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Travelling Wave Solution of the fisher-kolmogorov Equation with Non-Linear Diffusion

Applied Mathematics 2013年第8期4卷 148-160页

作者： Muhammad Shakeel Department of Mathematics Air University Islamabad Pakistan

In this paper we study one-dimensional fisher-kolmogorov equation with density dependent non-linear diffusion. We choose the diffusion as a function of cell density such that it is high in highly cell populated areas and it is small in the regions of fewer cells. The fisher equation with non-linear diffusion is known as modified fisher equation. We study the travelling wave solution of modified fisher equation and find the approximation of minimum wave speed analytically, by using the eigenvalues of the stationary states, and numerically by using COMSOL (a commercial finite element solver). The results reveal that the minimum wave speed depends on the parameter values involved in the model. We observe that when diffusion is moderately non-linear, the eigenvalue method correctly predicts the minimum wave speed in our numerical calculations, but when diffusion is strongly non-linear the eigenvalues method gives the wrong answer.

关键词： fisher-kolmogorov Equation Non-Linear Diffusion Travelling Wave Wave Speed Pulled Front Pushed Front

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A Direct Implementation of a Modified Boundary Integral Formulation for the Extended fisher-kolmogorov Equation

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Journal of Applied Mathematics and Physics 2015年第10期3卷 1262-1269页

作者： Okey Oseloka Onyejekwe Computational Science Program Addis Ababa University Arat Kilo Campus Addis Ababa Ethiopia

This study is concerned with the numerical approximation of the extended fisher-kolmogorov equation with a modified boundary integral method. A key aspect of this formulation is that it relaxes the domain-driven approach of a typical boundary element (BEM) technique. While its discretization keeps faith with the second order accurate BEM formulation, its implementation is element-based. This leads to a local solution of all integral equation and their final assembly into a slender and banded coefficient matrix which is far easier to manipulate numerically. This outcome is much better than working with BEM’s fully populated coefficient matrices resulting from a numerical encounter with the problem domain especially for nonlinear, transient, and heterogeneous problems. Faithful results of high accuracy are achieved when the results obtained herein are compared with those available in literature.

关键词： Boundary Element Method Extended fisher-kolmogorov Equation Boundary Integral Formulation Slender Coefficient Matrix Hybridization Domain-Driven

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