An explicit numerical method for the fractional cable equation. (English) Zbl 1237.65097

Summary: 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 the fractional equations. The convergence analysis is accomplished with a similar procedure. The von-Neumann stability analysis predicts very accurately the conditions under which the present explicit method is stable. This is thoroughly checked by means of extensive numerical integrations.


65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L10 Second-order hyperbolic equations
35R11 Fractional partial differential equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs


Full Text: DOI


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