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Numerical approximation of Lévy-Feller fractional diffusion equation via Chebyshev-Legendre collocation method

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Abstract.

This paper reports a new spectral algorithm for obtaining an approximate solution for the Lévy-Feller diffusion equation depending on Legendre polynomials and Chebyshev collocation points. The Lévy-Feller diffusion equation is obtained from the standard diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative. A new formula expressing explicitly any fractional-order derivatives, in the sense of Riesz-Feller operator, of Legendre polynomials of any degree in terms of Jacobi polynomials is proved. Moreover, the Chebyshev-Legendre collocation method together with the implicit Euler method are used to reduce these types of differential equations to a system of algebraic equations which can be solved numerically. Numerical results with comparisons are given to confirm the reliability of the proposed method for the Lévy-Feller diffusion equation.

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Authors and Affiliations

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

    N. H. Sweilam & M. M. Abou Hasan

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  1. N. H. Sweilam
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  2. M. M. Abou Hasan
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Correspondence to N. H. Sweilam.

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Sweilam, N.H., Abou Hasan, M.M. Numerical approximation of Lévy-Feller fractional diffusion equation via Chebyshev-Legendre collocation method. Eur. Phys. J. Plus 131, 251 (2016). https://doi.org/10.1140/epjp/i2016-16251-y

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