Efficient Jacobian Spectral Collocation Method for Spatio-Dependent Temporal Tempered Fractional Feynman-Kac Equation

doi:10.1007/s42967-024-00406-w

Communications on Applied Mathematics and Computation ›› 2026, Vol. 8 ›› Issue (1): 63-86.doi: 10.1007/s42967-024-00406-w

Efficient Jacobian Spectral Collocation Method for Spatio-Dependent Temporal Tempered Fractional Feynman-Kac Equation

Tinggang Zhao¹, Lijing Zhao^2,3

1. School of Mathematics and Statistics, Shandong University of Technology, Zibo, 255049, Shandong, China;
2. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an, 710129, Shaanxi, China;
3. Research & Development Institute of Northwestern Polytechnical University in Shenzhen, Shenzhen, 518057, Guangdong, China

收稿日期:2023-11-23 修回日期:2024-02-20 出版日期:2026-02-20 发布日期:2026-02-11
通讯作者: Lijing Zhao,E-mail:zhaolj17@nwpu.edu.cn E-mail:zhaolj17@nwpu.edu.cn
基金资助:
T.Z. is partly supported by the National Natural Science Foundation of China under Grant No. 11661048. L.Z. is partly supported by the Guangdong Basic and Applied Basic Research Foundation of China under Grant No. 2022A1515011332.

Efficient Jacobian Spectral Collocation Method for Spatio-Dependent Temporal Tempered Fractional Feynman-Kac Equation

Tinggang Zhao¹, Lijing Zhao^2,3

1. School of Mathematics and Statistics, Shandong University of Technology, Zibo, 255049, Shandong, China;
2. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an, 710129, Shaanxi, China;
3. Research & Development Institute of Northwestern Polytechnical University in Shenzhen, Shenzhen, 518057, Guangdong, China

Received:2023-11-23 Revised:2024-02-20 Online:2026-02-20 Published:2026-02-11
Contact: Lijing Zhao,E-mail:zhaolj17@nwpu.edu.cn E-mail:zhaolj17@nwpu.edu.cn
Supported by:
T.Z. is partly supported by the National Natural Science Foundation of China under Grant No. 11661048. L.Z. is partly supported by the Guangdong Basic and Applied Basic Research Foundation of China under Grant No. 2022A1515011332.

摘要/Abstract

摘要： Anomalous and non-ergodic diffusion is ubiquitous in the natural world. Fractional Feynman-Kac equations are used to characterize the functional distribution of the trajectories of the sub-diffusion particles which evolve slower than the Brownian motion. In this paper, by introducing the tempered fractional Jacobi functions (TFJFs), an efficient spectral collocation method is presented for solving the spatio-dependent temporal tempered fractional Feynman-Kac (STTFFK) equation. The tempered fractional differentiation matrix in the collocation scheme is generated by a recurrence relation which is fast and stable. The error estimate for the scheme is derived which shows that the proposed method is “spectral accuracy” if the solution is smooth. Several numerical experiments with different tempered factors are performed to support the theoretical results.

关键词: Fractional Feynman-Kac equation, Spatio-dependent tempered fractional derivative, Jacobian-collocation method, Tempered fractional differentiation matrix, Error estimate

Abstract: Anomalous and non-ergodic diffusion is ubiquitous in the natural world. Fractional Feynman-Kac equations are used to characterize the functional distribution of the trajectories of the sub-diffusion particles which evolve slower than the Brownian motion. In this paper, by introducing the tempered fractional Jacobi functions (TFJFs), an efficient spectral collocation method is presented for solving the spatio-dependent temporal tempered fractional Feynman-Kac (STTFFK) equation. The tempered fractional differentiation matrix in the collocation scheme is generated by a recurrence relation which is fast and stable. The error estimate for the scheme is derived which shows that the proposed method is “spectral accuracy” if the solution is smooth. Several numerical experiments with different tempered factors are performed to support the theoretical results.

Key words: Fractional Feynman-Kac equation, Spatio-dependent tempered fractional derivative, Jacobian-collocation method, Tempered fractional differentiation matrix, Error estimate

中图分类号:

Tinggang Zhao, Lijing Zhao. Efficient Jacobian Spectral Collocation Method for Spatio-Dependent Temporal Tempered Fractional Feynman-Kac Equation[J]. Communications on Applied Mathematics and Computation, 2026, 8(1): 63-86.

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Efficient Jacobian Spectral Collocation Method for Spatio-Dependent Temporal Tempered Fractional Feynman-Kac Equation

Efficient Jacobian Spectral Collocation Method for Spatio-Dependent Temporal Tempered Fractional Feynman-Kac Equation

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