Uniformly Stable Explicitly Solvable Finite Difference Method for Fractional Diffusion Equations

Uniformly Stable Explicitly Solvable Finite Difference Method for Fractional Diffusion Equations

Year:    2015

East Asian Journal on Applied Mathematics, Vol. 5 (2015), Iss. 1 : pp. 29–47

Abstract

A finite difference scheme for the one-dimensional space fractional diffusion equation is presented and analysed. The scheme is constructed by modifying the shifted Grünwald approximation to the spatial fractional derivative and using an asymmetric discretisation technique. By calculating the unknowns in differential nodal point sequences at the odd and even time levels, the discrete solution of the scheme can be obtained explicitly. We prove that the scheme is uniformly stable. The error between the discrete solution and the analytical solution in the discrete $l^2$ norm is optimal in some cases. Numerical results for several examples are consistent with the theoretical analysis.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/eajam.030614.051114a

East Asian Journal on Applied Mathematics, Vol. 5 (2015), Iss. 1 : pp. 29–47

Published online:    2015-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    19

Keywords:    Finite difference scheme fractional diffusion equation uniformly stable explicitly solvable method asymmetric technique error estimate.

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