论文标题
二维时间回火分数fokker-Planck方程的尖锐$α$ -Robust $ l1 $方案
A sharp $α$-robust $L1$ scheme on graded meshes for two-dimensional time tempered fractional Fokker-Planck equation
论文作者
论文摘要
在本文中，我们关注的是用于二维时间分数fokker-planck方程的数值解决方案，并具有频率$α$的回火分数导数。尽管在许多最近的数值分析论文中考虑了其某些变体，但仍然存在一些显着差异。在这里，我们首先提供解决方案的规律性估计。然后，采用了由中间矩形正交公式启发的修改后的1 $ 1方案，用于补偿$ t \ rightarrow 0^{+} $的解决方案的奇异性，而五点差异方案在太空中使用。稳定性和收敛性在$ l^{\ infty} $ norm的状态下得到证明，然后是急剧误差估计$ \ mathscr {o}（τ^{\ min \ {2- {2-α，rα\}}）$。此外，与以前的作品中证明的边界不同，分析中的恒定乘数不会爆炸，因为Caputo分数衍生物$α$接近1的经典值。最后，我们执行数值实验来验证提出的算法的有效性和收敛顺序。
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 $α$. 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 $\mathscr{O}(τ^{\min\{2-α, rα\}})$ 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 $α$ approaches the classical value of 1. Finally, we perform the numerical experiments to verify the effectiveness and convergence order of the presented algorithms.
