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

作　　者：Jiahui Hu Jungang Wang Yufeng Nie Yanwei Luo 胡嘉卉;王俊刚;聂玉峰;罗艳伟(Department of Applied Mathematics,Northwestern Polytechnical University,Xi’an 710129,China;College of Science,Henan University of Technology,Zhengzhou 450001,China)

机构地区：[1]Department of Applied Mathematics,Northwestern Polytechnical University,Xi’an 710129,China [2]College of Science,Henan University of Technology,Zhengzhou 450001,China

出　　处：《Chinese Physics B》2019年第10期226-236,共11页中国物理B（英文版）

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

摘　　要：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 nonlocal 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.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 nonlocal 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.

关 键 词：BACKWARD FRACTIONAL Feynman-Kac EQUATION FRACTIONAL substantial DERIVATIVE compact finite difference scheme numerical inversion of LAPLACE transforms 

分 类 号：O4[理学—物理]