Homotopic Approximate Solutions for the Perturbed CKdV Equation with Variable Coefficients

Dianchen Lu,Tingting Chen,Baojian Hong

doi:10.1155/2014/593642

Abstract

This work concerns how to find the double periodic form of approximate solutions of the perturbed combined KdV (CKdV) equation with variable coefficients by using the homotopic mapping method. The obtained solutions may degenerate into the approximate solutions of hyperbolic function form and the approximate solutions of trigonometric function form in the limit cases. Moreover, the first order approximate solutions and the second order approximate solutions of the variable coefficients CKdV equation in perturbation εu n are also induced.

Highlights

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
We focus on the perturbed CKdV equation with variable coefficients ut + a (t) uux + b (t) u2ux + c (t) u3x = f (x, t, u, ux, ut), (1)

Summary

Introduction

To solve the nonlinear partial differential equation (NPDE) has been an attractive research topic for mathematicians and physicists. Many researchers have developed various approaches for attaining the exact solutions and approximate of NPDE, such as the inverse scattering method [1], homogeneous balance method [2], elliptic function method [3], and perturbation method [4]. A recently reported analytic approximate method, the homotopic mapping method proposed in [5], has been applied to solve many nonlinear problems in engineering and technology effectively, like the nonlinear vibration of [6], boundary layer flow of [7], and so on [8,9,10,11,12]. We applied the homotopic mapping method to the variable coefficients perturbed CKdV equation and obtained the approximate solution of the Jacobi elliptic function form

Model and Homotopy Mapping

Approximate Solution

Conclusion