NUMERICAL SIMULATIONS FOR SPACE–TIME FRACTIONAL DIFFUSION EQUATIONS
NUMERICAL SIMULATIONS FOR SPACE–TIME FRACTIONAL DIFFUSION EQUATIONS
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DOI：
10.1142/s0219876213410016
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发表时间：
2013-03
影响因子：
1.7
通讯作者：
Leevan Ling;Masahiro Yamamoto
中科院分区：
文献类型：
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作者：
Leevan Ling;Masahiro Yamamoto
考虑区间[- 1,1]上的时空分数扩散方程的解。通过将二阶空间导数替换为1 ~ 2阶的Riemann-Liouville分数阶导数，将一阶时间导数替换为0 ~ 1阶的Caputo分数阶导数，得到了该方程。由于分数阶扩散方程的基本解是未知的（如果存在的话），采用本征函数法得到近似的基本解，然后用近似的基本解求解具有初值和边值的时空分数阶扩散方程。数值结果证明了该方法在长时间模拟中的有效性。
We consider the solutions of a space–time fractional diffusion equation on the interval [-1, 1]. The equation is obtained from the standard diffusion equation by replacing the second-order space derivative by a Riemann–Liouville fractional derivative of order between one and two, and the first-order time derivative by a Caputo fractional derivative of order between zero and one. As the fundamental solution of this fractional equation is unknown (if exists), an eigenfunction approach is applied to obtain approximate fundamental solutions which are then used to solve the space–time fractional diffusion equation with initial and boundary values. Numerical results are presented to demonstrate the effectiveness of the proposed method in long time simulations.