Volume 47, Issue 2
Finite Difference/Collocation Method for Two-Dimensional Sub-Diffusion Equation with Generalized Time Fractional Derivative

J. Math. Study, 47 (2014), pp. 173-189.

Published online: 2014-06

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• Abstract

In this paper, we propose a finite difference/collocation method for two-dimensional time fractional diffusion equation with generalized fractional operator. The main purpose of this paper is to design a high order numerical scheme for the new generalized time fractional diffusion equation. First, a finite difference approximation formula is derived for the generalized time fractional derivative, which is verified with order $2-\alpha$ $(0<\alpha<1)$. Then, collocation method is introduced for the two-dimensional space approximation. Unconditional stability of the scheme is proved. To make the method more efficient, the alternating direction implicit method is introduced to reduce the computational cost. At last, numerical experiments are carried out to verify the effectiveness of the scheme.

• Keywords

Time fractional diffusion equation, generalized fractional operator, collocation method, alternating direction implicit method.

35R11, 65M70, 65M12

qw_xu@hotmail.com (Qinwu Xu)

2009zhengzhoushun@163.com (Zhoushun Zheng)

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@Article{JMS-47-173, author = {Xu , Qinwu and Zheng , Zhoushun}, title = {Finite Difference/Collocation Method for Two-Dimensional Sub-Diffusion Equation with Generalized Time Fractional Derivative}, journal = {Journal of Mathematical Study}, year = {2014}, volume = {47}, number = {2}, pages = {173--189}, abstract = {

In this paper, we propose a finite difference/collocation method for two-dimensional time fractional diffusion equation with generalized fractional operator. The main purpose of this paper is to design a high order numerical scheme for the new generalized time fractional diffusion equation. First, a finite difference approximation formula is derived for the generalized time fractional derivative, which is verified with order $2-\alpha$ $(0<\alpha<1)$. Then, collocation method is introduced for the two-dimensional space approximation. Unconditional stability of the scheme is proved. To make the method more efficient, the alternating direction implicit method is introduced to reduce the computational cost. At last, numerical experiments are carried out to verify the effectiveness of the scheme.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v47n2.14.03}, url = {http://global-sci.org/intro/article_detail/jms/9953.html} }
TY - JOUR T1 - Finite Difference/Collocation Method for Two-Dimensional Sub-Diffusion Equation with Generalized Time Fractional Derivative AU - Xu , Qinwu AU - Zheng , Zhoushun JO - Journal of Mathematical Study VL - 2 SP - 173 EP - 189 PY - 2014 DA - 2014/06 SN - 47 DO - http://doi.org/10.4208/jms.v47n2.14.03 UR - https://global-sci.org/intro/article_detail/jms/9953.html KW - Time fractional diffusion equation, generalized fractional operator, collocation method, alternating direction implicit method. AB -

In this paper, we propose a finite difference/collocation method for two-dimensional time fractional diffusion equation with generalized fractional operator. The main purpose of this paper is to design a high order numerical scheme for the new generalized time fractional diffusion equation. First, a finite difference approximation formula is derived for the generalized time fractional derivative, which is verified with order $2-\alpha$ $(0<\alpha<1)$. Then, collocation method is introduced for the two-dimensional space approximation. Unconditional stability of the scheme is proved. To make the method more efficient, the alternating direction implicit method is introduced to reduce the computational cost. At last, numerical experiments are carried out to verify the effectiveness of the scheme.

Qinwu Xu & Zhoushun Zheng. (2019). Finite Difference/Collocation Method for Two-Dimensional Sub-Diffusion Equation with Generalized Time Fractional Derivative. Journal of Mathematical Study. 47 (2). 173-189. doi:10.4208/jms.v47n2.14.03
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