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A. Kolmogorov, I. Petrovsy and N. Piskounov, “Study of the Diffusion Equation with Growth of the Quantity of Matter and Its Applications to a Biological Problem,” Moscow University Mathematics Bulletin, 1989, p. 105.

has been cited by the following article:

• TITLE: Travelling Wave Solution of the Fisher-Kolmogorov Equation with Non-Linear Diffusion

  AUTHORS: Muhammad Shakeel

  KEYWORDS: Fisher-Kolmogorov Equation; Non-Linear Diffusion; Travelling Wave; Wave Speed; Pulled Front; Pushed Front

  JOURNAL NAME: Applied Mathematics, Vol.4 No.8A, July 29, 2013

  ABSTRACT: 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.