具有Caputo-Fabrizio分数阶导数的Fokker-Planck方程的高阶算法
High order algorithms for Fokker-Planck equation with Caputo-Fabrizio fractional derivative
摘要：
基于连续时间的随机游动，导出了具有Caputo-Fabrizio分数阶导数的Fokker-Planck方程，该方程可以有效地模拟各种物理现象，特别是不同尺度的材料非均匀性和结构。扩展分数实体微积分的离散化[Chen and Deng，\emph{ ESAIM: M2AN.} \textbf{49}，（2015），373--394]，我们首先提供具有全局截断误差$\mathcal{O}(τ^ν)$ $ (ν=1,2,3,4)$的Caputo-Fabrizio分数导数的数值离散化。然后我们利用所导出的格式求解Caputo-Fabrizio分数阶扩散方程。通过分析离散化Caputo-Fabrizio算子刚度矩阵的正定性，理论上证明了离散化Caputo-Fabrizio算子的无条件稳定性和具有全局截断误差$\mathcal{O}(τ^2+h^2)$的收敛性，并进行了数值验证.
英文摘要：
Based on the continuous time random walk, we derive the Fokker-Planck equations with Caputo-Fabrizio fractional derivative, which can effectively model a variety of physical phenomena, especially, the material heterogeneities and structures with different scales. Extending the discretizations for fractional substantial calculus [Chen and Deng, \emph{ ESAIM: M2AN.} \textbf{49}, (2015), 373--394], we first provide the numerical discretizations of the Caputo-Fabrizio fractional derivative with the global truncation error $\mathcal{O}(τ^ν)$ $ (ν=1,2,3,4)$. Then we use the derived schemes to solve the Caputo-Fabrizio fractional diffusion equation. By analysing the positive definiteness of the stiffness matrices of the discretized Caputo-Fabrizio operator, the unconditional stability and the convergence with the global truncation error $\mathcal{O}(τ^2+h^2)$ are theoretically proved and numerical verified.
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