On the computational modeling of traveling-wave solutions of a nonlinear population equation

Authors

  • Jorge Eduardo Macías Díaz Universidad Autónoma de Aguascalientes

DOI:

https://doi.org/10.33064/iycuaa2013574014

Keywords:

parabolic partial differential equation, reaction-difusion model, square-root law, finite-difference method, positivity and boundedness, monotonicity

Abstract

Departing from a diffusive partial differential equation with nonlinear reaction, we developed a non-standard, finite-difference scheme to approximate its solutions. The reaction term of the mathematical model follows a square-root regime, and the existence of traveling-wave solutions for this equation is a fact recently established in the specialized literature. In the present manuscript,
we propose a discretization which preserves most of the mathematical features of such solutions, namely, the positivity, the elevation mark, and the temporal and the spatial monotony. We provide here some illustrative simulations that evidence the preservation of such properties.

 

Downloads

Download data is not yet available.

Metrics

Metrics Loading ...

Author Biography

Jorge Eduardo Macías Díaz, Universidad Autónoma de Aguascalientes

Departamento de Matemáticas y Física, Centro de Ciencias Básicas

References

• FISHER, R.A., The wave of advance of advantageous genes. Annals of Eugenics, 7: 355-369, 1937.

• FUJIMOTO, T.; RANADE, R.R., Two characterizations of inverse-positive matrices: The Hawkins-Simon condition and the Le Chatelier-Braun principle. Electronic Journal of Linear Algebra, 11: 59-65, 2004.

• KOLMOGOROV, A.; PETROVSKY, I.; PISCOUNOV, N., Étude de l’équations de la diffusion avec croissance de la quantité de matiere et son application a un probléme biologique. Bulletin of the University of Moskou, International Series, 1A: 1-25, 1937.

• MACÍAS DÍAZ, J.E., On a boundedness-preserving semilinear discretization of a two-dimensional nonlinear diffusion-reaction model. International Journal of Computer Mathematics, 89: 1678-1688, 2012.

• MACÍAS DÍAZ, J.E.; PURI, A., On some explicit nonstandard methods to approximate nonnegative solutions of a weakly hyperbolic equation with logistic nonlinearity. International Journal of Computer Mathematics, 88: 3308-3323, 2011.

• MACÍAS DÍAZ, J.E.; RUIZ RAMÍREZ, J.; VILLA, J., The numerical solution of a generalized Burgers-Huxley equation through a conditionally bounded and symmetry-preserving method. Computer Mathematics with Applications, 61: 3330-3342, 2011.

• MICKENS, R., Wave front behavior of traveling wave solutions for a PDE having square-root dynamics. Mathematics and Computers in Simulation, 82: 1271-1277, 2012.

• POLYANIN, A.D.; ZAITSEV, V.F., Handbook of Nonlinear Partial Differential Equations. Estados Unidos de América: Chapman and Hall, 2004.

• TOMASIELLO, S., Numerical solutions of the BurgersHuxley equation by the IDQ method. International Journal of Computer Mathematics, 87: 129-140, 2010.

• WANG, X.Y.; ZHU, Z.S.; LU, Y.K., Solitary wave solutions of the generalized Burgers-Huxley equation. Journal of Physics A., 23: 57-79, 1990.

Published

2013-04-30

How to Cite

Macías Díaz, J. E. (2013). On the computational modeling of traveling-wave solutions of a nonlinear population equation. Investigación Y Ciencia De La Universidad Autónoma De Aguascalientes, (57), 27–31. https://doi.org/10.33064/iycuaa2013574014

Issue

Section

Artículos de Investigación

Categories