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**Numerical solution of hyperbolic telegraph equation using the Chebyshev tau method.**
*(English)*
Zbl 1186.65136

The authors give a numerical scheme to solve the one-dimensional hyperbolic telegraph equation. The approach consists of reducing the problem to a set of algebraic equations by expanding the approximate solution in terms of shifted Chebyshev polynomials with unknown coefficients. The operational matrices of the integral and the derivative are given and these matrices together with the tau method are then utilized to evaluate the unknown coefficients of shifted Chebyshev polynomials.

Reviewer: Nicolae Pop (Baia Mare)

### MSC:

65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |

35L05 | Wave equation |

### Keywords:

Chebyshev polynomials; hyperbolic equation; operational matrix; tau method; telegraph equation
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\textit{A. Saadatmandi} and \textit{M. Dehghan}, Numer. Methods Partial Differ. Equations 26, No. 1, 239--252 (2010; Zbl 1186.65136)

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### References:

[1] | Gao, Unconditionally stable difference schemes for a one-space-dimensional linear hyperbolic equation, Appl Math Comput 187 pp 1272– (2007) · Zbl 1114.65347 |

[2] | Dehghan, A numerical method for solving the hyperbolic telegraph equation, Numer Methods Partial Differential Eq 24 pp 1080– (2008) |

[3] | Mohebbi, High order compact solution of the one-space-dimensional linear hyperbolic equation, Numer Methods Partial Differential Eq 24 pp 1222– (2008) |

[4] | Saadatmandi, Numerical solution of the one-dimensional wave equation with an integral condition, Numer Methods Partial Differential Eq 23 pp 282– (2007) |

[5] | Dehghan, Variational iteration method for solving the wave equation subject to an integral conservation condition, Chaos Solitons Fractals · Zbl 1198.65202 |

[6] | 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 Comp Appl Math 72 pp 421– (1996) |

[7] | El-Azab, A numerical algorithm for the solution of telegraph equations, Appl Math Comput 190 pp 757– (2007) · Zbl 1132.65087 |

[8] | Metaxas, Industrial microwave, Heating (1993) |

[9] | Roussy, Foundations and industrial applications of microwaves and radio frequency fields (1995) |

[10] | Aloy, Computing the variable coefficient telegraph equation using a discrete eigenfunctions method, Comput Math Appl 54 pp 448– (2007) · Zbl 1127.65073 |

[11] | Mohanty, An unconditionally stable alternating direction implicit scheme for the two space dimensional linear hyperbolic equation, Numer Methods Partial Differential Eq 17 pp 684– (2001) |

[12] | Mohanty, An unconditionally stable ADI method for the linear hyperbolic equation in three space dimensions, Int J Comput Math 79 pp 133– (2002) · Zbl 0995.65093 |

[13] | Guo, The state of art in spectral methods (1996) |

[14] | Lanczos, Trigonometric interpolation of empirical and analytic functions, J Math Phys 17 pp 123– (1938) · Zbl 0020.01301 · doi:10.1002/sapm1938171123 |

[15] | Lanczos, Applied analysis (1957) |

[16] | Ortiz, The tau method, SIAM J Numer Anal Optim 12 pp 480– (1969) · Zbl 0195.45701 · doi:10.1137/0706044 |

[17] | Ortiz, Numerical solution of partial differential equations with variable coefficients with an operational approach to the tau method, Comput Math Appl 10 pp 5– (1984) · Zbl 0575.65118 |

[18] | Abadi, The algebraic kernel method for the numerical solution of partial differential equations, J Numer Funct Anal Optim 12 pp 339– (1991) · Zbl 0787.65080 |

[19] | Canuto, Spectral methods in fluid dynamic (1988) · doi:10.1007/978-3-642-84108-8 |

[20] | Cordero, The algebraic kernel method for the numerical solution of partial differential equations, J Numer Funct Anal Optim 12 pp 339– (1991) |

[21] | Dehghan, A tau method for the one-dimensional parabolic inverse problem subject to temperature overspecification, Comput Math Appl 52 pp 933– (2006) · Zbl 1125.65340 |

[22] | Saadatmandi, Hartley series approximations for the parabolic equations, Int J Comput Math 82 pp 1149– (2005) · Zbl 1075.65128 |

[23] | Saadatmandi, Numerical solution of a mathematical model for capillary formation in tumor angiogenesis via the tau method, Commun Numer Methods Eng 24 pp 1467– (2008) · Zbl 1151.92017 · doi:10.1002/cnm.1045 |

[24] | Elbarbary, Chebyshev finite difference approximation for the boundary value problems, Appl Math Comput 139 pp 513– (2003) · Zbl 1027.65098 |

[25] | Chen, Frobenius-Chebyshev polynomial approximations with a priori error bounds for nonlinear initial value differential problems, Comput Math Appl 41 pp 269– (2001) · Zbl 1001.65073 |

[26] | Chen, The truncation error of the two-variable Chebyshev series expansions, Comput Math Appl 45 pp 1647– (2003) · Zbl 1064.65085 |

[27] | Horng, Application of shifted Chebyshev series to the optimal control of linear distributed-parameter systems, Int J Control 42 pp 233– (1985) · Zbl 0566.93028 |

[28] | Momani, Analytic and approximate solutions of the space and time-fractional telegraph equations, Appl Math Comput 170 pp 1126– (2005) · Zbl 1103.65335 |

[29] | Dehghan, Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Math Comput Simul 71 pp 16– (2006) · Zbl 1089.65085 |

[30] | Dehghan, On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation, Numer Methods for Partial Differential Eq 21 pp 24– (2005) · Zbl 1059.65072 |

[31] | Dehghan, Identification of a time-dependent coefficient in a partial differential equation subject to an extra measurement, Numer Methods for Partial Differential Eq 21 pp 611– (2005) · Zbl 1069.65104 |

[32] | Dehghan, Implicit collocation technique for heat equation with non-classic initial condition, International Journal of Nonlinear Sciences and Numerical Simulation 7 pp 447– (2006) · Zbl 06942230 · doi:10.1515/IJNSNS.2006.7.4.461 |

[33] | Dehghan, A computational study of the one-dimensional parabolic equation subject to nonclassical boundary specifications, Numer Methods for Partial Differential Eq 22 pp 220– (2006) |

[34] | Dehghan, The one-dimensional heat equation subject to a boundary integral specification, Chaos, Solitons and Fractals 32 pp 661– (2007) · Zbl 1139.35352 |

[35] | Shakeri, Numerical solution of the Klein-Gordon equation via H’s variational iteration method, Nonlinear Dynamics 51 pp 89– (2008) · Zbl 1179.81064 |

[36] | Dehghan, The boundary integral equation approach for numerical solution of the one-dimensional Sine-Gordon equation, Numer Methods for Partial Differential Eq 24 pp 1405– (2008) |

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