Numerical algorithms for the forward and backward fractional Feynman-Kac equations

Abstract

The Feynman-Kac equations are a type of partial differential equations describing the distribution of functionals of diffusive motion. The probability density function (PDF) of Brownian functionals satisfies the Feynman-Kac formula, being a Schrödinger equation in imaginary time. The functionals of no-Brownian motion, or anomalous diffusion, follow the fractional Feynman-Kac equation [J. Stat. Phys. 141, 1071-1092, 2010], where the fractional substantial derivative is involved. Based on recently developed discretized schemes for fractional substantial derivatives [arXiv:1310.3086], this paper focuses on providing algorithms for numerically solving the forward and backward fractional Feynman-Kac equations; since the fractional substantial derivative is non-local time-space coupled operator, new challenges are introduced comparing with the general fractional derivative. Two ways (finite difference and finite element) of discretizing the space derivative are considered. For the backward fractional Feynman-Kac equation, the numerical stability and convergence of the algorithms with first order accuracy are theoretically discussed; and the optimal estimates are obtained. For all the provided schemes, including the first order and high order ones, of both forward and backward Feynman-Kac equations, extensive numerical experiments are performed to show their effectiveness.

27 pages, 8 figures

References in corpus (2)

• Anomalous Diffusion of Inertial, Weakly Damped Particles
• On distributions of functionals of anomalous diffusion paths

Cited by in corpus (15)

• High order schemes for the tempered fractional diffusion equations
• High order algorithms for the fractional substantial diffusion equation with truncated Lévy flights
• Variational formulation and efficient implementation for solving the tempered fractional problems
• Numerical schemes of the time tempered fractional Feynman-Kac equation
• Error estimates for backward fractional Feynman-Kac equation with non-smooth initial data
• Correction of BDFk for fractional Feynman-Kac equation with Lévy flight
• Discretized fractional substantial calculus
• Analysis of a WSGD scheme for backward fractional Feynman-Kac equation with nonsmooth data
• Numerical algorithms of the two-dimensional Feynman-Kac equation for reaction and diffusion processes
• A second-order difference scheme for the time fractional substantial diffusion equation
• Maximum-norm error analysis of compact difference schemes for the backward fractional Feynman-Kac equation
• High order algorithm for the time-tempered fractional Feynman-Kac equation
• Local discontinuous Galerkin methods for the time tempered fractional diffusion equation
• An advanced meshless approach for the high-dimensional multi-term time-space-fractional PDEs on convex domains
• Multiresolution Galerkin method for solving the functional distribution of anomalous diffusion described by time-space fractional diffusion equation