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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Vydané v:	Advances in computational mathematics Ročník 50; číslo 3; s. 58
Hlavní autori:	Nie, Daxin, Sun, Jing, Deng, Weihua
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.06.2024 Springer Nature B.V
Predmet:	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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Abstract	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.
AbstractList	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. 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., As0∂t1-α,x≠0∂t1-α,xAs, 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.
ArticleNumber	58
Author	Nie, Daxin Deng, Weihua Sun, Jing
Author_xml	– sequence: 1 givenname: Daxin surname: Nie fullname: Nie, Daxin organization: School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University – sequence: 2 givenname: Jing surname: Sun fullname: Sun, Jing organization: School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University – sequence: 3 givenname: Weihua orcidid: 0000-0002-8573-012X surname: Deng fullname: Deng, Weihua email: dengwh@lzu.edu.cn organization: School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University
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Cites_doi	10.1007/s10915-018-0640-y 10.1137/14097207X 10.1007/s10915-020-01256-3 10.1016/j.bulsci.2011.12.004 10.1007/BF01398686 10.1140/epjb/e2009-00126-3 10.1007/s10915-014-9873-6 10.1137/17M1118245 10.1515/fca-2019-0042 10.1103/PhysRevLett.103.190201 10.4310/ARKIV.2021.v59.n2.a2 10.1103/PhysRevE.84.061104 10.1007/s00211-020-01148-6 10.1007/s10915-021-01581-1 10.1007/s10915-017-0506-8 10.1016/j.cam.2016.09.006 10.1016/j.aim.2014.09.018 10.1007/s10955-010-0086-6 10.1007/BF01462237 10.1007/s00211-019-01025-x 10.1103/PhysRevE.89.030102
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Snippet	Fractional Feynman–Kac equation governs the functional distribution of the trajectories of anomalous diffusion. The non-commutativity of the integral...
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SubjectTerms	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
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Title	Numerical methods for forward fractional Feynman–Kac equation
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