Analytical Wave Solutions for Foam and KdV-Burgers Equations Using Extended Homogeneous Balance Method

U.M Abdelsalam,M G M Ghazal

doi:10.3390/math7080729

Abstract

In this paper, extended homogeneous balance method is presented with the aid of computer algebraic system Mathematica for deriving new exact traveling wave solutions for the foam drainage equation and the Kowerteg-de Vries–Burgers equation which have many applications in industrial applications and plasma physics. The method is effective to construct a series of analytical solutions including many types like periodical, rational, singular, shock, and soliton wave solutions for a wide class of nonlinear evolution equations in mathematical physics and engineering sciences.

Highlights

The nonlinear evolution equations and their solutions are very important for understanding many physical phenomena, for example studying the waves observed in plasma, fluids, optical fibers, laser, astrophysics, water waves and other areas of engineering and science [1,2,3,4,5]
The solutions (49) and (72) are shock wave solutions as depicted in Figure 3, note here that the shock wave solution of the foam drainage equation has an important role to describing the motion in the foam while the shock wave solution of the KdV-Burgers equation is very important for studying the acoustic waves in plasma physics as in [29,30] where the dust acoustic shock waves have been studied
The extended homogeneous balance (HB) method has been developed and applied with the help of Mathematics to deal with nonlinear partial differential equations (PDEs)

Summary

Introduction

The nonlinear evolution equations and their solutions are very important for understanding many physical phenomena, for example studying the waves observed in plasma, fluids, optical fibers, laser, astrophysics, water waves and other areas of engineering and science [1,2,3,4,5]. There are several methods have been constructed and developed for finding the traveling wave solutions to the nonlinear partial differential equations. The homogeneous balance (HB) method [21,22] is a powerful method for deriving the exact traveling wave solutions, the HB method [23,24] was extended to give different types of exact solutions. The foam drainage Equation [26] is very important to describe the drainage of liquid in a foam

Methods

Discussion

Conclusion