INTRODUCTION
Most scientific problems are inherently nonlinear as their functional
equations are nonlinear such as parabolic equations. The nonlinear parabolic
equations arise in various fields of mechanics, physics, statistics and
material science, for instance: transverse vibrations of uniform flexible
beams (Arshad et al., 2005), optimization of the trade-off between
noise removal and edge preservation that may minimize a cost functional
(You and Kaveh, 2000), the epitaxial growth of nanoscale thin films (Belinda
et al., 2003) and waves of the steady natural convection in a vertical
fluid layer (Tang and Christov, 2007). We know that most of these types
of equations do not have analytical solution as these are functioning
nonlinear. Therefore, those should be solved using numerical techniques,
although; perturbation methods had been used for the analytical solutions
of some especial cases before. In the numerical methods, stability and
convergence should be considered so as to avoid divergence or inappropriate
results. In the perturbation methods, we should exert the small parameter
in the equation. Therefore, finding the small parameter and exerting it
into the equation are difficulties of this method. Since there are some
limitations with the common perturbation method, also because the basis
of the common perturbation method is upon the existence of a small parameter,
developing this method for different applications is very difficult.
Indeed, many different methods have been recently introduced some ways
to eliminate the small parameter, such as the variational iteration method
(He, 1998, 1999, 2000; Khatami et al., 2008), the homotopy perturbation
method (He, 2003, 2005, 2006; Tolou et al., 2008) and the Exp-function
method (He and Wu, 2006; Mahmoudi et al., 2008). In this letter,
we will apply He`s homotopy perturbation method in order to drive the
explicit solution of fourth-order parabolic equations analytically. In
order to evaluate the benefits of proposed method, five illustrating examples
have been used. Wazwaz (2001, 2002) obtained exact solution of these equations
and (Biazar and Ghazvini, 2007) previously used variational iteration
method to solve these equations. This research is motivated to extend
these works by implementing homotopy perturbation method that shall be
more efficient. The results of homotopy perturbation method have been
compared and verified with those of exact solutions from the study of
Wazwaz (2001, 2002).
The fourth-order parabolic partial differential equation with variable
coefficients reads as following (Wazwaz, 2001, 2002):
where, μ (x, y, z), λ (x, y, z) and η (x, y, z) are positive.
Subject to the initial conditions (Wazwaz, 2002):
And the boundary conditions (Wazwaz, 2002):
where, the functions fi, gi, ki, hi,
,
i = 0.1 are continuous.
THE BASIC IDEA OF HPM
To illustrate the basic idea of this method, we 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. 2 can,
therefore, be rewritten as:
By the homotopy technique, we construct a homotopy as 
which
satisfies:
where, p ε [0, 1] is an embedding parameter and u0
is an initial approximation of Eq. 4 which satisfies
the boundary conditions. Obviously, considering Eq. 4
we will have:
The changing process of p from zero to unity is just that of v (r, p)
from u0 (r) to u (r). In topology, this is called deformation
and L(v)οL(u0) and A (v)οf (r) are called homotopy.
According to HPM, we can first use the embedding parameter p as small
parameter and assume that the solution of Eq. 5 can
be written as a power series in p:
Setting p = 1 results in the approximate solution of Eq.
5:
The combination of the perturbation method and the homotopy method is
called the HPM, which lacks the limitations of the traditional perturbation
methods, although this technique has full advantages of the traditional
perturbation techniques. The series (8) is convergent for most cases.
However, the convergence rate depends on the nonlinear operator A(v).
IMPLIMENTATION OF HPM
In order to illustrate the solution procedure and to show the capability
of the method, five examples of different kind of fourth-order non-linear
parabolic partial differential equations is presented here.
Example one: Consider the following one dimensional, variable
coefficient fourth-order parabolic partial differential equations (Wazwaz,
2002).
Subject to the initial conditions:
And the boundary conditions:
Substituting Eq. 7 and 9 into Eq.
5, after some simplification and substitution and rearranging based
on powers of p-terms we have:
Accuracy of solution shall give rise as n in Eq. 7
and power of p increasing. Solving Eq. 11 subject to
boundary conditions will result in:
So on substituting Eq. 12 into Eq. 7
gives the approximate solution in the following form:
Driving Eq. 7 for n>2 gives
While this is the same as the exact solution presented (Wazwaz, 2001,
2002).
Example two: Consider the following parabolic equation (Wazwaz,
2002).
Subject to the following initial conditions
And the boundary conditions of
As stated in example one, Increasing the n in Eq. 7,
give rise to the accuracy of the solution.
Substituting v(r) from Eq. 7 and 9
into Eq. 5, after some simplification and substitution
and rearranging based on powers of p-terms up to second orders of p, we
have:
In the same manner, the rest of component can be obtained in order to
obtain better approximation.
The solution of set of Eq. 16 gives:
Substituting the Eq. 17a-c into Eq.
7 gives the approximate solution for n = 2 as:
Therefore if we continue,
This is as the exact solution that has been obtained previously by Wazwaz
(2001, 2002).
Example three: Now we solve the following one dimensional non-homogeneous
fourth-order equation (Wazwaz, 2002).
In the same manner to previous examples by implementing HPM to Eq.
20 we have:
Thus,
And so,
Equation 23 is as the exact solution that has been
obtained before (Wazwaz, 2002).
Example four: Consider the fourth-order parabolic equation in
two space variables (Wazwaz, 2001).
And the initial conditions are:
Also, boundary conditions are:
Afterwards, implementing HPM to Eq. 24 result in:
Then, solving Eq. 25 we have:
So on substituting Eq. 26 into Eq. 7
gives the approximate solution in the following form:
Thus,
Equation 28 is the same as the result obtained previously
(Wazwaz, 2001, 2002).
Example five: Finally, we solve the following partial differential
equation in three space variables (Wazwaz, 2001).
Subject to the initial conditions:
And the boundary conditions:
In the same manner of previous examples, after some manipulation and
rearranging based on powers of p-terms we have:
Solving Eq. 29 subject to initially condition give:
So,
It is known that as 
as n → ∞; thus an exact solution is obtained which reads.
And this is the exact solution (Wazwaz, 2001, 2002).
CONCLUSION
In this study, for the first time a kind of analytical method called,
HPM has been successfully applied to find the solution of the parabolic
equations. This method has been used for solving five examples of parabolic
equations. The results show that this method provides excellent approximations
to the solution of this nonlinear systems with high accuracy. It is worth
pointing out that this method presents a rapid convergence for the solutions
with out the difficulties that have been arisen in traditional analytical
methods. As shown, the homotopy perturbation method doesn`t need a small
parameter. Finally, it has been attempted to show the capabilities and
wide-range applications of the homotopy perturbation method.
