Numerical Solution for the Fractional Wave Equation Using Pseudo-Spectral Method Based on the Generalized Laguerre Polynomials

Affiliation(s)

^{1}
Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt.

^{2}
Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Kingdom of Saudi Arabia.

^{3}
Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt.

Abstract

In this paper, an efficient numerical method is considered for solving the fractional wave equation (FWE). The fractional derivative is described in the Caputo sense. The method is based on Laguerre approximations. The properties of Laguerre polynomials are utilized to reduce FWE to a system of ordinary differential equations, which is solved by the finite difference method. An approximate formula of the fractional derivative is given. Special attention is given to study the convergence analysis and estimate an error upper bound of the presented formula. Numerical solutions of FWE are given and the results are compared with the exact solution.

In this paper, an efficient numerical method is considered for solving the fractional wave equation (FWE). The fractional derivative is described in the Caputo sense. The method is based on Laguerre approximations. The properties of Laguerre polynomials are utilized to reduce FWE to a system of ordinary differential equations, which is solved by the finite difference method. An approximate formula of the fractional derivative is given. Special attention is given to study the convergence analysis and estimate an error upper bound of the presented formula. Numerical solutions of FWE are given and the results are compared with the exact solution.

Keywords

Fractional Wave Equation, Caputo Derivative, Finite Difference Method, Laguerre Polynomials, Convergence Analysis

Fractional Wave Equation, Caputo Derivative, Finite Difference Method, Laguerre Polynomials, Convergence Analysis

Cite this paper

Sweilam, N. , Khader, M. and Adel, M. (2015) Numerical Solution for the Fractional Wave Equation Using Pseudo-Spectral Method Based on the Generalized Laguerre Polynomials.*Applied Mathematics*, **6**, 647-654. doi: 10.4236/am.2015.64058.

Sweilam, N. , Khader, M. and Adel, M. (2015) Numerical Solution for the Fractional Wave Equation Using Pseudo-Spectral Method Based on the Generalized Laguerre Polynomials.

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