Piecewise linear approximation to Fisher's equation

Abstract

A simple method is presented which allows the replacement of a nonlinear differential equation with a piecewise linear differential equation. The method is based on the idea that a curve of the nonlinear terms of the dependent variable in a differential equation can be replaced by an approximate curve consisting of a set of line segments tangent to the original curve. We apply this method to the ubiquitous Fisher's equation and demonstrate that accurate solutions are obtained with a relatively small number of line segments. References R. A. Fisher. The wave of advance of advantageous genes. Ann. of Eugenics. , 7 , 355--369, 1937. V. M. Kenkre and M. N. Kuperman. Applicability of the Fisher equation to bacterial population dynamics. Phys. Rev. E. , 67 , 2003, 051921. doi:10.1103/PhysRevE.67.051921 D. A. Kessler, Z. Ner and L. M. Sander. Front propagation: precursors, cutoffs, and structural stability. Phys. Rev. E. , 58 , 1998, 107--114. P. K. Maini, D. L. S. McElwain and D. I. Leavesley. Traveling wave model to interpret a wound-healing cell migration assay for human peritoneal mesothelial cells. Tissue Eng. , 10 , 2004, 475--482. M. J. Ablowitz and A. Zeppetella. Explicit solutions of Fisher's equation for a special wave speed. Bull. Math. Biol. , 41 , 1979, 835--840. X. Y. Wang. Exact and explicit solitary wave solutions for the generalised Fisher equation. Phys. Lett. A , 131 , 1988, 277--279. J. Rinzel and J. B. Keller. Traveling wave solutions of a nerve conduction equation. Biophys. J. , 13 , 1973, 1313--1337. Z. Jovanoski and S. Soo. Piecewise linear approximation of nonlinear ordinary differential equations. ANZIAM J. (E) , 51 , 2010, C570--C585. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/2622 T. Hagstrom and H. B. Keller. The numerical calculation of traveling wave solutions of nonlinear parabolic equations. SIAM J. Sci. Stat. Comput. , 7 , 1986, 978--988.