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Numerical solution of Fisher's equation

Published online by Cambridge University Press:  14 July 2016

Jenö Gazdag  and
José Canosa
José Canosa
Affiliation:
IBM Corporation, Palo Alto, California
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Abstract

The accurate space derivative (ASD) method for the numerical treatment of nonlinear partial differential equations has been applied to the solution of Fisher's equation, a nonlinear diffusion equation describing the rate of advance of a new advantageous gene, and which is also related to certain water waves and plasma shock waves described by the Korteweg-de-Vries-Burgers equation. The numerical experiments performed indicate how from a variety of initial conditions, (including a step function, and a wave with local perturbation) the concentration of advantageous gene evolves into the travelling wave of minimal speed. For an initial superspeed wave this evolution depends on the cutting off of the right-hand tail of the wave, which is physically plausible; this condition is necessary for the convergence of the ASD method. Detailed comparisons with an analytic solution for the travelling waves illustrate the striking accuracy of the ASD method for other than very small values of the concentration.

Keywords

NONLINEAR DIFFUSIONEPIDEMIC WAVESNUMERICAL SIMULATION

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 11 , Issue 3 , September 1974 , pp. 445 - 457
Copyright
Copyright © Applied Probability Trust 1974 

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