Numerical and approximate solutions for two-dimensional hyperbolic telegraph equation via wavelet matrices

TL;DR

This paper introduces a Legendre wavelet operational matrix method to efficiently and accurately solve two-dimensional hyperbolic telegraph equations, transforming the problem into an algebraic form and validating its effectiveness through numerical experiments.

Contribution

The paper develops a novel LWOMM approach for 2D hyperbolic telegraph equations, including convergence analysis and comparison with existing methods.

Findings

The proposed method is accurate and fast.

Numerical results confirm the method's efficiency.

Convergence of the wavelet approximation is established.

Abstract

The present article is devoted to developing the Legendre wavelet operational matrix method (LWOMM) to find the numerical solution of two-dimensional hyperbolic telegraph equations (HTE) with appropriate initial time boundary space conditions. The Legendre wavelets series with unknown coefficients have been used for approximating the solution in both of the spatial and temporal variables. The basic idea for discretizing two-dimensional HTE is based on differentiation and integration of operational matrices. By implementing LWOMM on HTE, HTE is transformed into algebraic generalized Sylvester equation. Numerical experiments are provided to illustrate the accuracy and efficiency of the presented numerical scheme. Comparisons of numerical results associated with the proposed method with some of the existing numerical methods confirm that the method is easy, accurate and fast experimentally. Moreover, we have investigated the convergence analysis of multidimensional Legendre wavelet approximation. Finally, we have compared our result with the research article of Mittal and Bhatia (see [1]).