Numerical approximation of Lévy-Feller fractional diffusion equation via Chebyshev-Legendre collocation method
Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt
* e-mail: email@example.com
Revised: 12 June 2016
Accepted: 16 June 2016
Published online: 4 August 2016
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.
© Società Italiana di Fisica and Springer-Verlag Berlin Heidelberg, 2016