In this paper, we are concerned with the numerical solution for the two-dimensional time fractional Fokker-Planck equation with tempered fractional derivative of order $\alpha$. Although some of its variants are considered in many recent numerical analysis papers, there are still some significant differences. Here we first provide the regularity estimates of the solution. And then a modified $L$1 scheme inspired by the middle rectangle quadrature formula on graded meshes is employed to compensate for the singularity of the solution at $t\rightarrow 0^{+}$, while the five-point difference scheme is used in space. Stability and convergence are proved in the sence of $L^{\infty}$ norm, then a sharp error estimate $\mathscr{O}(\tau^{\min\{2-\alpha, r\alpha\}})$ is derived on graded meshes. Furthermore, unlike the bounds proved in the previous works, the constant multipliers in our analysis do not blow up as the Caputo fractional derivative $\alpha$ approaches the classical value of 1. Finally, we perform the numerical experiments to verify the effectiveness and convergence order of the presented algorithms.
翻译：在本文中, 我们对二维时间分解法克-普朗克方程式的数值解决方案表示担忧, 该方程式中含有低温的分衍生物 $\ alpha$。 虽然最近许多数字分析文件中考虑了其中的一些变量, 但仍有一些显著的差异。 我们首先提供解决方案的规律性估计值。 然后, 我们根据分级的 meshes 的中矩形二次方程式, 采用了一个修改后的 $1 美元方案, 以补偿以 $t\ rightror 0. $ $ $ 美元为单位的解决方案的单一性, 而五点差异方案则在空间中使用。 稳定和趋同以 $ $ / infty} 标准为例, 其稳定性和趋同性得到了证明, 然后一个严重错误估计 $\ mathscr{O} (\\\\\\ min\ 2-\\\\\\\ \\ ALpha} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
相关内容
微信扫码咨询专知VIP会员