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International Journal of Differential Equations
Volume 2011 (2011), Article ID 231920, 12 pages
http://dx.doi.org/10.1155/2011/231920
Research Article

An Explicit Numerical Method for the Fractional Cable Equation

Departamento de Física, Universidad de Extremadura, 06071 Badajoz, Spain

Received 27 April 2011; Accepted 30 June 2011

Academic Editor: Fawang Liu

Copyright © 2011 J. Quintana-Murillo and S. B. Yuste. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

An explicit numerical method to solve a fractional cable equation which involves two temporal Riemann-Liouville derivatives is studied. The numerical difference scheme is obtained by approximating the first-order derivative by a forward difference formula, the Riemann-Liouville derivatives by the Grünwald-Letnikov formula, and the spatial derivative by a three-point centered formula. The accuracy, stability, and convergence of the method are considered. The stability analysis is carried out by means of a kind of von Neumann method adapted to fractional equations. The convergence analysis is accomplished with a similar procedure. The von-Neumann stability analysis predicted very accurately the conditions under which the present explicit method is stable. This was thoroughly checked by means of extensive numerical integrations.