A Comparative Study of Adomain Decompostion Method and He-Laplace Method

In this paper, we present a comparative study between the He-Laplace and Adomain decomposition method. The study outlines the significant features of two methods. We use the two methods to solve the nonlinear Ordinary and Partial differential equations. Laplace transformation with the homotopy method is called He-Laplace method. A comparison is made among Adomain decomposition method and He-Laplace. It is shown that, in He-Laplace method, the nonlinear terms of differential equation can be easy handled by the use He’s polynomials and provides better results.


Introduction
This paper outlines a reliable Comparison between two powerful methods that were recently developed.The first is Adomain decomposition method (ADM) developed by Adomain in [1] [2], and used heavily in the literature in [3]- [10] and the references therein.The second is He-Laplace method, an elegant combination of the Laplace transformation, the homotopy perturbation method and He's polynomials.The use of He's polynomial in nonlinear term was first introduced by Ghorbani [11].The proposed algorithm provides the solution in a rapid convergent series which may lead to the solution in a closed form.The two methods give rapidly convergent series with specific significant features for each scheme.Some of the classical analytic methods are lyapunov's ar-tificial small parameter method [12] perturbation techniques [13] [14] and Hiroa bilinear method [15] [16].In recent years, many authors have paid attention to study the solution of nonlinear partial differential equation by using various methods.Variational iteration method, He's semi inverse method [17] and the differential transform method, etc. are among these.The main objective is to introduce a comparative study to nonlinear ordinary differential and partial differential equations by using adomain decomposition method and He-Laplace method.
This paper contains basic idea of homotopy pertaturbation method and He-Laplace method in Section 2, Adomain decomposition method in 3, Application in 4 and conclusion and discussions in 5 respectively.

Homotopy Perturbation Method
Consider the following nonlinear differential equation with boundary conditions of , 0, y B y r n where A, B, ( ) f r and Γ are a general differential operator, a boundary operator, a known analytic function and the boundary of the domain Ω , respectively.
The operator A can generally be divided into a linear part L and a nonlinear part M. Equation ( 1) may therefore be written as: By the homotopy technique, we construct a homotopy ( ) [ ] which satisfies: or where is an embedding parameter, while 0 y is an initial approximation of Equation ( 1), which sa- tisfies the boundary conditions.Obviously, from Equatons (4) and ( 5), we will have: The changing process of  from zero to unity is just that of ( ) are called homotopy.If the embedding parameter pis considered as a small parameter, applying the classical perturbation technique, we can assume that the solution of Equations ( 4) and ( 5) can be written as a power series in : p Setting 1 p = in Equations (8), we have The combination of the perturbation method and the homotopy method is called the HPM, which eliminates the drawbacks of the traditional perturbation methods while keeping all its advantages.The series (9) is convergent for most cases.However, the convergent rate depends on the nonlinear operator ( ) A v .Moreover, He [18] made the following suggestions: 1) The second derivative of ( ) M v with respect to must be small because the parameter may be relatively large, i.e.

p →
2) The norm of must be smaller than one so that the series converges.

He-Laplace Method
Consider the following nonlinear differential equation (IVP): ( ) ( ) ( ) ( ) where 1 2 3 , , , , p p p α β are constant.( ) f y is a nonlinear function and ( ) f x is the source term.Taking Laplace transformation (denoted throughout this paper by L ) on both side of Equation ( 10), we have By using linearity of Laplace transformation, the result is Applying the formula on Laplace transform, we obtain Using initial conditions in Equation ( 14), we have Or Taking inverse Laplace transform, we have where ( ) F x represents the term arising from the source term and the prescribed initial conditions.Now, we apply homotopy perturbation method [12], ( ) ( ) where the term n y are to recursively calculated and the nonlinear term ( ) for some He's polynomial n H (see [11] [19]) that are given by ( ) ( ) Substituting Equations ( 18) and ( 19) in (17), we get which is the coupling of the Laplace transformation and the homotopy perturbation method using He's polynomials.Comparing the coefficient of like powers of , p the following approximations are obtained:

A Domain Decomposition Method
A domain decomposition method [3] [4] define the unknown function ( ) where the components are usually determined recurrently.The nonlinear operator ( ) F u can be decomposed into an infinite series of polynomials given by ( ) where n A are the so-called Adomain polynomial of 0 1 2 , , , n u u u u  defined by ( ) or equivalently It is now well known that these polynomials can be generated for all classes of nonlinear according to specific algorithms defined by (24).Recently, an alternative algorithm for constructing Adomain polynomials has been developed by Wazwaz [6].
This powerful technique handles both linear and nonlinear equations in unified manner without any need for the so-called Adomain polynomials.However, Adomin decomposition method provides the component of the exact solution, where these components should follow the summation given in ( 22), whereas ADM requires the evaluation of the Adomain polynomials that mostly require tedious algebraic work.

Example 1
Consider the following nonlinear PDE [20]: with the following conditions: By applying the Laplace transform to both sides of Equation ( 24) subject to the initial condition, we have The inverse of the Laplace transform implies that ( ) Now, we apply the homotopy perturbation method, we have ( ) ( ) Comparing the coefficient of like powers of p, we have ( ) , 30 , 0

B. A. A. Adam
So that the solution ( ) , u x y is given by ( ) , 0 0 30 30 which is the exact solution of the problem.

Adomain Decomposition Method
We first rewrite Equation (26) in an operator L is .
The inverse This can be rewrite at the form In view of (39), the following recursive relation According to Adomain [19], and approximate solution can be obtained [12].

Example 2
Consider the following non-homogeneous nonlinear PDE [20]: with the following condition: ( )

Using He-Laplace Method
By applying the Laplace transform method subject to the initial condition, we have ( ) 1 , 4 x L x y x s L u s s The inverse of the Laplace transform implies that ( ) H u are He's polynomials.The first few com-ponents of He's polynomials are given by

1 xxLL
− are assumed as an integral operator given by − on both sides of (35) and using initial condition we find ) into the function Equation (38) gives