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.
J. Quintana-Murillo. S. B. Yuste. "An Explicit Numerical Method for the Fractional Cable Equation." Int. J. Differ. Equ. 2011 (SI1) 1 - 12, 2011. https://doi.org/10.1155/2011/231920