Homotopy Perturbation Transform Method for Solving Third Order Korteweg-DeVries (KDV) Equation

In this paper, we develop a method to calculate approximate solution of some Third-order Korteweg-de Vries equations with initial condition with the help of a new method called Aboodh transform homotopy perturbation method (ETHPM). This method is a combination of the new integral transform “Aboodh transform” and the homotopy perturbation method. The nonlinear term can be easily handled by homotopy perturbation method. The results reveal that the combination of Aboodh transform and homotopy perturbation method is quite capable, practically well appropriate for use in such problems and can be applied to other nonlinear problems. This method is seen as a better alternative method to some existing techniques for such realistic problems.


Introduction
The exact solutions of nonlinear partial differential equations are difficult, therefore many various approximate methods have recently been developed such as homotopy perturbation method [1][2][3][4][5][6], Adomian's decomposition method, differential transform method and projected differential transform method to solve linear and nonlinear differential equations. The homotopy perturbation method has the merits of simplicity and easy execution. Homotopy theory becomes a powerful mathematical tool also ; homotopy theory can overcome the difficulties arising in calculation of Adomian's polynomials in Adomian's decomposition method. The KDV equation plays an important role in diverse areas of engineering and scientific applications, and therefore, the enormous amount of research work has been invested in the study of KDV equations [7][8][9][10][11][12]. The Aboodh transform [13][14][15][16] is totally incapable of handling nonlinear equations because of the difficulties that are caused by the nonlinear terms. Furthermore, the homotopy perturbation method is also combined with the well-known Aboodh transform method and the variational iteration method to produce a highly effective technique for handling many nonlinear problems.
In this paper, we shall deal with the KDV equation in the following form, Where ( , ) is the displacement.KDV equation Was first derived by Korteweg and Vries (1895) to the water waves in shallow canal, when the study of water waves was of vital interest for applications in naval architecture and for the knowledge of tides and floods. The purpose of this paper is to extend the (HPTM) for the solution of Korteweg-DeVries (KDV) Equation. The method has been successfully applied for obtaining exact solutions for nonlinear equations. Proof: To obtain transforms of partial derivatives we use integration by parts as follows:

Aboodh Transform
We assume that $ is piecewise continuous and it is of exponential order. Let We can easily extend this result to the nth partial derivative by using mathematical induction.

Homotopy Perturbation Method
Let F and G be the topological spaces. If $ and A are continuous maps of the space F into G , it is said that $ is homotopic to A , if thereis continuous map H: F × #0,1% → G such that H ( , 0) = $ ( )andH ( , 1) = A ( ),foreach ∈ F,then the map is called homotopy between $andA.
To explain the homotopy perturbation method, we consider a general equation of the type, We assume that Eq. (8) has a unique solution. The comparisons of like powers of N give solutions of various orders, for more details see [17][18][19][20].

Basic Idea
To illustrate the basic idea of this method, we consider a general form of nonlinear non homogeneous partial differential equation as the follow: with the following initial conditions ( , 0) = ℎ( ), ( , 0) = $ ( ) Where D is the second order linear differential operator V = 1 B 1 B , is the linear differential operator ofless order than D, N represents the general non-linear differential operator and A ( , )is the source term.
Taking Aboodh transform (denoted throughout this paper by" (. )) on both sides of Eq. (10), to get: Where Z ( , ) represents the term arising from the source term and the prescribed initial condition. Now, we apply the homotopy perturbation method And the nonlinear term can be decomposed as:

Applications
In this section, the effectiveness and the usefulness of

Conclusions
In this paper, we have applied the homotopy perturbation transform method to Korteweg-DeVries (KDV) Equation. It can be concluded that the HPTM is a very powerful and efficient technique in finding exact and approximates solutions for nonlinear problems. By using this method we obtain a new efficient recurrent relation to solve (KDV) Equation.