Javidi, M. (2013). Chebyshev Spectral Collocation Method for Computing Numerical Solution of Telegraph Equation. Computational Methods for Differential Equations, 1(1), 16-29.

M. Javidi. "Chebyshev Spectral Collocation Method for Computing Numerical Solution of Telegraph Equation". Computational Methods for Differential Equations, 1, 1, 2013, 16-29.

Javidi, M. (2013). 'Chebyshev Spectral Collocation Method for Computing Numerical Solution of Telegraph Equation', Computational Methods for Differential Equations, 1(1), pp. 16-29.

Javidi, M. Chebyshev Spectral Collocation Method for Computing Numerical Solution of Telegraph Equation. Computational Methods for Differential Equations, 2013; 1(1): 16-29.

Chebyshev Spectral Collocation Method for Computing Numerical Solution of Telegraph Equation

Receive Date: 12 December 2013,
Accept Date: 12 December 2013

Abstract

In this paper, the Chebyshev spectral collocation method(CSCM) for one-dimensional linear hyperbolic telegraph equation is presented. Chebyshev spectral collocation method have become very useful in providing highly accurate solutions to partial differential equations. A straightforward implementation of these methods involves the use of spectral differentiation matrices. Firstly, we transform telegraph equation to system of partial differential equations with initial condition. Using Chebyshev differentiation matrices yields a system of ordinary differential equations. Secondly, we apply fourth order Runge-Kutta formula for the numerical integration of the system of ODEs. Numerical results verified the high accuracy of the new method, and its competitive ability compared with other newly appeared methods.

[1] A. Mohebbi, M. Dehaghan, High order compact solution of the one dimensional lin- ear hyperbolic equation, Numerical method for partial differential equations, 24 (2008) 11221135.

[2] F. Gao, C. Chi, Unconditionally stable difference scheme for a one-space dimensional linear hyperbolic equation, Applied Mathematics and Computation 187 (2007) 12721276.

[3] A. Saadatmandi, M. Dehghan, Numerical solution of hyperbolic telegraph equation using the Chebyshev Tau method, Numer. Methods Partial Differential Equations 26 (1) (2010) 239-252.

[4] S.A. Yousefi, Legendre multi wavelet Galerkin method for solving the hyperbolic telegraph equation, Numerical Method for Partial Differential Equations, (2008). doi:10.1002/num.

[5] M. Dehghan, A. Ghesmati, Solution of the second-order one-dimensional hyperbolic telegraph equation by using the dual reciprocity boundary integral equation (DRBIE) method, Engineering Analysis with Boundary Elements 34 (2010) 5159.

[6] S. Das, P.K. Gupta, Homotopy analysis method for solving fractional hyperbolic par- tial differential equations, International Journal of Computer Mathematics 88 (2011) 578588.

[7] M.A. Abdou, Adomian decomposition method for solving the telegraph equation in charged particle transport, J. Quant. Spectrosc. Radiat. Transfer 95 (2005) 407-414.

[8] M. Lakestani, B. N. Saray, Numerical solution of telegraph equation using interpolating scaling functions, Computers Mathematics with Applications, 60(2010) 1964-1972.

[9] R.K. Mohanty, An unconditionally stable difference scheme for the one-space dimen- sional linear hyperbolic equation, Appl. Math. Lett. 17 (2004) 101-105.

[10] R.K. Mohanty, An unconditionally stable finite difference formula for a linear second order one space dimensional hyperbolic equation with variable coefficients, Appl. Math. Comput. 165 (2005) 229-236.

[11] L. Lapidus, G.F. Pinder, Numerical Solution of Partial Differential Equations in Science and Engineering, Wiley, New York, 1982.

[12] I. Hashim, O. Abdulaziz, S. Momani, Homotopy analysis method for fractional IVPs, Communications in Nonlinear Science and Numerical Simulation 14 (2009) 674684.

[13] A. Borhanifar, Reza Abazari, An unconditionally stable parallel difference scheme for telegraph equation scheme for telegraph equation, Math. Probl. Eng. (2009) Article ID 969610

[14] M. Dehghan, A. Shokri, A numerical method for solving the hyperbolic telegraph equa- tion, Numer. Methods Partial Differential Equations 24 (2008) 10801093.

[15] M. Dehghan, M. Lakestani, The use of Chebyshev cardinal functions for solution of the

[16] J. Biazar, M. Eslami, Analytic solution for Telegraph equation by differential transform method, Physics Letters A, 374(29)(2010) 2904-2906.

[17] L.N. Trefethen, Spectral methods in MATLAB, SIAM, Philadelphia(2000).

[18] W.S. Don and A. Solomonoff, Accuracy and speed in computing the Chebyshev collo- cation derivative, SIAM J. of Sci. Coput., 16 No. 4(1995) 1253-1268.

[19] C. Canuto ,A. Quarteroni, M.Y. Hussaini and T. Zang, Spectral method in fluied me- chanics, Springer-Verlag, New York (1988).

[20] J.P. Boyd, Chebyshev and Fourier spectral methods, Lecture notes in engineering, 49, Springer-verlag, Berlin(1989).

[21] R. Baltensperger and M.R. Trummer, Spectral differencing with a twist, SIAM J. of Sci. Comp., 24,no. 5(2003),1465-1487.

[22] R. Baltensperger and J.P. Berrut, The errors in calculating the pseudospectral differen- tiation matrices for Chebyshev-Gauss-Lobatto point, Comput. Math. Appl., 37(1999),41-48.