高阶算法用于时间-温控分数费米-卡克方程
High order algorithm for the time-tempered fractional Feynman-Kac equation
AI总结：
本文提出并分析了高阶算法，用于描述粒子进行异常运动时的功能分布模型，该模型具有幂律跳跃长度和温控幂律等待时间。算法具有无条件稳定性，全局截断误差为O(τ²+h²)，并通过复杂空间中的理论证明和数值验证。
摘要：
我们为描述粒子进行异常运动时的功能分布模型提供了并分析了高阶算法。该模型来源于[Wu, Deng, and Barkai, Phys. Rev. E., 84 (2016), 032151]，被称为时间-温控分数费米-卡克方程。设计算法的关键步骤是离散化时间温控分数物质导数，定义为$${^S\!}D_t^{γ,\widetildeλ} G(x,p,t)\!=\!D_t^{γ,\widetildeλ} G(x,p,t)\!-\!λ^γG(x,p,t) ~{\rm with}~\widetildeλ=λ+ pU(x),\, p=ρ+Jη,\, J=\sqrt{-1}$$，其中$$D_t^{γ,\widetildeλ} G(x,p,t) =\frac{1}{Γ(1-γ)} \left[\frac{\partial}{\partial t}+\widetildeλ \right] \int_{0}^t{\left(t-z\right)^{-γ}}e^{-\widetildeλ\cdot(t-z)}{G(x,p,z)}dz$$，其中$λ\ge 0$，$0<γ<1$，$ρ>0$，且$η$是实数。设计的方案具有无条件稳定性，全局截断误差为$\mathcal{O}(τ^2+h^2)$，并在复杂空间中理论证明和数值验证。此外，还进行了首次通过时间分布的模拟，并且在解决'物理'方程（无人工源项）时也获得了二阶收敛性。
英文摘要：
We provide and analyze the high order algorithms for the model describing the functional distributions of particles performing anomalous motion with power-law jump length and tempered power-law waiting time. The model is derived in [Wu, Deng, and Barkai, Phys. Rev. E., 84 (2016), 032151], being called the time-tempered fractional Feynman-Kac equation. The key step of designing the algorithms is to discretize the time tempered fractional substantial derivative, being defined as $${^S\!}D_t^{γ,\widetildeλ} G(x,p,t)\!=\!D_t^{γ,\widetildeλ} G(x,p,t)\!-\!λ^γG(x,p,t) ~{\rm with}~\widetildeλ=λ+ pU(x),\, p=ρ+Jη,\, J=\sqrt{-1},$$ where $$D_t^{γ,\widetildeλ} G(x,p,t) =\frac{1}{Γ(1-γ)} \left[\frac{\partial}{\partial t}+\widetildeλ \right] \int_{0}^t{\left(t-z\right)^{-γ}}e^{-\widetildeλ\cdot(t-z)}{G(x,p,z)}dz,$$ and $λ\ge 0$, $0<γ<1$, $ρ>0$, and $η$ is a real number. The designed schemes are unconditionally stable and have the global truncation error $\mathcal{O}(τ^2+h^2)$, being theoretically proved and numerically verified in {\em complex} space. Moreover, some simulations for the distributions of the first passage time are performed, and the second order convergence is also obtained for solving the `physical' equation (without artificial source term).
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