Numerical Algorithms of the Two-dimensional Feynman–Kac Equation for Reaction and Diffusion Processes

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., 51, 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 onΩ rather than the whole space. Lastly, we verify the predicted convergence orders and the effectiveness of the presented schemes by numerical examples.

Finite difference method for inhomogeneous fractional Dirichlet problem

“…Compared with the discretizations in [9,10], our scheme can deal with the inhomogeneous fractional Dirichlet problem more easily and accurately. Different from the discretizations proposed in [16,18], the current discretization can produce a Toeplitz matrix in one-dimensional case and a block-Toeplitz-Toeplitz-block for two-dimensional case; so fast Fourier Transform can be directly used to speed up the evaluation [19]. Besides, we use some examples to verify the effectiveness of the designed scheme, including truncation errors, convergence, and the simulation of the mean exit time of Lévy motion with generator A = ∇P (x) • ∇ + (−∆) s ; the detailed results can refer to Section 5.…”

“…Since the singularity and non-locality, numerical approximation of fractional Laplacian is still a challenging topic. In the past few decades, finite difference method has been widely used to approximate fractional derivatives [6,7,8,9,10,11,12,13,14,15,16,17,18]. Among them, [12,13,14,15] discretize time fractional Caputo derivative by L 1 method and convolution quadrature method; [8,17] provide weighted and shifted Grünwald difference method to discretize fractional Riesz derivative; as for fractional Laplacian, [6,9,10,11] propose the finite difference scheme for solving d-dimensional (d = 1, 2, 3) fractional Laplace equation with homogeneous Dirichlet boundary condition; moreover, the finite difference schemes provided in [16,18] for tempered fractional Laplacian with λ = 0 still apply to fractional Laplacian.…”

Error estimates for backward fractional Feynman-Kac equation with non-smooth initial data

“…So far there have been many works for fractional partial differential equations, including the finite difference method, finite element method, spectral method, and so on [1,3,7,8,9,10,14,19,25], but there are relatively less researches on solving fractional Feynman-Kac equation numerically [6,11,12,13,23]. The main reasons are that fractional substantial derivative is a time-space coupled non-local operator and the equation covers the complex parameters which bring about many challenges on regularity and numerical analyses.…”