A numerical algorithm for the solution of telegraph equations M.S. El-Azab * , Mohamed El-Gamel Mathematical and Physical Sciences Department, Faculty of Engineering, Mansoura University, Mansoura 35516, Egypt Abstract In this paper, we present a new competitive numerical scheme to solve nonlinear telegraph equations. The method is based on Rothe’s approximation in time discretization and on the Wavelet–Galerkin in the spatial discretization. The approximate solutions converge in the space C ð0; T Þ; L 2 ðXÞ ð Þ\ L 2 ð0; T Þ; W 1;2 0 ðXÞ   to the variational solution. A full error analysis is performed and a numerical experiment is given to illustrate the good convergence behavior of the approximate solution. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Hyperbolic partial differential equations; Rothe–Wavelet–Galerkin method; Error analysis 1. Introduction In the present work we are dealing with the numerical approximation of the following second order hyper- bolic problem: o 2 u ot 2 þ ou ot þ Au ¼ f ðx; t; uÞ in Q  X  I ; ð1:1Þ uðx; 0Þ¼ u 0 ðxÞ; u t ðx; 0Þ¼ u 1 ðxÞ in X; ð1:2Þ u ¼ 0 on R  oX  I : ð1:3Þ Here X  R N ðN ¼ 1; 2; 3Þ is a simply connected bounded domain with boundary oX 2 C 0;1 , I ð0; T Þ is a time interval and uðx; tÞ denotes the dependent variable representing the magnitude that is modeled. We introduce the elliptic differential operator A defined by Au :¼ r  AðxÞru ð Þþ aðxÞu; ð1:4Þ where $ and $Æ denote the gradient and divergence operators, respectively and A(x) is a symmetric matrix with entries that are uniformly bounded and measurable. The functions f ; u 0 ; u 1 and A(x) are given data for the problem and will be assumed as regular as necessary. 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.01.091 * Corresponding author. E-mail address: ms_elazab@hotmail.com (M.S. El-Azab). Applied Mathematics and Computation 190 (2007) 757–764 www.elsevier.com/locate/amc