Adomian decomposition method for Burger's–Huxley and Burger's–Fisher equations

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Abstract

The approximate solutions for the Burger's–Huxley and Burger's–Fisher equations are obtained by using the Adomian decomposition method [Solving Frontier Problems of Physics: the Decomposition Method, Kluwer, Boston, 1994]. The algorithm is illustrated by studying an initial value problem. The obtained results are presented and only few terms of the expansion are required to obtain the approximate solution which is found to be accurate and efficient.

Introduction

Generalized Burger's–Huxley equation shows a prototype model for describing the interaction between reaction mechanisms, convection effects and diffusion transports. The equation was investigated by Satsuma [4] in 1986. Approximate solutions of nonlinear differential equations are of importance in physical problems. So far there exists no general method for finding solutions of nonlinear diffusion equations. The Adomian decomposition method [1], [2], [3] has been applied to a wide class of stochastic and deterministic problems in physics, biology and chemical reactions. The Adomian method is powerful and can be used in applications of nonlinear evolution models.

Section snippets

The governing equations

Consider the generalized Burger's–Huxley equationut+αuδux−uxx=βu(1−uδ)(uδ−γ)∀0⩽x⩽1,t⩾0with the initial conditionu(x,0)=γ2+γ2Tanh[A1x]1/δ.The exact solution of Eq. (1) is derived in [5] asu(x,t)=γ2+γ2Tanh[A1(x−A2t)]1/δ,whereA1=−αδ+δα2+4β(1+δ)4(1+δ)γ,A2=γα(1+δ)−(1+δ−γ)(−α+α2+4β(1+δ))2(1+δ),where α, β, γ and δ are parameters, β⩾0, δ>0, γ∈(0,1).

And consider the generalized Burger's–Fisher equationut+αuδux−uxx=βu(1−uδ)∀0⩽x⩽1,t⩾0with the initial conditionu(x,0)=f(x)and exact solution isu(x,t)=12+12

Analysis of the method

We begin with the equationLu+R(u)+N(u)=g(t),where L is the operator of the highest-ordered derivatives and R is the remainder of the linear operator. The nonlinear term is represented by N(u). Thus we getLu=g(t)−R(u)−N(u).The inverse,L−1=∫0t(·)dt,operating with the operator L−1 on both sides of Eq. (6) we haveu=f0+L−1(g(t)−R(u)−N(u)),where f0 is the solution of homogeneous equationLu=0involving the constants of integration. The integration constants involved in the solution of homogeneous

Application

In this section we will apply ADM for the two problems: The first is the generalized Burger's–Huxley equation (1) and the second is the generalized Burger's–Fisher equation (4).

References (6)

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    Solving Frontier Problems of Physics: the Decomposition Method

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    A review of the decomposition method in applied mathematics

    J. Math. Anal. Appl

    (1988)
  • G. Adomian

    New approach to nonlinear partial differential equations

    J. Math. Anal. Appl

    (1984)
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