A modified exponential method that preserves structural properties of the solutions of the Burgers–Huxley equation

ABSTRACT

In this work, we consider the classical Burgers–Huxley partial differential equation defined on a closed and bounded interval of the real line. For this model, theorems on the existence and uniqueness of positive and bounded solutions are readily at hand, whence the design of positivity- and boundedness-preserving numerical methods is pragmatically justified. In this manuscript, we propose a monotone and explicit numerical technique that preserves the structure of such solutions. The method proposed here is a correction of the well-known Bhattacharya method to avoid the presence of singularities. The scheme is an explicit technique, and the simulations confirm in the practice that the structural properties of interest are preserved. Moreover, the computer results yield good approximations to the exact travelling-wave solutions considered here.

KEYWORDS:

• Exponential technique
• modified Bhattacharya method
• structure-preserving method
• Burgers–Huxley equation
• positivity preservation
• boundedness preservation
• monotonicity preservation

2010 AMS Subject Classifications:

• 65M06
• 65M22
• 65Q10

Acknowledgements

The author would like to thank the anonymous reviewers and the editor in charge of handling this manuscript for their invaluable comments and suggestions.

Disclosure statement

The author declares that there is no conflict of interest regarding the publication of this paper.

ORCID

J. E. Macías-Díaz  http://orcid.org/0000-0002-7580-7533