An integro-differential generalization and dynamically consistent discretizations of some hyperbolic models with nonlinear damping

Abstract

This note presents a weak generalization of a time-delayed partial differential equation which, in turn, generalizes the well-known Burger–Fisher and Burgers–Huxley models. In this work, we provide a full discretization which is consistent with the integro-differential equation under consideration. The main analytical result of this note establishes that the discrete temporal rate of change of the discretization yields a consistent approximation to the differential form of the integro-differential equation investigated. Some numerical examples are provided in order to assess the efficiency and effectiveness of our methodology.

Keywords:

• hyperbolic partial differential equation
• Burgers–Fisher equation
• Burgers–Huxley equation
• integro-differential generalization
• dynamical consistency
• finite-difference scheme

2010 AMS Subject Classifications::

• 65M06
• 35R09
• 35L70

Acknowledgements

The author would like to thank the anonymous reviewers and the anonymous editor in charge of handling this manuscript for all their invaluable comments, suggestions and criticisms. He also wishes to acknowledge enlightening discussions with Dr José Villa-Morales, professor of the Department of Mathematics and Physics at the Universidad Autónoma de Aguascalientes.