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Uniformly Stable Explicitly Solvable Finite Difference Method for Fractional Diffusion Equations

Published online by Cambridge University Press:  06 March 2015

Hongxing Rui  and
Jian Huang
Hongxing Rui*
Affiliation:
School of Mathematics, Shandong University, Jinan 250100, China
Jian Huang
Affiliation:
School of Mathematics, Shandong University, Jinan 250100, China
*
*Corresponding author. Email addresses: hxrui@sdu.edu.cn (H.-X. Rui), yghuangjian@sina.com (J. Huang)
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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 l2 norm is optimal in some cases. Numerical results for several examples are consistent with the theoretical analysis.

Keywords

65M0665M1265M15Finite difference schemefractional diffusion equationuniformly stableexplicitly solvable methodasymmetric techniqueerror estimate

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Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 5 , Issue 1 , February 2015 , pp. 29 - 47
Copyright
Copyright © Global-Science Press 2015 

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