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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.

MSC:

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

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