Numerical Algorithms for the Forward and Backward Fractional Feynman–Kac Equations

Weihua Deng, Minghua Chen, Eli Barkai

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34 Scopus citations

Abstract

The Feynman–Kac equations are a type of partial differential equations describing the distribution of functionals of diffusive motion. The probability density function (PDF) of Brownian functionals satisfies the Feynman–Kac formula, being a Schrödinger equation in imaginary time. The functionals of non-Brownian motion, or anomalous diffusion, follow the fractional Feynman–Kac equation (Carmi et al. in J Stat Phys 141:1071–1092, 2010), where the fractional substantial derivative is involved. Based on recently developed discretized schemes for fractional substantial derivatives (Chen and Deng arXiv:1310.3086), this paper focuses on providing algorithms for numerically solving the forward and backward fractional Feynman–Kac equations; since the fractional substantial derivative is non-local time-space coupled operator, new challenges are introduced compared with the ordinary fractional derivative. Two ways (finite difference and finite element) of discretizing the space derivative are considered. For the backward fractional Feynman–Kac equation, the numerical stability and convergence of the algorithms with first order accuracy are theoretically discussed; and the optimal estimates are obtained. For all the provided schemes, including the first order and high order ones, of both forward and backward Feynman–Kac equations, extensive numerical experiments are performed to show their effectiveness.

Original languageEnglish
Pages (from-to)718-746
Number of pages29
JournalJournal of Scientific Computing
Volume62
Issue number3
DOIs
StatePublished - Mar 2014

Bibliographical note

Publisher Copyright:
© 2014, Springer Science+Business Media New York.

Funding

This work was supported by the National Natural Science Foundation of China under Grant No. 11271173, and the Fundamental Research Funds for the Central Universities under Grant No. lzujbky-2014-228. EB thanks the Israel Science Foundation.

FundersFunder number
National Natural Science Foundation of China
National Natural Science Foundation of China11271173
Israel Science Foundation
Fundamental Research Funds for the Central Universitieslzujbky-2014-228

    Keywords

    • Fractional Feynman–Kac equation
    • Fractional substantial derivative
    • Numerical inversion of Laplace transforms
    • Numerical stability and convergence
    • Optimal convergent order

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