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**Legendre multiwavelet Galerkin method for solving the hyperbolic telegraph equation.**
*(English)*
Zbl 1189.65231

Summary: Recently, it is found that telegraph equation is more suitable than ordinary diffusion equation in modeling reaction diffusion for such branches of sciences. In this article a numerical method for solving the one-dimensional hyperbolic telegraph equation is presented. The method is based upon Legendre multiwavelet approximations. The properties of Legendre multiwavelet are first presented. These properties together with the Galerkin method are then utilized to reduce the telegraph equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

### MSC:

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

35L15 | Initial value problems for second-order hyperbolic equations |

65T60 | Numerical methods for wavelets |

### Keywords:

Galerkin method; Legendre multiwavelet; second-order hyperbolic telegraph equation; numerical examples
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\textit{S. A. Yousefi}, Numer. Methods Partial Differ. Equations 26, No. 3, 535--543 (2010; Zbl 1189.65231)

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### References:

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

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

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

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

[5] | Mohanty, An unconditionally stable alternating direction implicit scheme for the two space dimensional linear hyperbolic equation, Numer Methods Partial Differential Equations 17 pp 684– (2001) · Zbl 0990.65101 |

[6] | 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 |

[7] | Cascaval, Fractional telegraph equations, J Math Anal Appl 276 pp 145– (2002) · Zbl 1038.35142 |

[8] | Eckstein, The mathematics of suspensions: kac walks and asymptotic analyticity, Electron J Differential Equations Conf 3 pp 39– (1999) · Zbl 0963.76090 |

[9] | Eckstein, Linking theory and measurements of tracer particle position in suspension flows, Proc ASME FEDSM 251 pp 1– (2000) |

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

[11] | Alonso, Bounded solutions of second order semilinear evolution equations and applications to the telegraph equation, J Math Pure Appl 78 pp 49– (1999) · Zbl 0927.34048 |

[12] | 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 |

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

[14] | Dehghan, The use of Chebyshev cardinal functions for solution of the second-order one-dimensional telegraph equation, Numer Methods Partial Differential Equations |

[15] | Chui, Wavelets: a mathematical tool for signal analysis (1997) · Zbl 0903.94007 |

[16] | Ming, The computation of wavelet-Galerkin approximation on a bounded interval, Int J Numer Methods Eng 39 pp 2921– (1996) · Zbl 0884.76058 |

[17] | Beylkin, Fast wavelet transforms and numerical algorithms, Commun Pure Appl Math 44 pp 141– (1991) · Zbl 0722.65022 |

[18] | Gu, The Haar wavelets operational matrix of integration, Int J Sys Sci 27 pp 623– (1996) · Zbl 0875.93116 |

[19] | Razzaghi, Legendre wavelets direct method for variational problems, Math Comput Simul 53 pp 185– (2000) |

[20] | Yousefi, Legendre wavelets method for the nonlinear Volterra-Fredholm integral equations, Math Comput Simul 70 pp 1– (2005) · Zbl 1205.65342 |

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