Numerical schemes of the time tempered fractional Feynman-Kac equation

TL;DR

This paper develops and analyzes finite difference and finite element schemes for numerically solving the backward time tempered fractional Feynman-Kac equation, demonstrating their stability, convergence, and effectiveness through numerical experiments.

Contribution

It introduces discretization methods for the tempered fractional derivative and designs stable, convergent numerical schemes for the equation, advancing computational approaches for this model.

Findings

Numerical schemes are stable and convergent.

Schemes effectively solve the tempered fractional Feynman-Kac equation.

Numerical experiments confirm the accuracy and efficiency of the methods.

Abstract

This paper focuses on providing the computation methods for the backward time tempered fractional Feynman-Kac equation, being one of the models recently proposed in [Wu, Deng, and Barkai, Phys. Rev. E, 84 (2016) 032151]. The discretization for the tempered fractional substantial derivative is derived, and the corresponding finite difference and finite element schemes are designed with well established stability and convergence. The performed numerical experiments show the effectiveness of the presented schemes.