A numerical method for solving hyperbolic partial differential equations with piecewise constant arguments and variable coefficients

Abstract

This article deals with hyperbolic partial differential equations with piecewise constant arguments and variable coefficients. This study, therefore, with the aid of the finite difference technique, aims at presenting a numerical solution scheme for solving such types of equations. The stability, consistency, convergence, and convergence rate of our proposed numerical method are investigated. Moreover, the process of the computation of the analytical solution is studied. In order to support and confirm our theoretical results, some numerical examples are also presented. The figures of the numerical and analytical solutions and also the tables of errors are provided to demonstrate the validity of our proposed scheme.

Keywords:

• Finite difference method
• piecewise constant arguments
• asymptotic stability
• consistency
• convergence
• convergence rate

2010 Mathematics Subject Classifications:

• 65M06
• 65N06
• 65L12
• 65L05

Acknowledgments

The authors would like to thank Dr Javad Alavi for his helps and valuable comments about the computer programs of the manuscript. Authors also wish to express their sincere thanks to the referees for their valuable comments and suggestions to improve the quality of the article.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

Note: The results show that as m increases, $r_m$ tends to 1.

Note: The results show that as p increases, $r_p$ tends to 2.