New Tchebyshev-Galerkin Operational Matrix Method for Solving Linear and Nonlinear Hyperbolic Telegraph Type Equations W. M. Abd-Elhameed, 1,2 E. H. Doha, 2 Y. H. Youssri, 2 M. A. Bassuony 3 1 Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia 2 Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt 3 Department of Mathematics, Faculty of Science, Fayoum University, Fayoum 63514, Egypt Received 13 July 2015; accepted 18 April 2016 Published online in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/num.22074 The telegraph equation describes various phenomena in many applied sciences. We propose two new effi- cient spectral algorithms for handling this equation. The principal idea behind these algorithms is to convert the linear/nonlinear telegraph problems (with their initial and boundary conditions) into a system of lin- ear/nonlinear equations in the expansion coefficients, which can be efficiently solved. The main advantage of our algorithm in the linear case is that the resulting linear systems have special structures that reduce the computational effort required for solving them. The numerical algorithms are supported by a careful conver- gence analysis for the suggested Chebyshev expansion. Some illustrative examples are given to demonstrate the wide applicability and high accuracy of the proposed algorithms. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 000: 000–000, 2016 Keywords: Chebyshev polynomials; Galerkin and collocation methods; hyperbolic telegraph equation; operational matrix I. INTRODUCTION The telegraph equation is one of the most important hyperbolic partial differential equations (PDEs) as it describes various phenomena in many applied sciences. For example, it arises in wave propagation along transmission lines in electrical circuit theory [1, 2], and it describes energetic particle transport in the interplanetary medium [1, 3]. The telegraph equation has two time scales, namely dispersive and diffusive. In the case where one is only interested in slow diffusive phenomena, the idealized model is much easier to solve numerically than the complete physical model [1, 4]. Correspondence to: Y. H. Youssri, Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt (e-mail: youssri@sci.cu.edu.eg) © 2016 Wiley Periodicals, Inc.