International Journal of Differential Equations

An Explicit Numerical Method for the Fractional Cable Equation

J. Quintana-Murillo and S. B. Yuste

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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.

Article information

Int. J. Differ. Equ., Volume 2011Special Issue (2011), Article ID 231920, 12 pages.

Received: 27 April 2011
Accepted: 30 June 2011
First available in Project Euclid: 26 January 2017

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Quintana-Murillo, J.; Yuste, S. B. An Explicit Numerical Method for the Fractional Cable Equation. Int. J. Differ. Equ. 2011 (2011), Article ID 231920, 12 pages. doi:10.1155/2011/231920.

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