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Numerical Algorithms of the Two-dimensional Feynman–Kac Equation for Reaction and Diffusion Processes

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Abstract

This paper provides a finite difference discretization for the backward Feynman–Kac equation, governing the distribution of functionals of the path for a particle undergoing both reaction and diffusion (Hou and Deng in J Phys A Math Theor 51:155001, 2018). Numerically solving the equation with the time tempered fractional substantial derivative and tempered fractional Laplacian consists in discretizing these two non-local operators. Here, using convolution quadrature, we provide the first-order and second-order schemes for discretizing the time tempered fractional substantial derivative, which doesn’t require the assumption of the regularity of the solution in time; we use the finite difference method to approximate the two-dimensional tempered fractional Laplacian, and the accuracy of the scheme depends on the regularity of the solution on \({\bar{\varOmega }}\) rather than the whole space. Lastly, we verify the predicted convergence orders and the effectiveness of the presented schemes by numerical examples.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant No. 11671182, and the Fundamental Research Funds for the Central Universities under Grants No. lzujbky-2018-ot03.

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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

    Daxin Nie, Jing Sun & Weihua Deng

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  1. Daxin Nie
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  2. Jing Sun
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  3. Weihua Deng
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Correspondence to Weihua Deng.

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Nie, D., Sun, J. & Deng, W. Numerical Algorithms of the Two-dimensional Feynman–Kac Equation for Reaction and Diffusion Processes. J Sci Comput 81, 537–568 (2019). https://doi.org/10.1007/s10915-019-01027-9

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