Chebyshev Pseudo-Spectral Method for Solving Fractional Advection-Dispersion Equation

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, KSA.

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

Abstract

Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional Advection-dispersion equation (ADE) is considered. The fractional derivative is described in the Caputo sense. The method is based on Chebyshev approximations. The properties of Chebyshev polynomials are used to reduce ADE to a system of ordinary differential equations, which are solved using the finite difference method (FDM). Moreover, the convergence analysis and an upper bound of the error for the derived formula are given. Numerical solutions of ADE are presented and the results are compared with the exact solution.

Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional Advection-dispersion equation (ADE) is considered. The fractional derivative is described in the Caputo sense. The method is based on Chebyshev approximations. The properties of Chebyshev polynomials are used to reduce ADE to a system of ordinary differential equations, which are solved using the finite difference method (FDM). Moreover, the convergence analysis and an upper bound of the error for the derived formula are given. Numerical solutions of ADE are presented and the results are compared with the exact solution.

Keywords

Fractional Advection-Dispersion Equation, Caputo Fractional Derivative, Finite Difference Method, Chebyshev Pseudo-Spectral Method, Convergence Analysis

Fractional Advection-Dispersion Equation, Caputo Fractional Derivative, Finite Difference Method, Chebyshev Pseudo-Spectral Method, Convergence Analysis

Cite this paper

Sweilam, N. , Khader, M. and Adel, M. (2014) Chebyshev Pseudo-Spectral Method for Solving Fractional Advection-Dispersion Equation.*Applied Mathematics*, **5**, 3240-3248. doi: 10.4236/am.2014.519301.

Sweilam, N. , Khader, M. and Adel, M. (2014) Chebyshev Pseudo-Spectral Method for Solving Fractional Advection-Dispersion Equation.

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