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The homotopy analysis method is applied to solve the variable coefficient KdV-Burgers equation. With the aid of generalized elliptic method and Fourier’s transform method, the approximate solutions of double periodic form are obtained. These solutions may be degenerated into the approximate solutions of hyperbolic function form and the approximate solutions of trigonometric function form in the limit cases. The results indicate that this method is efficient for the nonlinear models with the dissipative terms and variable coefficients.

To solve the nonlinear partial differential equation (NPDE) has been an attractive research topic for mathematicians and physicists. Nonlinear evolution equations with variable coefficients can describe the physical phenomenon more accurately, and it is of great significance to study how to find the solutions of nonlinear evolution equations with variable coefficients. The nonlinear partial differential equations are generally difficult to solve and their exact solutions are difficult to obtain. In recent years, some various approximate methods have been developed such as homotopy analysis method [

This paper is arranged in the following manner. In Section

To explain the basic idea of the homotopy analysis method, we consider the following nonlinear differential equation:

Generally speaking, the operator

By (

As can be seen from 0 to 1 of

Assume that the solution of

In this section, we focus on the variable coefficient KdV-Burgers equation

It is fascinating to observe that, when

Next, we applied the homotopy analysis method to study the approximate solution of (

In order to get the solution of (

To aim at (

By using the generalized elliptic method [

When

When

One can easily prove that

Let

In order to obtain the approximate solution of (

From (

By using the Fourier transform, one can obtain the solution of (

where

By comparing the higher power coefficients of

This work studies the variable coefficients KdV-Burgers equations by using the homotopy analysis method, and the two-degree approximate solution of the Jacobi elliptic function form is obtained, which can degenerate to solitary wave approximate solution and trigonometric function approximate solution in the limit case. Our results show that the homotopy analysis method is applicable to the variable solution equations; how to apply this method to high-degree and high-dimensional system remains to be further studied.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Nature Science Foundation of China (no. 61070231), the Outstanding Personal Program in Six Fields of Jiangsu Province, China (Grant no. 2009188), and the Graduate Student Innovation Project of Jiangsu Province (Grant no. CXLX13_673).