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F. Gao and C. Chi, “Unconditionally Stable Difference Schemes for a One-Space-Dimensional Linear Hyperbolic Equation,” Applied Mathematics and Computation, Vol. 187, No. 2, 2007, pp. 1272-1276. doi:10.1016/j.amc.2006.09.057

has been cited by the following article:

• TITLE: Legendre Approximation for Solving Linear HPDEs and Comparison with Taylor and Bernoulli Matrix Methods

  AUTHORS: Emran Tohidi

  KEYWORDS: Legendre Operational Matrix of Differentiation; Hyperbolic Partial Differential Equations; Legendre Polynomial Solutions; Double Legendre Series

  JOURNAL NAME: Applied Mathematics, Vol.3 No.5, May 16, 2012

  ABSTRACT: The aim of this study is to give a Legendre polynomial approximation for the solution of the second order linear hyper-bolic partial differential equations (HPDEs) with two variables and constant coefficients. For this purpose, Legendre matrix method for the approximate solution of the considered HPDEs with specified associated conditions in terms of Legendre polynomials at any point is introduced. The method is based on taking truncated Legendre series of the functions in the equation and then substituting their matrix forms into the given equation. Thereby the basic equation reduces to a matrix equation, which corresponds to a system of linear algebraic equations with unknown Legendre coefficients. The result matrix equation can be solved and the unknown Legendre coefficients can be found approximately. Moreover, the approximated solutions of the proposed method are compared with the Taylor [1] and Bernoulli [2] matrix methods. All of computations are performed on a PC using several programs written in MATLAB 7.12.0.