Efficient Jacobian Spectral Collocation Method for Spatio-Dependent Temporal Tempered Fractional Feynman-Kac Equation

Anomalous and non-ergodic diffusion is ubiquitous in the natural world. Fractional Feynman-Kac equations are used to characterize the functional distribution of the trajectories of the sub-diffusion particles which evolve slower than the Brownian motion. In this paper, by introducing the tempered fractional Jacobi functions (TFJFs), an efficient spectral collocation method is presented for solving the spatio-dependent temporal tempered fractional Feynman-Kac (STTFFK) equation. The tempered fractional differentiation matrix in the collocation scheme is generated by a recurrence relation which is fast and stable. The error estimate for the scheme is derived which shows that the proposed method is “spectral accuracy” if the solution is smooth. Several numerical experiments with different tempered factors are performed to support the theoretical results.