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A Bounded Finite-Difference Discretization Of A Two-Dimensional Diffusion Equation With Logistic Nonlinear Reaction

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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)

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.

Suggested Citation

  • J. E. Macías-Díaz, 2011. "A Bounded Finite-Difference Discretization Of A Two-Dimensional Diffusion Equation With Logistic Nonlinear Reaction," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 22(09), pages 953-966.
    • Handle: RePEc:wsi:ijmpcx:v:22:y:2011:i:09:n:s0129183111016713
    DOI: 10.1142/S0129183111016713
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