Numerical Solution and Stability Analysis for Burger's-Huxley Equation

Saad Manaa,Farhad Saleem

doi:10.33899/csmj.2009.163797

Abstract

The Burger's-Huxley equation has been solved numerically by using two finite difference methods, the explicit scheme and the Crank-Nicholson scheme. A comparison between the two schemes has been made and it has been found that, the first scheme is simpler while the second scheme is more accurate and has faster convergent. Also, the stability analysis of the two methods by using Fourier (Von Neumann) method has been done and the results were that, the explicit scheme is stable under the condition 2

Highlights

Nowadays engineers and scientists in all fields of their research are using partial differential equations to describe their problems and such partial differential equations arise in the study of heat transfer, boundarylayer flow, fluid flow problems, vibrations elasticity, circular and rectangular wave guides, in applied mathematics and so on
We have to find the numerical solution of these problems using computers which came into existence[7]
For time-dependent problems considerable progress in Finite difference methods was made during the period of, and immediately following, the Second World War, when large-scale practical applications became possible with the aid of computers

Summary

Introduction

Nowadays engineers and scientists in all fields of their research are using partial differential equations to describe their problems and such partial differential equations arise in the study of heat transfer, boundarylayer flow, fluid flow problems, vibrations elasticity, circular and rectangular wave guides, in applied mathematics and so on. Finding the exact solution for the above problems which involve partial differential equations is difficult in some cases. Parabolic PDEs describe practically useful phenomena such as transport chemistry problems of the advection-diffusion-reaction type and problem of this type plays an important role in the modeling of pollution of the atmosphere, ground water and surface water [3]. For time-dependent problems considerable progress in Finite difference methods was made during the period of, and immediately following, the Second World War, when large-scale practical applications became possible with the aid of computers. 2. Mathematical model We consider The generalized Burger's-Huxley equation [10] of the form u t. We define the point p as the point having the coordinate p( x) , and denote that point x p ; that is, xp = p( x). The approximating formula for the second derivative is[6]:

Grid points

Method

Conclusion