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

    Authors

    R. A. Van Gorder

    Comments

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    Abbreviated Journal Title

    Commun. Theor. Phys.

    Keywords

    density dependent diffusion equation; nonlinear partial differential; equation; perturbation method; delta-expansion method; HOMOTOPY ANALYSIS METHOD; PERTURBATIVE APPROACH; MODELS; Physics, Multidisciplinary

    Abstract

    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 delta-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 - > +/-infinity. 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 alpha 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.

    Journal Title

    Communications in Theoretical Physics

    Volume

    58

    Issue/Number

    1

    Publication Date

    1-1-2012

    Document Type

    Article

    Language

    English

    First Page

    5

    Last Page

    11

    WOS Identifier

    WOS:000306569800002

    ISSN

    0253-6102

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