The use of Chebyshev cardinal functions for solution of the second-order one-dimensional telegraph equation. (English) Zbl 1169.65102

Dehghan, Mehdi; Lakestani, Mehrdad

Numer. Methods Partial Differ. Equations 25, No. 4, 931-938 (2009).

Summary: A numerical technique is presented for the solution of the second order one-dimensional linear hyperbolic equation. This method uses the Chebyshev cardinal functions. The method consists of expanding the required approximate solution as the elements of Chebyshev cardinal functions. Using the operational matrix of the derivative, the problem is reduced to a set of algebraic equations. Some numerical examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces very accurate results.

MSC:

65M70	Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35L15	Initial value problems for second-order hyperbolic equations

Keywords:

Chebyshev cardinal functions; operational matrix of derivative; telegraph equation; second order hyperbolic equation; numerical examples

References:

[1]	Dehghan, On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation, Numer Methods Partial Differential Eq 21 pp 24– (2005)
[2]	Mohanty, On the use of high order difference methods for the system of one space second order non-linear hyperbolic equations with variable coefficients, J Comput Appl Math 72 pp 421– (1996)
[3]	Twizell, An explicit difference method for the wave equation with extended stability range, BIT 19 pp 378– (1979) · Zbl 0441.65066
[4]	Mohanty, An unconditionally stable difference scheme for the one-space dimensional linear hyperbolic equation, Appl Math Lett 17 pp 101– (2004) · Zbl 1046.65076
[5]	Mohanty, An unconditionally stable finite difference formula for a linear second order one space dimensional hyperbolic equation with variable coefficients, Appl Math Comput 165 pp 229– (2005) · Zbl 1070.65076
[6]	Lapidus, Numerical solution of partial differential equations in science and engineering (1982) · Zbl 0584.65056
[7]	Mohebbi, High order compact solution of the one-space-dimensional linear hyperbolic equation, Numer Methods Partial Differential Eq 24 pp 1222– (2008)
[8]	Dehghan, Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Math Comput Simulation 71 pp 16– (2006) · Zbl 1089.65085
[9]	Dehghan, Implicit collocation technique for heat equation with non-classic initial condition, Int J Non-Linear Sci Numer Simul 7 pp 447– (2006) · Zbl 06942230
[10]	Dehghan, A numerical method for solving the hyperbolic telegraph equation, Numer Methods Partial Differential Eq 24 pp 1080– (2008)
[11]	Boyd, Chebyshev and Fourier spectral methods (2000)
[12]	Dehghan, Parameter determination in a partial differential equation from the overspecified data, Math Comput Model 41 pp 196– (2005) · Zbl 1080.35174
[13]	Dehghan, A computational study of the one-dimensional parabolic equation subject to nonclassical boundary specifications, Numer Methods Partial Differential Eq 22 pp 220– (2006)
[14]	Dehghan, Time-splitting procedures for the solution of the two-dimensional transport equation, Kybernetes 36 pp 791– (2007) · Zbl 1193.93013
[15]	Dehghan, The one-dimensional heat equation subject to a boundary integral specification, Chaos, Solitons and Fractals 32 pp 661– (2007) · Zbl 1139.35352