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On a Nonlinear Diffusion Equation Describing Population Growth

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1 Author(s)
Jose Canosa ; IBM Data Processing Division Scientgc Center, 2670 Hanover Street, Palo Alto, California 94304

A nonlinear eigenvalue problem is solved analytically to obtain the shock-like traveling waves of Fisher's nonlinear diffusion equation, with which he described the wave of advance of advantageous genes. A phase-plane analysis of the wave profiles shows that the propagation speed of the waves is linearly proportional to their thickness. The analytic solution is asymptotically accurate in the limit of infinitely large characteristic speeds. However, as they have a minimum threshold value which is not zero, the asymptotic solution turns out to be highly accurate for all propagation speeds. The wave profiles of Fisher's equation are shown to be identical to the steady state solutions of the Korteweg-de Vries-Burgers equation that are obtained when dissipative effects are dominant over dispersive effects.

Note: The Institute of Electrical and Electronics Engineers, Incorporated is distributing this Article with permission of the International Business Machines Corporation (IBM) who is the exclusive owner. The recipient of this Article may not assign, sublicense, lease, rent or otherwise transfer, reproduce, prepare derivative works, publicly display or perform, or distribute the Article.  

Published in:

IBM Journal of Research and Development  (Volume:17 ,  Issue: 4 )