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A nonstandard finite difference scheme for a Fisher PDE having nonlinear diffusion. (English) Zbl 1036.65071

Summary: A number of important phenomena in ecology can be modeled by one-dimensional, nonlinear reaction-diffusion partial differential equations (PDEs). This paper considers a modified Fisher PDE for which the diffusion term is nonlinear. A nonstandard finite difference scheme is constructed using methods generated by the previous work of R. E. Mickens [Numer. Methods Partial Differ. Equations 15, 201–214 (1999; Zbl 0926.65085)]. As a check on the mathematical properties of this scheme, a linear stability analysis is carried out for the two fixed-points appearing in the differential and difference equations. The finite difference scheme is shown to have solutions which satisfy a positivity condition as well as the requirement of boundedness.
Further, the scheme is explicit and a functional relationship is obtained between the space and time step-sizes. A numerical test of the scheme is done for a particular initial/boundary value problem. A brief discussion of how the work can be extended and/or generalized is also given.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K57 Reaction-diffusion equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs

Citations:

Zbl 0926.65085
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References:

[1] Edelstein-Keshet, L., Mathematical Models in Biology (1987), McGraw-Hill: McGraw-Hill New York · Zbl 0674.92001
[2] Murray, J. D., Mathematical Biology (1989), Springer-Verlag: Springer-Verlag Berlin · Zbl 0682.92001
[3] Smoller, J., Shock Waves and Reaction-Diffusion Equations (1983), Springer-Verlag: Springer-Verlag New York · Zbl 0508.35002
[4] Oran, E. S.; Boris, J. P., Numerical Simulation of Reactive Flow (1987), Elsevier: Elsevier New York · Zbl 0762.76098
[5] Mickens, R. E., Nonstandard finite difference schemes for reaction-diffusion equations, Numer. Methods Partial Diff. Eq., 15, 201-214 (1999) · Zbl 0926.65085
[6] Mickens, R. E., Exact solutions to a finite-difference model of a nonlinear reaction-advection equation: Implications for numerical analysis, Numer. Methods Partial Diff. Eq., 5, 313-325 (1989) · Zbl 0693.65059
[7] Mickens, R. E., Nonstandard Finite Difference Models of Differential Equations (1994), World Scientific: World Scientific Singapore · Zbl 0925.70016
[8] Mickens, R. E.; Smith, A., Finite-difference models of ODE’s: Influence of denominator functions, J. Franklin Institute, 327, 143-149 (1990) · Zbl 0695.93063
[9] Anguelov, R.; Lubuma, J. M.-S., Contributions to the mathematics of the nonstandard finite difference method and applications, Numer. Methods Partial Diff. Eq., 17, 518-543 (2001) · Zbl 0988.65055
[10] Logan, J. D., Nonlinear Differential Equations (1994), Wiley-Interscience: Wiley-Interscience New York
[11] Mickens, R. E., Construction of a novel finite-difference scheme for a nonlinear diffusion equation, Numer. Methods Partial Diff. Eq., 7, 299-302 (1991) · Zbl 0736.65065
[12] Mickens, R. E., A nonstandard finite difference scheme for a nonlinear PDE having diffusive shock wave solutions (1999), Physics Department, Clark Atlanta University: Physics Department, Clark Atlanta University Atlanta, GA, (unpublished manuscript)
[13] Debnath, L., Nonlinear Partial Differential Equations (1997), Birkhäuser: Birkhäuser Boston, MA
[14] Newman, W. I., Some exact solutions to a nonlinear diffusion problem in population genetics and combustion, J. Theor. Bio., 85, 325-334 (1980)
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