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

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

### 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.### References

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

[2] 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].

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

[4] 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.

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

[6] 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.

[7] 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)

[8] 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).

[2] 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].

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

[4] 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.

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

[6] 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.

[7] 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)

[8] 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).

Published

2016-04-24

Issue

Section

Articles

Authors who submit papers with this journal agree to the following terms.