Bilinear Equation of the Nonlinear Partial Differential Equation and Its Application

Xiao-Feng Yang,Yi Wei

doi:10.1155/2020/4912159

Abstract

The homogeneous balance of undetermined coefficient method is firstly proposed to derive a more general bilinear equation of the nonlinear partial differential equation (NLPDE). By applying perturbation method, subsidiary ordinary differential equation (sub-ODE) method, and compatible condition to bilinear equation, more exact solutions of NLPDE are obtained. The KdV equation, Burgers equation, Boussinesq equation, and Sawada-Kotera equation are chosen to illustrate the validity of our method. We find that the underlying relation among theG′/G-expansion method, Hirota’s method, and HB method is a bilinear equation. The proposed method is also a standard and computable method, which can be generalized to deal with other types of NLPDE.

Highlights

The nonlinear partial differential equation (NLPDE) is known to describe a wide variety of phenomena in physics and in biology, chemistry, and several other fields [1,2,3]
By improving some key steps in the homogeneous balance (HB) method [26], we propose a new method, HB of undetermined coefficient method, which can be used to derive the bilinear equation of NLPDE
We show the underlying relations among the ðG′/GÞ-expansion method, Hirota’s method, and HB method

Summary

Introduction

The nonlinear partial differential equation (NLPDE) is known to describe a wide variety of phenomena in physics and in biology, chemistry, and several other fields [1,2,3]. The ðG′/GÞ-expansion method, Hirota’s method, and HB method are very effective for constructing the exact solutions of NLPDE. Fan improved the HB method to investigate the BT, Lax pairs, symmetries, and exact solutions for some NLPDE [31, 32]. He showed that there are many links among the HB method, Weiss-Tabor-Carnevale method, and Clarkson-Kruskal method. By improving some key steps in the HB method [26], we propose a new method, HB of undetermined coefficient method, which can be used to derive the bilinear equation of NLPDE.

Description of the HB of Undetermined Coefficient Method

Application to the KdV Equation

Application to the Burgers Equation

Bilinear Equation of the Boussinesq Equation and Sawada-Kotera Equation

Conclusions