Applied Mathematics, 2011, 2, 1091-1095
doi:10.4236/am.2011.29150 Published Online September 2011 (http://www.SciRP.org/journal/am)
Copyright © 2011 SciRes. AM
Application of He’s Variational Iteration Method for the
Analytical Solution of Space Fractional Diffusion Equation
Mehdi Safari
Department of Mechanical Engineering, Islamic Azad University, Aligoodarz Branch, Aligoodarz, Iran
E-mail: ms_safari2005@yahoo.com
Received July 1, 2011; revised August 3, 2011; accepted August 11, 2011
Abstract
Spatially fractional order diffusion equations are generalizations of classical diffusion equations which are
increasingly used in modeling practical super diffusive problems in fluid flow, finance and others areas of
application. This paper presents the analytical solutions of the space fractional diffusion equations by varia-
tional iteration method (VIM). By using initial conditions, the explicit solutions of the equations have been
presented in the closed form. Two examples, the first one is one-dimensional and the second one is
two-dimensional fractional diffusion equation, are presented to show the application of the present tech-
niques. The present method performs extremely well in terms of efficiency and simplicity.
Keywords: He’s Variational Iteration Method, Fractional Derivative, Fractional Diffusion Equation
1. Introduction
Fractional diffusion equations are used to model prob-
lems in Physics [1-3], Finance [4-7], and Hydrology
[8-12]. Fractional space derivatives may be used to for-
mulate anomalous dispersion models, where a particle
plume spreads at a rate that is different than the classical
Brownian motion model. When a fractional derivative of
order 1 <
< 2 replaces the second derivative in a
diffusion or dispersion model, it leads to a super diffu-
sive flow model. Nowadays, fractional diffusion equa-
tion plays important roles in modeling anomalous diffu-
sion and subdiffusion systems, description of fractional
random walk , unification of diffu sion and w ave prop aga-
tion phenomenon, see, e.g. the reviews in [1-16], and
references therein. Consider a one-dimensional fractional
diffusion equation considered in [17]
(,) (,)
() (,)
uxt uxt
dx qxt
tx


(1)
on a finite domain
L
R
x
xx with 12
. We
assume that the diffusion coefficient (or diffusivity)
. We also assume an initial condition
for

0dx
(, 0uxt) ()sx
L
R
x
xx
(,
L
ux tand Dirichlet bound-
ary conditions of the form and
R
. Equation (1) uses a Riemann fractional
derivative of order
)0
(,)
R
uxt ()b t
.
Consider a two-dimensional fractional diffusion equa-
tion considered in [18]
(,,) (,,)
(,)
(,,)
(,)(,,)
uxytuxyt
dxy
tx
ux yt
exy qxyt
x


(2)
on a finite rectangular domain
L
H
x
xx and
L
R
yyy
, with fractional orders 1 < 2
and
1 < 2
, where the diffusion coefficients
, 0dxy
and
exy, 0. The “forcing” function
t, , qxy
can be used to represent sources and sinks. We will as-
sume that this fractional diffusion equation has a unique
and sufficiently smooth solution under the following
initial and boundary conditions. Assume the initial con-
dition u(x, y, t = 0) = f(x, y) for
L
H
x
xx and
L
R
yyy
, and Dirichlet boundary condition
, , yt, , uxytBx on the boundary (perimeter) of
the rectangular region
L
H
x
xx ,
L
R
y
(, ,)
LL
Bxyt
yy
(, ,)Bx yt, with
the additional restriction that .
In physical applications, this means that the left/lower
boundary is set far away enough from an evolving plume
that no significant concentrations reach that boundary.
The classical dispersion equation in two-dimensions is
given by
0
2
. The values of 1 < 2
and
1 < 2
model a super diffusive process in that coor-
dinate. Equation (2) also uses Riemann fractional deriva-
tives of order
and
. In this paper, we use the v ar ia-
tional iteration method (VIM) to obtain the solutions of
M. SAFARI
1092
the fractional diffusion Equations (1) and (2). The varia-
tional iteration method (VIM) established in (1999) by
He in [19-22] is thoroughly used by many researchers to
handle linear and non linear models. The reliability of the
method and the reduction in the size of computational
domain gave this method a wider applicability. The
method has been proved by many authors [23-26], and
the references therein, to be reliable and efficient for a
wide variety of scientific applications, linear and nonlin-
ear as well. The method gives rapidly convergent suc-
cessive approximations of the exact solution if such a
solution exists. For co ncrete problems, a few numbers of
approximations can be used for numerical purposes with
high degree of accuracy. The VIM does not require spe-
cific transformations or nonlinear terms as required by
some existing techniques. However, we use the VIM to
solve fractional diffusion Equations (1) and (2) and fi-
nally the results are illustrated in graphical figures.
2. Mathematical Aspects
The mathematical definition of fractional calculus has
been the subject of several different approaches [27,28].
The most frequently enco untered defin ition of an integral
of fractional order is the Riemann-Liouville integral, in
which the fractional order integral is defined as
1
0
d() 1()d
() ()
d(
qt
q
tqq
)
f
tf
Dft q
tt


tx
x
(3)
while the definition of fractional order derivative is
()
() 1
0
d d()1d()d
() ()
dd d()
nnq n
t
q
tnnqn nq
f
t
Dft nq
tt ttx

 




 ftx
n
(6)
where ( and ) is the order of the opera-
tion and n is an integer that satisfies .
q0qqR1
nq 
3. Basic Idea of He’s Variational Iteration
Method
To clarify the basic ideas of VIM, we consider the fol-
lowing differential equation:

LuNugt (5)
where is a linear operator, a nonlinear operator
and L

N
g
t an inhomogeneous term. According to VIM,
we can write down a correction functional as follows:
 
10d
t
nnn n
ututLu Nu g

 
(6)
where
is a general Lagrangian multiplier which can
be identified optimally via the variational theory. The
subscript indicates the nth approximation and
is considered as a restricted variation
nn
u
0
n
u
.
4. The Fractional Diffusion Equation Model
and Its Solution by VIM
Now we adopt variational iteration method for solving
Equation (1). In the light of this method we assume that
 
()
10() (,)d
tx
nn nn
ututudxu qxt
 
(7)
where ()
x
indicates a differential with respect to x
and dot denotes a differential with respect to t,
is
general Lagrangian multiplier.Similarly, fo r Equation (2)
using variational iteration method, we can obtain

1
() ()
0(, )(, )(, ,)d
nn
txy
nn n
utut
u dxyuexyuqxyt

 
(8)
5. Numerical Illustrations
5.1. Example 1
Let us consider a one-dimensional fractional diffusion
equation for the Equation (1), as taken in [17]
1.8
1.8
(,) (,)
() (,)
uxt uxt
dx qxt
tx


(9)
on a finite domain 01
x
, with the diffusion coeffi-
cient
2.8 2.8
( )(2.2)60.183634dx xx  (10)
the source/sink function 3
(,)(1) t
qxtxex
  (11)
the initial condition 3
(,0) ,ux x (12) for 01x
and the boundary conditions
(0,)0,(1, ),0
t
utut e fort
 (13)
Implementation of Variational Iteration Method for
Example 1
Now we consider the application of VIM to one-
dimensional fractional diffusion equation with the initial
condition of:
3
(,0) ,ux x (14) for 01x
Its correction variational functional in x and t can be
expressed, respectively, as follows:
(1.8 )
10
(,) (,)()(,)d
tx
nn nn
uxtu xtudxuqxt
 
(15)
where (1.8 )
x
indicates a differential with respect to x
and dot denotes a differential with respect to t,
is
general Lagrangian multiplier. After some calculations,
we obtain the following stationary conditions:
0

(16)
1
t

0
(17)
Copyright © 2011 SciRes. AM
M. SAFARI1093
Equation (16) is called Lagrange-Euler equation and
Equation (17) is natural boundary condition. The La-
grange multiplier can therefore, be identified as 1
and the variational iteration formula is obtained in the
form of:

(1.8 )
10
(,) (,)()(,)d
tx
nn nn
uxtuxtudxuqxt

(18)
We start with the initial approximation of
given by Equation (14). Using the above iteration for-
mula (18), we can directly obtain the other components
as follows:
(,0)ux
3
0(,) ,uxtx (19)
44
1(, )1.000001369(),
t
uxtxxt exx
 
34
(20)
(1.8 )
21 11
0
(,) (,)()(,)d
tx
uxtuxtudxuqxt

(21)
In Figure 1 we can see the 3-D result of approximate
solution of the one-dimensional fractional diffusion
equation by VIM.
5.2. Example 2
Let us consider a two-dimensional fractional diffusion
equation for the Equation (2), considered in [18].
1.8
1.8
1.6
1.6
(,,) (,,)
(, )
(,,)
+(,)(,,),
uxytuxyt
dxy
tx
ux yt
exy qxyt
y

(22)
on a finite rectangular domain 01
x
, , for
with the diffusion coefficients 01y
end
0Tt
Figure 1. For the one-dimensional fractional diffusion equa-
tion with the initial condition (12) of Equation (9), VIM
result for .
(,)uxt
2.8
(,)(2.2)/ 6dxyx y (23)
and
2.6
( ,)2/(4.6)exyx y (24)
and the forcing function
33.6
(,,)(1 2) t
qxytxyexy
  (25)
with the initial cond ition
33.6
(,,0)uxy xy (26)
and Dirichlet bo undary cond itions on the r ectangle in the
form and
, for all .
3
(,0,)(0,,)0,(,1, ),
t
uxtuytuxte x
 
3.6t
ey
0t(1,, )uyt
Implementation of Variational Iteration
Method for Example 2
Again we consider the application of VIM fractional
diffusion equation with the initial condition of:
33.6
(,0) ,ux xy (27) for 01, 0y1x 
Its correction variational functional in x and t can be
expressed, respectively, as follows:

1
(1.8 )(1.6)
0
(,,) (,,)
(, )(, )(, ,)d
nn
txy
nn n
uxytuxyt
u dxyuexyuqxyt
 
(28)
where (1.8 )
x
indicates a differential with respect to x,
indicates a differential with respect to y and dot
denotes a differential with respect to t, also
(1.6 )y
is gen-
eral Lagrangian multiplier. After some calculations, we
obtain the following stationary conditions:
0

(29)
1
t

0
(30)
Equation (29) is called Lagrange-Euler equation and
Equation (30) is natural boundary condition. The La-
grange multiplier can therefore, be identified as 1
and the variational iteration formula is obtained in the
form of:

1
(1.8 )(1.6)
0
(,,) (,,)
(, )(, )(,,)d
nn
txy
nn n
uxytuxyt
u dxyuexyuqxyt
 
(31)
We start with the initial approximation of
given by Equation (27). Using the above iteration for-
mula (31), we can directly obtain the other components
as follows:
(,0)ux
33.6
0(,,) ,uxyt xy (32)
23 23
44
55
1
18 23
34
55
(,,)22
2,
tt
u xytxyxyt
exy exy

 

(33)
Copyright © 2011 SciRes. AM
M. SAFARI
1094
Figure 2. For the two-dimensional fractional diffusion equa-
tion with the initial condition (26) of Equation (22), VIM
result for with . ,
()xtu1y
21
(1.8 )(1.6)
11 1
0
(,,) (,,)
(, )(, )(, ,)d
txy
uxyt uxyt
u dxyuexyuqxyt
 
(34)
In Figure 2 we can see the 3-D result of approximate
solution of the one-dimensional fractional diffusion equa-
tion by VIM.
6. Conclusions
In this paper, He’s variational iteration method has been
successfully applied to find the solution of space frac-
tional diffusion equation. All cases show that the results
of the VIM method are very good and the obtained solu-
tions are shown graphically. In our work, we use the
Maple Package to calculate the functions obtained from
the He’s variational iteration method.
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