A numerical procedure based on Hermite wavelets for two-dimensional hyperbolic telegraph equation

Abstract

In this paper, Hermite wavelets are used to develop a numerical procedure for numerical solutions of two-dimensional hyperbolic telegraph equation. In first stage, we rewrite the second order hyperbolic telegraph equation as a system of partial differential equations by introducing a new variable and then using finite difference approximation we discretized time-dependent variables. After that, Hermite wavelets series expansion is used for discretization of space variables. With this approach, finding the solution of two-dimensional hyperbolic telegraph equation is transformed to finding the solution of two algebraic system of equations. The solution of these systems of algebraic equations gives Hermite wavelet coefficients. Then by inserting these coefficients into Hermite wavelet series expansion numerical solutions can be acquired consecutively. The main goal of this paper is to indicate that Hermite wavelet-based method is suitable and efficient for two-dimensional hyperbolic telegraph equation as well as other type of hyperbolic partial differential equations such as wave and sinh-Gordon equations. Six test problems are chosen and $$L_2$$ , $$L_{\infty }$$ and root mean squared (RMS) error norms are measured for comparison of current numerical results with exact results and with the results of previous studies based on such as meshless, B-spline and differential quadrature methods. The obtained results corroborate the applicability and efficiency of the proposed method.