Travelling Waves for a Density Dependent Diffusion Nagumo Equation over the Real Line  

作　　者：Robert A. Van Gorder 

机构地区：[1]Department of Mathematics,University of Central Florida,P.O.Box 161364,Orlando,FL 32816-1364,USA

出　　处：《Communications in Theoretical Physics》2012年第7期5-11,共7页理论物理通讯（英文版）

基　　金：R.A.V.supported in part by a National Science Foundation research fellowship

摘　　要：We consider the density dependent diffusion Nagumo equation, where the diffusion coefficient is a simple power function. This equation is used in modelling electrical pulse propagation in nerve axons and in population genetics (amongst other areas). In the present paper, the δ-expansion method is applied to a travelling wave reduction of the problem, so that we may obtain globally valid perturbation solutions (in the sense that the perturbation solutions are valid over the entire infinite domain, not just locally; hence the results are a generalization of the local solutions considered recently in the literature). The resulting boundary value problem is solved on the real line subject to conditions at z →±∞. Whenever a perturbative method is applied, it is important to discuss the accuracy and convergence properties of the resulting perturbation expansions. We compare our results with those of two different numerical methods (designed for initial and boundary value problems, respectively) and deduce that the perturbation expansions agree with the numerical results after a reasonable number of iterations. Finally, we are able to discuss the influence of the wave speed c and the asymptotic concentration value α on the obtained solutions. Upon recasting the density dependent diffusion Nagumo equation as a two-dimensional dynamical system, we are also able to discuss the influence of the nonlinear density dependence (which is governed by a power-law parameter m) on oscillations of the travelling wave solutions.

关 键 词：density dependent diffusion equation nonlinear partial differential equation perturbation method δ-expansion method 

分 类 号：O411.1[理学—理论物理] O175[理学—物理]