Homotopy Perturbation Method for the Generalized Hirota-Satsuma Coupled KdV Equation ()

Dalal A. Maturi

Department of Mathematics, Faculty of Science, King AbdulAziz University, Jeddah, Saudi Arabia.

**DOI: **10.4236/am.2012.312273
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Department of Mathematics, Faculty of Science, King AbdulAziz University, Jeddah, Saudi Arabia.

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.

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Maturi, D. (2012) Homotopy Perturbation Method for the Generalized Hirota-Satsuma Coupled KdV Equation. *Applied Mathematics*, **3**, 1983-1989. doi: 10.4236/am.2012.312273.

1. Introduction

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 [23]. An elementary introduction to the homotopy perturbation method can be found in [24]. Improved homotopy perturbation method is given in [25-29]. Some applications of He’s homotopy perturbation method [1] are proposed in [30-35]. Homotopy perturbation method is useful for solving many different kinds of linear and nonlinear problems as explored in [36-49] and for numerical solution of 12th order boundary value problems as in [50]. It can be said that He’s homotopy perturbation method is a universal approach and that is able to solve various kinds of nonlinear equations. For example, it was applied to nonlinear Burger’s equation [51-53], to the Fisher’s equation [54-57], and solitary wave solutions for a generalized Hirota-Satsuma coupled KdV equation [58-60]. Solution of the Hirota-Satsuma KdV equation with the aid of homotopy perturbation method, adomian decomposition method, variational iteration method and homotopy analysis method can be found in [61-66].

2. Homotopy Perturbation Method (HPM)

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

(1)

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:

(2)

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),

3. Method of Solution

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

(3)

with the following initial conditions:

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

(4)

(5)

(6)

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

(7)

(8)

(9)

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 Figure 1.

Figure 1. The 3D exact solution of u(x, t), for k = 0.1, β = 1.

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 Figure 2.

Figure 2. The 3D exact solution of v(x, t), for c_{0} = 1, c_{1} = 1, k = 0.1, β = 1.

Figure 3. The 3D exact solution of w(x, t), for c_{0} = 1, c_{1} = 1, k = 0.1, β = 1.

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

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

4. Conclusion

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.

Conflicts of Interest

The authors declare no conflicts of interest.

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