Numerical algorithms of the two-dimensional Feynman-Kac equation for reaction and diffusion processes

TL;DR

This paper develops finite difference schemes for the two-dimensional backward Feynman-Kac equation involving non-local operators, enabling accurate numerical solutions for particle reaction and diffusion processes without requiring high regularity assumptions.

Contribution

It introduces first- and second-order convolution quadrature schemes for discretizing the time tempered fractional derivative, and finite difference methods for the tempered fractional Laplacian, with verified convergence.

Findings

Numerical schemes achieve predicted convergence orders.

Schemes effectively handle non-local operators in reaction-diffusion models.

Numerical examples confirm scheme accuracy and efficiency.

Abstract

This paper provides a finite difference discretization for the backward Feynman-Kac equation, governing the distribution of functionals of the path for a particle undergoing both reaction and diffusion [Hou and Deng, J. Phys. A: Math. Theor., {\bf51}, 155001 (2018)]. Numerically solving the equation with the time tempered fractional substantial derivative and tempered fractional Laplacian consists in discretizing these two non-local operators. Here, using convolution quadrature, we provide a first-order and second-order schemes for discretizing the time tempered fractional substantial derivative, which doesn't require the assumption of the regularity of the solution in time; we use the finite difference method to approximate the two-dimensional tempered fractional Laplacian, and the accuracy of the scheme depends on the regularity of the solution on $\bar{\Omega}$ rather than the whole space. Lastly, we verify the predicted convergence orders and the effectiveness of the presented schemes by numerical examples.