Compact finite difference schemes for the backward fractional Feynman-Kac equation with fractional substantial derivative

doi:10.1088/1674-1056/ab3af3

Abstract The fractional Feynman-Kac equations describe the distributions of functionals of non-Brownian motion, or anomalous diffusion, including two types called the forward and backward fractional Feynman-Kac equations, where the non-local time-space coupled fractional substantial derivative is involved. This paper focuses on the more widely used backward version. Based on the newly proposed approximation operators for fractional substantial derivative, we establish compact finite difference schemes for the backward fractional Feynman-Kac equation. The proposed difference schemes have the q-th (q=1,2,3,4) order accuracy in temporal direction and fourth order accuracy in spatial direction, respectively. The numerical stability and convergence in the maximum norm are proved for the first order time discretization scheme by the discrete energy method, where an inner product in complex space is introduced. Finally, extensive numerical experiments are carried out to verify the availability and superiority of the algorithms. Also, simulations of the backward fractional Feynman-Kac equation with Dirac delta function as the initial condition are performed to further confirm the effectiveness of the proposed methods.

Keywords: backward fractional Feynman-Kac equation fractional substantial derivative compact finite difference scheme numerical inversion of Laplace transforms

Received: 15 June 2019
Revised: 23 July 2019
Accepted manuscript online:

PACS:	02.60.-x	(Numerical approximation and analysis)
	02.60.Cb	(Numerical simulation; solution of equations)
	02.70.Bf	(Finite-difference methods)

Fund: Project supported by the National Natural Science Foundation of China (Grant No. 11471262) and Henan University of Technology High-level Talents Fund, China (Grant No. 2018BS039).

Corresponding Authors: Yufeng Nie
E-mail: yfnie@nwpu.edu.cn

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Jiahui Hu(胡嘉卉), Jungang Wang(王俊刚), Yufeng Nie(聂玉峰), Yanwei Luo(罗艳伟) Compact finite difference schemes for the backward fractional Feynman-Kac equation with fractional substantial derivative 2019 Chin. Phys. B 28 100201

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Compact finite difference schemes for the backward fractional Feynman-Kac equation with fractional substantial derivative

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