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Two L1 Schemes on Graded Meshes for Fractional Feynman-Kac Equation

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Abstract

In this paper, we study the following time-fractional Feynman-Kac equation

$$\begin{aligned} {_\sigma ^CD_t^{\alpha }G(x,t)}-\Delta G(x,t)=f(x,t),~~~ 0<\alpha <1,~~\sigma > 0. \end{aligned}$$

As is well known, the optimal rate of convergence \(\mathcal {O}\left( \tau ^{\min \{2-\alpha ,~r\alpha \}}\right) \) with \(\sigma =0\) on graded meshes has been proved in [Stynes et al., SIAM J. Numer. Anal. 55, 1057–1079 (2017)] by L1 scheme. However, there are still some significant differences when \(\sigma >0\). More concretely, it shall drop down to the \(\mathcal {O}\left( \tau ^{\min \{1,~r\alpha \}}\right) \) by the implicit L1 scheme. This motivates us to design the implicit-explicit L1 scheme, which recovers a convergence rate \(\mathcal {O}\left( \tau ^{\min \{2-\alpha ,~r\alpha \}}\right) \) on graded meshes. Finally, numerical experiments are given to illustrate theoretical results.

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Acknowledgements

This work was supported by NSFC 11601206, the Research Foundation of Education Commission of Hunan Province of China (No. 19B565), the Project of Scientific Research Fund of Hunan Provincial Science and Technology Department (No. 2018WK4006), and the Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University. We would like to thank the anonymous reviewers for suggesting to simulate the two-dimensional case in Example 3, and for several suggestions and comments that led to much better results and an improved presentation.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Minghua Chen & Suzhen Jiang

  2. School of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Hunan, 411105, China

    Weiping Bu

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  1. Minghua Chen
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Correspondence to Minghua Chen.

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Chen, M., Jiang, S. & Bu, W. Two L1 Schemes on Graded Meshes for Fractional Feynman-Kac Equation. J Sci Comput 88, 58 (2021). https://doi.org/10.1007/s10915-021-01581-1

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