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 $${}^S{D}_t^{\gamma ,\tilde{\lambda }}G(x,p,t)=D_t^{\gamma ,\tilde{\lambda }}G(x,p,t)-{\lambda }^{\gamma }G(x,p,t)\mathrm{with}\tilde{\lambda }=\lambda +pU(x),p=\rho +J\eta ,J=\sqrt{-1},$$ where $$D_t^{\gamma ,\tilde{\lambda }}G(x,p,t)=\frac{1}{\Gamma (1-\gamma )}\left[\frac{\partial }{\partial t}+\tilde{\lambda }\right]{\int }_0^t{\left(t-z\right)}^{-\gamma }{e}^{-\tilde{\lambda }\cdot (t-z)}G(x,p,z)dz,$$ and $\lambda \ge 0$, $0<\gamma <1$, $\rho >0$, and $\eta$ is a real number. The designed schemes are unconditionally stable and have the global truncation error $\mathcal{O}({\tau }^2+h^2)$, 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).

arXiv comment21 pages, 4 figures

References in corpus (5)

• Tempered Fractional Feynman-Kac Equation
• A fractional Feynman-Kac equation for weak ergodicity breaking
• Numerical algorithms for the forward and backward fractional Feynman-Kac equations
• Mean exit time and escape probability for the anomalous processes with the tempered power-law waiting times
• Numerical schemes of the time tempered fractional Feynman-Kac equation