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Numerical Solution of the Time-Fractional Sub-Diffusion Equation on an Unbounded Domain in Two-Dimensional Space

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Partial differential equations, boundary value problems

Published online by Cambridge University Press:  07 September 2017

Hongwei Li ,
Xiaonan Wu  and
Jiwei Zhang
Hongwei Li*
Affiliation:
School of Mathematics and Statistics, Shandong Normal University, Ji'nan 250014, China
Xiaonan Wu*
Affiliation:
Department of Mathematics, Hong Kong Baptist University, Hong Kong, China
Jiwei Zhang*
Affiliation:
Beijing Computational Science Research Center, Beijing 100193, China
*
*Corresponding author. Email addresses:hwli@sdnu.edu.cn (H. Li), xwu@hkbu.edu.hk (X. Wu), jwzhang@csrc.ac.cn (J. Zhang)
*Corresponding author. Email addresses:hwli@sdnu.edu.cn (H. Li), xwu@hkbu.edu.hk (X. Wu), jwzhang@csrc.ac.cn (J. Zhang)
*Corresponding author. Email addresses:hwli@sdnu.edu.cn (H. Li), xwu@hkbu.edu.hk (X. Wu), jwzhang@csrc.ac.cn (J. Zhang)
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Abstract

The numerical solution of the time-fractional sub-diffusion equation on an unbounded domain in two-dimensional space is considered, where a circular artificial boundary is introduced to divide the unbounded domain into a bounded computational domain and an unbounded exterior domain. The local artificial boundary conditions for the fractional sub-diffusion equation are designed on the circular artificial boundary by a joint Laplace transform and Fourier series expansion, and some auxiliary variables are introduced to circumvent high-order derivatives in the artificial boundary conditions. The original problem defined on the unbounded domain is thus reduced to an initial boundary value problem on a bounded computational domain. A finite difference and L1 approximation are applied for the space variables and the Caputo time-fractional derivative, respectively. Two numerical examples demonstrate the performance of the proposed method.

Keywords

Fractional sub-diffusion equationunbounded domainlocal artificial boundary conditionsfinite difference methodCaputo time-fractional derivative

MSC classification

Secondary: 65M06: Finite difference methods 65M12: Stability and convergence of numerical methods 65N06: Finite difference methods
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 3 , August 2017 , pp. 439 - 454
Copyright
Copyright © Global-Science Press 2017 

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