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Analysis of a WSGD scheme for backward fractional Feynman-Kac equation with nonsmooth data

Liyao Hao, Wenyi Tian

TL;DR

The paper tackles the numerical solution of the backward fractional Feynman-Kac equation with nonsmooth data, a problem that involves time-space nonlocal operators and complex parameters. It introduces a second-order time discretization based on the weighted and shifted Grünwald difference (WSGD) for the fractional substantial derivative, augmented with initial corrections to handle inhomogeneous terms. The authors establish rigorous error estimates showing $O(\tau^2)$ temporal convergence under mild data regularity for both homogeneous and certain inhomogeneous cases, and corroborate these results with numerical experiments in 1D and 2D. This work provides a robust, high-accuracy numerical tool for simulating path functionals of anomalously diffusing particles with nonsmooth data, with potential extensions to related nonlocal fractional problems.

Abstract

In this paper, we propose and analyze a second-order time-stepping numerical scheme for the inhomogeneous backward fractional Feynman-Kac equation with nonsmooth initial data. The complex parameters and time-space coupled Riemann-Liouville fractional substantial integral and derivative in the equation bring challenges on numerical analysis and computations. The nonlocal operators are approximated by using the weighted and shifted Grünwald difference (WSGD) formula. Then a second-order WSGD scheme is obtained after making some initial corrections. Moreover, the error estimates of the proposed time-stepping scheme are rigorously established without the regularity requirement on the exact solution. Finally, some numerical experiments are performed to validate the efficiency and accuracy of the proposed numerical scheme.

Analysis of a WSGD scheme for backward fractional Feynman-Kac equation with nonsmooth data

TL;DR

The paper tackles the numerical solution of the backward fractional Feynman-Kac equation with nonsmooth data, a problem that involves time-space nonlocal operators and complex parameters. It introduces a second-order time discretization based on the weighted and shifted Grünwald difference (WSGD) for the fractional substantial derivative, augmented with initial corrections to handle inhomogeneous terms. The authors establish rigorous error estimates showing temporal convergence under mild data regularity for both homogeneous and certain inhomogeneous cases, and corroborate these results with numerical experiments in 1D and 2D. This work provides a robust, high-accuracy numerical tool for simulating path functionals of anomalously diffusing particles with nonsmooth data, with potential extensions to related nonlocal fractional problems.

Abstract

In this paper, we propose and analyze a second-order time-stepping numerical scheme for the inhomogeneous backward fractional Feynman-Kac equation with nonsmooth initial data. The complex parameters and time-space coupled Riemann-Liouville fractional substantial integral and derivative in the equation bring challenges on numerical analysis and computations. The nonlocal operators are approximated by using the weighted and shifted Grünwald difference (WSGD) formula. Then a second-order WSGD scheme is obtained after making some initial corrections. Moreover, the error estimates of the proposed time-stepping scheme are rigorously established without the regularity requirement on the exact solution. Finally, some numerical experiments are performed to validate the efficiency and accuracy of the proposed numerical scheme.
Paper Structure (10 sections, 14 theorems, 125 equations, 4 tables)

This paper contains 10 sections, 14 theorems, 125 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A second-order WSGD scheme
  4. Error estimates
  5. Homogeneous case
  6. Inhomogeneous case
  7. Numerical examples
  8. One dimensional case
  9. Two dimensional case
  10. Conclusion

Key Result

Lemma 2.1

The Laplace transform of the Riemann-Liouville fractional substantial integral with $\alpha\in(0,1)$ is the Laplace transform of the Riemann-Liouville fractional substantial derivative with $\alpha\in(0,1)$ is and the Laplace transform of the Caputo fractional substantial derivative with $\alpha\in(0,1)$ is where '$\sim$' means taking the Laplace transform and ${}_0I_t^{1-\alpha}$ stands for t

Theorems & Definitions (28)

  • Lemma 2.1: LiDZ:2019
  • Lemma 2.2: DengLQW:2018
  • Theorem 2.1: SunND:2021a
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 18 more