A BOUNDED FINITE-DIFFERENCE DISCRETIZATION OF A TWO-DIMENSIONAL DIFFUSION EQUATION WITH LOGISTIC NONLINEAR REACTION

J. E. Macías-Díaz ()
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J. E. Macías-Díaz: Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, Aguascalientes, Ags. 20131, Mexico

International Journal of Modern Physics C (IJMPC), 2011, vol. 22, issue 09, 953-966

Abstract: In the present manuscript, we introduce a finite-difference scheme to approximate solutions of the two-dimensional version of Fisher's equation from population dynamics, which is a model for which the existence of traveling-wave fronts bounded within (0,1) is a well-known fact. The method presented here is a nonstandard technique which, in the linear regime, approximates the solutions of the original model with a consistency of second order in space and first order in time. The theory ofM-matrices is employed here in order to elucidate conditions under which the method is able to preserve the positivity and the boundedness of solutions. In fact, our main result establishes relatively flexible conditions under which the preservation of the positivity and the boundedness of new approximations is guaranteed. Some simulations of the propagation of a traveling-wave solution confirm the analytical results derived in this work; moreover, the experiments evince a good agreement between the numerical result and the analytical solutions.

Keywords: Fisher's equation; two-dimensional model; finite-difference scheme; boundedness; positivity (search for similar items in EconPapers)
Date: 2011
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DOI: 10.1142/S0129183111016713

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