This paper focuses on the design of an efficient numerical approach for solving a two-dimensional multi-term Caputo time fractional Fokker-Planck (TFFP) model. The solution of such problem, in general, shows a weak singularity at the time origin. A numerical technique based on a graded time mesh is proposed to handle the singular behavior of the solution. The multi-term Caputo time fractional derivatives in the TFFP model are discretized by means of the L1 scheme on the nonuniform mesh, while a high-order compact alternating direction implicit finite difference scheme is designed to approximate the spatial derivatives. Convergence and stability analysis of the suggested method is analyzed. Two numerical examples subjected to smooth and nonsmooth exact solutions are presented to demonstrate the applicability and accuracy of the method. The results obtained by the proposed graded mesh technique are compared with the results obtained by the uniform mesh technique.

中文翻译：

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一种渐变网格技术，用于在二维空间中对多项 Caputo 时间分数 Fokker-Planck 方程进行数值逼近


本文重点介绍了一种求解二维多项 Caputo 时间分数法 Fokker-Planck （TFFP） 模型的高效数值方法的设计。一般来说，这个问题的解在时间原点处显示出一个弱奇点。提出了一种基于分级时间网格的数值技术来处理解的奇异行为。TFFP 模型中的多项 Caputo 时间分数阶导数通过非均匀网格上的 L1 方案进行离散化，同时设计了高阶紧凑交替方向隐式有限差分方案来近似空间导数。分析了所建议方法的收敛性和稳定性分析。给出了两个受 smooth 和非smooth 精确解的数值算例，以验证该方法的适用性和准确性。将所提出的分级网格技术获得的结果与均匀网格技术获得的结果进行了比较。