the space- and time-fractional
telegraph equations, Appl. Math. Comput. 170 (2), (2005), 1126-1134.
[4]
S. Das, Solution of Fractional Vibration Equation by the Variational Iteration Method
and Modified Decomposition Method, Int. J. Nonl. Sci. Numer. Simul. 9 (2008) 361365.
[5]
S. Momani and Z. Odibat, A. Alawneh, Variational iteration method for solving the
space-fractional and time- fractional KdV equation, Num. Meth. For Part. Diff. Equ.
2007.
[6]
Z. Odibat, S. Momani and A. Alawneh, Analytic study on time-fractional Schrödinger
equations: Exact solutions by GDTM, J. Phy.: Conference Series 96 (2008), 012066.
[7]
G. Jumarie, Stochastic differential equations with fractional Brownian motion input,
Int. J. Syst. Sci. 6 (1993), 1113-1132.
[8]
G. Jumarie, New stochastic fractional models for Malthusian growth, the Poissonian
birth process and optimal management of populations, Math. Comput. Model. 44
(2006), 231-254.
[9]
G.Jumarie, Table of some basic fractional calculus formulae derived from a modified
Riemann-Liouville derivative for non-differentiable functions, Appl. Math. Lett. 22
(2009).
[10]
G. Jumarie, Laplace transform of fractional order via the Mittag–Leffler function and
modified Riemann- Liouville derivative, Appl. Math. Lett. 22 (2009), 1659-1664.
[11]
S. J. Liao, Beyond perturbation: Introduction to the Homotopy Analysis Method ,
Chapman and Hall/CRC Press, Boca Raton, 2003.
[12]
S. J. Liao, on the analytic solution of magneto hydro dynamic flows of non-Newtonian
fluids over a stretching sheet. J. Fluid Mech. 488 (2003), 189–212.
[13]
S.J. Liao, Notes on the Homotopy Analysis Method : some definitions and theorems
Commun. Nonlinear Sci. Numer. Simul. 14 (2009), 983–997.
[14]
M. M. Meerschaert, H. Scheffler and C. Tadjeran, Finite difference methods for twodimensional fractional dispersion equation, J.Comput. Phys. 211 (2006), 249–261.
[15]
C. Tadjeran and M. M. Meerschaert and H. Scheffler, A second-order accurate
numerical approximation for the fractional diffusion equation, J. Comput. Phys. 213
(2006), 205-213.
[16]
S. S. Ray, K.S. Chaudhuri and R. K. Bera, Application of modified decomposition
method for the analytical solution of space-time diffusion equation, Appl. Math.
Comput. 196 (2008), 294–302.