A Positivity-Preserving Nonstandard Finite Difference Scheme for Parabolic System with Cross-Diffusion Equations and Nonlocal Initial Conditions

Marc E. Songolo

Abstract


An important number of ecological phenomena can be modeled using nonlinear diffusion partial differential equations. This paper considers a system of cross-diffusion equations with nonlocal initial conditions. Such equations arise as steady-state equations in an age-structured predator-prey model with diffusion. We use the nonstandard finite difference method developed by Mickens. These types of schemes are made by the following two rules: first, renormalization of step size for the denominator function of representations of derivatives, and secondly, nonlocal representations of nonlinear terms. We obtained a scheme that preserves the positivity of solutions. Furthermore, this scheme is explicit and functional relationship is obtained between time, space, and age step sizes.

Keywords


Cross-diffusion equations; finite difference methods; nonlocal initial conditions; nonstandard finite difference schemes; positivity of solutions; predator-prey model.

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References


Mickens R.E., “Nonstandard finite difference schemes for reaction-diffusion Equations,” Numerical Methods for Partial Differential Equation, vol 15, pp. 201-214, 1999.

Walker C., Positive solutions of some system of cross-diffusion equations and nonlocal initial conditions. 2010. [Online] Available: http://www.arxiv.org [November 15, 2010].

Mickens R.E., Nonstandard finite difference models of differential equations, world scientific, 1994.

Mickens R.E., “A Nonstandard finite difference schemes for a Fischer PDE Having Nonlinear Diffusion,” Computers and Mathematics with Applications, vol 45, pp. 429-436, 2003.

Anguelov R. and Lubuma JMS., “Nonstandard finite difference models by nonlocal approximation,” mathematical and computer in simulation, vol 61 pp. 465-475, 2003.

Mickens R.E., “Calculation of Denominator functions in Nonstandard finite Difference schemes for Differential Equations Satisfying a Positivity condition,” Num. Meth. Diff. Eqs, vol 23, pp. 672-691, 2007.

Walker C., On positive solutions of some system of reaction-diffusion equations with nonlocal initials conditions. 2010. [Online] Available: http://www.arxiv.org (March 24, 2010)

Songolo M. E., “A positivity-preserving nonstandard finite difference scheme for a system of reaction-diffusion equations with nonlocal initial conditions,” (To be appear in International Journal of Innovation and Applied Sciences in May 2016).


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