A Study of Some Nonlinear Partial Differential Equations by Using Adomian Decomposition Method and Variational Iteration Method

In this paper, a numerical solution of nonlinear partial differential equation, Benjamin-Bona-Ma-hony (BBM) and Cahn-Hilliard equation is presented by using Adomain Decomposition Method (ADM) and Variational Iteration Method (VIM). The results reveal that the two methods are very effective, simple and very close to the exact solution.


Introduction
In this paper, we discuss the solution of the nonlinear BBM equation [1] , subject to the initial condition The Cahn-Hilliard equation [2]: subject to the initial condition ( )
x u x = + (4) [3] [4] derived a variety of exact travelling wave solutions of distinct physical structures for the BBM equation, where the Tanh and the sine-cosine methods were used.Also [5] is devoted to analyzing the physical structures of the nonlinear dispersive variants of the BBM equation, where new exact solutions with compact and noncompact structures for BBM are derived.[6] applied the decomposition method to obtain explicit and numerical solutions of different types of generalized BBM.Many articles have investigated Cahn-Hilliard equation analytically and numerically, [7] applied the finite difference method to obtain the numerical solution of Cahn-Hilliard equation.[8]- [10] used the Exp-function method to obtain exact solutions of Cahn-Hilliard.[11] solved these equations by Differential Transform Method.
In the beginning of the 1980, a so-called Adomiande composition method (ADM), which appeared in [12]- [15] used (ADM) to solve coupled kdv equation.

The Adomian Decomposition (ADM)
In this section, ADM is explained.For this, we consider a general nonlinear partial differential equation in the following form.
where ( ) , N u x t presents the nonlinear term and g is the source term.Appling the inverse operator ( ) to both sides of (5) and using the given conditions we obtain using the given conditions, the ADM defines the solution u by the series in the following form.

( )
N u presents by an infinite series of the so-called Adomian's , , 0 n u x t n ≥ are the components of ( ) , u x t that will be easy determined and n A are called Ado- mian's polynomials and defined by ( ) ( ) From the above considerations, the decomposition method defines the components ( ) Finally, the approximate solution for u(x, t) is obtained by truncating the series For more details about ADM and its convergence, see [12]- [14] [16] Now, we first consider a general form of nonlinear equation ( ) , , , , , , , 0.

Application of ADM
In this article, we investigate some example of the nonlinear partial differential equations

Benjamin-Bona-Mahony
subject to the initial condition with the exact solution is ( ) By using (15), Equations ( 18) and (19) converted to the ODE ( ) subject to the initial conditions ( ) ( ) Applying the ADM to (21)-( 22), we obtain ( ) we apply the inverse operator 1 L − on both sides (24) we get where the components of k A are the so-called Adomian polynomials, for each , k k A depends on ( ) ( ) ( ) The components Then approximation solution of Equation ( 18) is ( ) 0 1 2 , u x t u u u = + + + with third-order approximation.Now we compare exact solution with Adomain Decomposition Method (ADM) solution in Figure 1.where the exact solution is ( )

The Cahn-Hilliard Equation
By using (15), Equations ( 30) and (31) converted to the ODE subject to the initial conditions ( ) ( ) Applying the ADM to (33)-(34), we get ( ) we apply the inverse operator 1 L − on both sides (36) we have ( ) ( ) where the components of k A are the so-called Adomian polynomials, for each , k k A depends on The components ( ) Then approximation solution of Equation ( 30) is ( )

Variational Iteration Method (VIM) [17]
Let consider the differential equation ( ) where L and N are linear and nonlinear operators, respectively, and ( ) f t is the inhomogeneous term.In the