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An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations. (English) Zbl 1119.65379

Summary: A numerical method for solving the fractional diffusion equation, which could also be easily extended to other fractional partial differential equations, is considered. In this paper we combine the forward time centered space (FTCS) method, well known for the numerical integration of ordinary diffusion equations, with the Grünwald-Letnikov discretization of the Riemann-Liouville derivative to obtain an explicit FTCS scheme for solving the fractional diffusion equation. The stability analysis of this scheme is carried out by means of a powerful and simple new procedure close to the well-known von Neumann method for nonfractional partial differential equations. The analytical stability bounds are in excellent agreement with numerical test. A comparison between exact analytical solutions and numerical predictions is made.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
26A33 Fractional derivatives and integrals
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