Solution of Pattern Waves for Diffusive Fisher-like Non-linear Equations with Adaptive Methods

Abstract

AbstractIn this paper, we investigate some simple numerical methods for the solution of one-dimensional reaction–diffusion problems in biological context to study the rate of diffusivity, travelling wave patterns and mechanism of logistic growth in Fisher and Nagumo equations. Hitherto, most computations in the last decade have been restricted to lower order method due to the difficulty involved in the combination of non-linearity and stiffness. In this paper, we have adopted matrix formulation techniques based on finite difference scheme of order four for the spatial discretization of the partial differential equation. For the time evolution, fourth-order exponential time-differencing (ETD) Runge–Kutta method is considered. This method provides an order of magnitude improvement over its fourth-order counterparts such as fourth-order ETD method, fourth-order ETD method of Adam-type as well as fifth-order ETD and sixth-order ETD methods whose formulations are based on ETD techniques. Applicability and suitability of our approach is demonstrated with some numerical experiments and comparison is equally made with the existing software packages.