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 - <p style="text-align: justify;">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 $2-\alpha$ $(0&lt;\alpha&lt;1)$. 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.</p>