Existence and uniqueness of monotone and bounded solutions for a finite-difference discretization à la Mickens of the generalized Burgers–Huxley equation

Abstract

Departing from a generalized Burgers–Huxley partial differential equation, we provide a Mickens-type, nonlinear, finite-difference discretization of this model. The continuous system is a nonlinear regime for which the existence of travelling-wave solutions has been established previously in the literature. We prove that the method proposed also preserves many of the relevant characteristics of these solutions, such as the positivity, the boundedness and the spatial and the temporal monotonicity. The main results provide conditions that guarantee the existence and the uniqueness of monotone and bounded solutions of our scheme. The technique was implemented and tested computationally, and the results confirm both a good agreement with respect to the travelling-wave solutions reported in the literature and the preservation of the mathematical features of interest.

Keywords::

• existence and uniqueness of solutions
• Mickens-type finite-difference scheme
• generalized Burgers–Huxley model
• preservation of positivity and boundedness
• monotone method
• preservation of skew-symmetry

Keywords::

• 65Q10
• 65M06
• 65Q30
• 35K55

Acknowledgements

This manuscript was completed when the first author visited the Department of Analysis and Numerical Mathematics of the Gdańsk University of Technology, Poland, with the partial financial support of the Federal Government of Mexico and the Universidad Autónoma de Aguascalientes. He wishes to acknowledge the hospitality and the kindness that he enjoyed during his stay in this university. In particular, he wishes to thank Prof. Karol Dziedziul for all his kindness and support. Finally, the authors want to thank the anonymous reviewers and the anonymous editor who handled this work for all their invaluable suggestions and criticisms, which substantially improved the overall quality of this work.