High order algorithm for the time-tempered fractional Feynman-Kac equation

Abstract

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&#39; equation (without artificial source term).