INTRODUCTION
One of the most frequent problems in the physical sciences is to obtain
the time solution of a (linear or nonlinear) partial differential equation
which satisfies a set of boundary conditions on a rectangular boundary.
For instance, let us consider the following problem (GarcíaOlivares,
2002a):
With the following boundary conditions defined on a rectangle (GarcíaOlivares,
2002a):
For instance, if have R = 1 and f (x, y) = 0 in Eq. 1
subject to the following initial and boundary conditions (GarcíaOlivares,
2002a):
This kind of Partial Differential Equations (PDE) appears frequently
coupled with others. For example, in the incompressible fluid flow problem,
the equation above would be slightly completed to become the first component
of the NavierStokes equations which should be solved in parallel with
a poisson equation for the pressure.
Some methods to obtain an analytical solution of PDE with boundary conditions
by means of power series have been explored in GarciaOlivares (2002b)
and Mahmodi et al. (2008). Those works are based on the method
proposed by George Adomian called, decomposition method (Adomian, 1998)
that uses analytic functions to approximate the problem solution.
In order to develop these efforts, He`s variational iteration method and homotopy
perturbation method (He, 2000, 2004; Abdou and Soliman, 2005; Ganji and Rafei,
2006; Tolou et al., 2008) also, homotopy analysis method (Khatami et
al., 2008) have been used to conduct an analytical investigation on the
solution of timedependent partial differential equations. In order to assess
benefits of the methods, firstly, fundamentals of the proposed method have been
presented and some illustrating examples have been used. Afterwards, the results
obtained by aforementioned methods have been shown and compared graphically.
Finally, conclude with some discussion.
MATERIALS AND METHODS
Variational Iteration Method (VIM)
Fundamentals: To illustrate the basic concepts of variational
iteration method, consider the following deferential:
where, L is a linear operator, N a nonlinear operator and g (x)
a heterogeneous term. According to VIM, can construct a correction functional
as follows:
where, λ is a general Lagrangian multiplier (He, 1998a, 2005), which
can be identified, optimally via the variational theory (He, 1998b), the
subscript n indicates the nth order approximation, ũ_{n}
which is considered as a restricted variation, i.e., δ ũ_{n}
= 0.
Application: Considering timedependent partial differential equations
as (GarcíaOlivares, 2002a):
Subject to the following initial condition (GarcíaOlivares, 2002a):
Solve Eq. 13 and 14 using VIM, have
the correction functional as:
where, indicates
the restricted variations; i.e.,
Making the above correction functional stationary, obtain the following
stationary conditions:
The Lagrangian multiplier can therefore be identified as:
Substituting Eq. 19 into the correction functional
equation system (16) results in the following iteration formula:
Each result obtained from Eq. (20) is u (x, y, t) with
its own error relative to the exact solution, but higher number iterations
leads to better approximation, even to the exact solution. Using the iteration
formula (20) and the initial condition as u_{0}, two iterations
were made as follows:
The first iteration results in:
The second iteration results in:
In the same manner the rest of the component of the iteration formula
can be obtained.
Homotopy Perturbation Method (HPM)
Fundamentals: To clarify the basic ideas of HPM, consider the
following nonlinear differential equation:
Considering the boundary conditions of:
where, A is a general differential operator, B a boundary operator, f(r)
a known analytical function and Γ is the boundary of the domain Ω.
The operator A can be divided into two parts of L and N, where L is the
linear part, while N is a nonlinear one. Eq. 23 can,
therefore, be rewritten as:
By the homotopy technique, construct a homotopy as which satisfies:
where, p ε [0, 1] is an embedding parameter and u_{0}
is an initial approximation of Eq. 26 which satisfies
the boundary conditions. Obviously, considering Eq. 26
will have:
The changing process of P from zero to unity is just that of v (r, p)
from u_{0} (r) to u (r). In topology, this is called deformation
and L(v)L(u_{0}) and A(v)f(r) are called homotopy. According
to HPM, can first use the embedding parameter P as small parameter and
assume that the solution of Eq. 26 can be written as
a power series in p:
Setting p = 1 result in the approximate solution of Eq.
26:
The combination of the perturbation method and the homotopy method is
called the HPM, which eliminates the limitations of the traditional perturbation
methods while it has full advantages of the traditional perturbation techniques.
The series (29) is convergent for most cases. However, the convergence
rate depends on the nonlinear operator A (v). The following opinions are
suggested by He (2004):
• 
The second derivative of N (v) with respect to v must
be small because the parameter p may be relatively large, i.e., p→1. 
• 
The norm of L^{1} ∂N / ∂v must be smaller than
one so that the series converges. 
Application: With the same first example as mentioned previously,
the equation is as:
With the initial condition of:
Substituting Eq. 29 into 26 and then
substituting v from Eq. 28 and rearranging based on
power series of P, have an equation system including n+1 equations to
be simultaneously solved; n is the order of P in Eq. 28.
Assuming 3, the system is as follows Eq. 32.
Now try to obtain a solution for equation system (32), in the form of
(33):
Finally, u (x, y, t) = u_{0} (x, y, t)+u_{1} (x, y, t)+u_{2}
(x, y, t)+ u_{3}(x, y, t)
Homotopyz Analysis Method (HAM)
Fundamentals: Consider the following differential equation:
where, N is a nonlinear operator, τ denotes an independent variable,
u(τ) is an unknown function. For simplicity, ignore all boundary
or initial conditions, which can be treated in the similar way. By means
of generalizing the traditional homotopy method, Liao (2003) constructed
the socalled zeroorder deformation equation as:
where, p ε [0, 1] is the embedding parameter, ħ ≠ 0 a nonzero
auxiliary parameter, H (τ)≠ 0 an auxiliary function, L an auxiliary
linear operator, u_{0}(τ) an initial guess of u (τ)
and φ(τ; p) is an unknown function. It is important to have
enough freedom to choose auxiliary unknowns in HAM. Obviously, when p
= 0 and p = 1, it holds: φ(τ; 0) = u_{0}(τ) and
φ(τ; 1) = u(τ).
Thus, as p increases from 0 to 1, the solution φ(τ; p) varies
from the initial guess, u_{0}(τ) to the solution u(τ).
Expanding φ(τ; p) in Taylor series with respect to p, have:
Where:
If the auxiliary linear operator, the initial guess, the auxiliary parameter
ħ and the auxiliary function are quite properly chosen, the series (37)
converges at p = 1 then have:
This must be one of the solutions of the original nonlinear equation,
as proved by Liao (2003). As ħ = 1 and H(τ) = 1, Eq.
35 becomes:
This is mostly used in HPM, whereas the solution can be obtained directly
without using Taylor series. According to the Eq. 35,
the governing equation can be deduced from the zeroorder deformation
Eq. 40. The vector is defined as:
Differentiating Eq. 35 for m times with respect to
the embedding parameter p and then setting p = 0 and finally dividing
them by m!, will have the socalled mthorder deformation equation as:
Where:
and
It should be emphasized that u_{m}(τ) for m≥1 is governed
by the linear Eq. 41 with the linear boundary conditions
coming from the original problem, which can be easily solved using symbolic
computation software.
Application: Consider Eq. 30, 31
and let us solve them through HAM with proper assignment of H(τ)
= 1 subject to the initial condition and assuming m = 2.
Finally, u (x, y, t) = u_{0} (x, y, t)+u_{1} (x, y,
t)+u_{2} (x, y, t).
RESULTS AND DISCUTION
In this study, new kind of analytical methods, VIM, HPM, HAM, have been
used in order to obtain the solution of timedependent nonlinear partial
differential equations. Figure 1ac
shows the behavior of u(x, y, t) versus x and y from VIM, HPM HAM respectively
for t = 0.001. This figure clearly shows the well agreement between results
of these methods. For further verification, the crosssection of u(x,
y, t) is shown in Fig. 2a ,b fort
= 0.001 and t = 0.002 while y assumed to be constant at value o f zero.
Figure 3 a, b shows the crosssection
of u(x, y, t) for t = 0.001 and t = 0.003 while the constant value of
x is zero. Figure 1 as well as Fig. 2
and 3 are obtained for R = 1, y_{l} = 0.4, x_{l}
= 0.4. Figure 2, 3 approve once more
an excellent agreement between methods.

Fig. 1: 
The behavior of u (x, y, t) versus x and y
evaluates by VIM (a), HPM (b), HAM (c) at t = 0.001: R = 1, y_{1}
= 0.4, x_{l} = 0.4 

Fig. 2: 
Crosssection of u(x, y, t) at t = 0.001 (a), t = 0.002
(b), y = 0: R = 1, y_{l} = 0.4, x_{l} = 0.4 

Fig. 3: 
Crosssection of u(x, y, t) at t = 0.001 (a), t =
0.003 (b), x = 0: R = 1, y_{l} = 0.4, x_{l} = 0.4 
CONCLUSION
In this survey, VIM, HPM and VIM have been successfully applied to obtain
the analytical solution of nonlinear timedepended partial differential
equations. As a clear conclusion, these methods provide successive rapidly
convergent approximations without any restrictive assumptions or transformations
causing changes in the physical properties of the problem. Also adding
up the number of iterations leads to the explicit solution for the problem.
Moreover, the VIM, HPM and HAM do not require small parameters in the
equation so that it overcomes the limitations that have arisen in traditional
perturbation methods. The approximations are valid not only for small
parameters but also for larger ones. The VIM, HPM and HAM are all efficient
and powerful mathematical method to overcome this kind of problem and
can be appropriate substitutions for each other. However, since the HPM
has got shorter equations, the related results converge more rapidly.
VIM including internalization calculations, takes a longer time and more
difficulty arising in calculations. HAM is a new method that can be use
for wide rang of nonlinear equation and the auxiliary parameter, h, provides
a convenient way to adjust and control convergence region and rate of
solution series so, it may leads to obtain the solution for fewer approximations.
Moreover the solution of given nonlinear problem can be expressed by many
base function and thus, can be more efficiently approximated by letter
set of base function. All the aforementioned methods give rapidly convergent
successive approximations of the exact solution if such a solution exists,
otherwise approximations can be used for numerical purposes.