Volume 7, Issue 3
Numerical Solution of the Time-Fractional Sub-Diffusion Equation on an Unbounded Domain in Two-Dimensional Space

Hongwei Li, Xiaonan Wu & Jiwei Zhang

East Asian J. Appl. Math., 7 (2017), pp. 439-454.

Published online: 2018-02

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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 equation, unbounded domain, local artificial boundary conditions, finite difference method, Caputo time-fractional derivative.

  • AMS Subject Headings

65M06, 65M12, 65N06

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-7-439, author = {}, title = {Numerical Solution of the Time-Fractional Sub-Diffusion Equation on an Unbounded Domain in Two-Dimensional Space}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {7}, number = {3}, pages = {439--454}, 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.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.031116.080317a}, url = {http://global-sci.org/intro/article_detail/eajam/10758.html} }
TY - JOUR T1 - Numerical Solution of the Time-Fractional Sub-Diffusion Equation on an Unbounded Domain in Two-Dimensional Space JO - East Asian Journal on Applied Mathematics VL - 3 SP - 439 EP - 454 PY - 2018 DA - 2018/02 SN - 7 DO - http://doi.org/10.4208/eajam.031116.080317a UR - https://global-sci.org/intro/article_detail/eajam/10758.html KW - Fractional sub-diffusion equation, unbounded domain, local artificial boundary conditions, finite difference method, Caputo time-fractional derivative. AB -

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.

Hongwei Li, Xiaonan Wu & Jiwei Zhang. (2020). Numerical Solution of the Time-Fractional Sub-Diffusion Equation on an Unbounded Domain in Two-Dimensional Space. East Asian Journal on Applied Mathematics. 7 (3). 439-454. doi:10.4208/eajam.031116.080317a
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