Numerical methods for forward fractional Feynman–Kac equation

Fractional Feynman–Kac equation governs the functional distribution of the trajectories of anomalous diffusion. The non-commutativity of the integral fractional Laplacian and time-space coupled fractional substantial derivative, i.e., A s 0 ∂ t 1 - α , x ≠ 0 ∂ t 1 - α , x A s , brings about huge cha...

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Bibliographic Details
Published in:Advances in computational mathematics Vol. 50; no. 3; p. 58
Main Authors: Nie, Daxin, Sun, Jing, Deng, Weihua
Format: Journal Article
Language:English
Published: New York Springer US 01.06.2024
Springer Nature B.V
Subjects:
Applied mathematics
Commutativity
Computational mathematics
Computational Mathematics and Numerical Analysis
Computational Science and Engineering
Derivatives
Diffusion
Error analysis
Estimates
Finite element analysis
Finite element method
Integrals
Laplace transforms
Mathematical and Computational Biology
Mathematical Modeling and Industrial Mathematics
Mathematics
Mathematics and Statistics
Methods
Numerical analysis
Numerical methods
Quadratures
Regularity
Sobolev space
Visualization
Convolution quadrature
Finite element method
Error estimate
35R11
65M60
Integral fractional Laplacian
Fractional substantial derivative
65M12
Forward fractional Feynman–Kac equation
ISSN:1019-7168, 1572-9044
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Summary:Fractional Feynman–Kac equation governs the functional distribution of the trajectories of anomalous diffusion. The non-commutativity of the integral fractional Laplacian and time-space coupled fractional substantial derivative, i.e., A s 0 ∂ t 1 - α , x ≠ 0 ∂ t 1 - α , x A s , brings about huge challenges on the regularity and spatial error estimates for the forward fractional Feynman–Kac equation. In this paper, we first use the corresponding resolvent estimate obtained by the bootstrapping arguments and the generalized Hölder-type inequalities in Sobolev space to build the regularity of the solution, and then the fully discrete scheme constructed by convolution quadrature and finite element methods is developed. Also, the complete error analyses in time and space directions are respectively presented, which are consistent with the provided numerical experiments.
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ISSN:1019-7168
1572-9044
DOI:10.1007/s10444-024-10152-5