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In this paper, we consider the homotopy perturbation method (HPM) to obtain the exact solution of Hirota-Satsuma Coupled KdV equation. The results reveal that the proposed method is very effective and simple and can be applied to other nonlinear mathematical problems.

A number of methods have been proposed in the literature recently for solving different kinds of physical and mathematical problems. Among those methods are: the homotopy perturbation method [1-7], the variational iteration method [8-22] and the domain decomposition method [

To illustrate the basic idea of this method, we consider the following general non-linear differential equation:

with the following boundary conditions:

where is a general differential operator, is a boundary operator, is a known analytical function and is the boundary of the domain.

The operator can be decomposed into a linear part and a non-linear one, designated as and respectively. Hence Equation (1) can be written as the following form:

Using homotopy technique, we construct a homotopy which satisfies:

where is an embedding parameter and is an initial approximation of Equation (1) which satisfies the boundary conditions. Obviously, from Equation (2) we have

By changing the value of from zero to unity, changes from to, in topology this is called Deformation and and are called Homotopic. Due to the fact that can be considered as a small parameter, hence we considered as a small parameter, hence we consider the solution of Equation (2) as a power series in as the following:

setting results in the approximate solution for Equation (1),

In this section, we consider the generalized Hirota-Satsuma Coupled KdV equation,

with the following initial conditions:

Using homotopy perturbation method, we construct a homotopy in the following from:

Suppose the solution of Equations (4), (5) and (6) has the form

where are functions yet to be determined. Substituting Equations (7), (8) and (9) into Equations (4), (5) and (6), respectively, and equating the terms with identical powers of, we have

Therefore, the exact solution of Equation (3) can be obtained by setting, i.e.

Solving the systems accordingly with using Matlab7.8, thus we obtain,

and so on for other components. The solution in a closedform is given by

The 3D exact solution of, for, obtained by HPM is given in

and so on for other components. The solution in a closedform is given by

The 3D exact solution of, for , obtained by HPM is given in

and so on. The solution in a closed-form is given by

The 3D exact solution of, for , obtained by HPM is given in

In this paper, the homotopy perturbation method was used for finding solutions of a generalized Hirota-Satsuma coupled KdV equation with initial conditions. It can be concluded that the homotopy perturbation method is very powerful and efficient technique in finding exact solutions for wide classes of problems. In our work we use the MATLAB to calculate the series obtained from the homotopy perturbation method.