nach oben

Journal of Scientific Computing

Erschienen in:

08.08.2019

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

verfasst von: Daxin Nie, Jing Sun, Weihua Deng

Erschienen in: Journal of Scientific Computing | Ausgabe 1/2019

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

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.

Vorheriger Artikel A Tangential Block Lanczos Method for Model Reduction of Large-Scale First and Second Order Dynamical Systems

Nächster Artikel Numerical Approximations for the Tempered Fractional Laplacian: Error Analysis and Applications

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Acosta, G., Bersetche, F.M., Borthagaray, J.P.: Finite element approximations for fractional evolution problems. arXiv:1705.09815 [math.NA]

Agmon, N.: Residence times in diffusion processes. J. Chem. Phys. 81, 3644–3647 (1984)MathSciNetCrossRef

Baeumer, B., Meerschaert, M.M.: Tempered stable Lévy motion and transient super-diffusion. J. Comput. Appl. Math. 233, 2438–2448 (2010)MathSciNetCrossRefMATH

Buschman, R.: Decomposition of an integral operator by use of Mikusski calculus. SIAM J. Math. Anal. 3, 83–85 (1972)MathSciNetCrossRefMATH

Cairoli, A., Baule, A.: Anomalous processes with general waiting times: functionals and multipoint structure. Phys. Rev. Lett. 115, 110601 (2015)CrossRef

Cairoli, A., Baule, A.: Feynman–Kac equation for anomalous processes with space- and time-dependent forces. J. Phys. A 50, 164002 (2017)MathSciNetCrossRefMATH

Carmi, S., Turgeman, L., Barkai, E.: On distributions of functionals of anomalous diffusion paths. J. Stat. Phys. 141, 1071–1092 (2010)MathSciNetCrossRefMATH

Cartea, A., del-Castillo-Negrete, D.: Fluid limit of the continuous-time random walk with general Lévy jump distribution functions. Phys. Rev. E 76, 041105 (2007)CrossRef

Chen, K.H.: Matrix Preconditioning Techniques and Applications. Cambridge University Press, Cambridge (2005)CrossRefMATH

10.

Chen, M.H., Deng, W.H.: High order algorithm for the time-tempered fractional Feynman–Kac equation. J. Sci. Comput. 76, 867–887 (2018)MathSciNetCrossRefMATH

11.

Chen, M.H., Deng, W.H., Serra-Capizzano, S.: Uniform convergence of V-cycle multigrid algorithms for two-dimensional fractional Feynman–Kac equation. J. Sci. Comput. 74, 1034–1059 (2018)MathSciNetCrossRefMATH

12.

Deng, W.H., Chen, M.H., Barkai, E.: Numerical algorithms for the forward and backward fractional Feynman–Kac equations. J. Sci. Comput. 62, 718–746 (2015)MathSciNetCrossRefMATH

13.

Deng, W.H., Zhang, Z.J.: High Accuracy Algorithm for the Differential Equations Governing Anomalous Diffusion. World Scientific, Singapore (2019)CrossRefMATH

14.

Deng, W.H., Li, B.Y., Tian, W.Y., Zhang, P.W.: Boundary problems for the fractional and tempered fractional operators. Multiscale Model. Simul. 16, 125–149 (2018)MathSciNetCrossRefMATH

15.

Deng, W.H., Zhang, Z.J.: Numerical schemes of the time tempered fractional Feynman–Kac equation. Comput. Math. Appl. 73, 1063–1076 (2017)MathSciNetCrossRefMATH

16.

Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer, Berlin (1993)MATH

17.

Hou, R., Deng, W.H.: Feynman–Kac equations for reaction and diffusion processes. J. Phys. A Math. Theor. 51, 155001 (2018)MathSciNetCrossRefMATH

18.

Jin, B.T., Lazarov, R., Zhou, Z.: Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data. SIAM J. Sci. Comput. 38, A146–A170 (2016)MathSciNetCrossRefMATH

19.

Kac, M.: On distributions of certain Wiener functionals. Trans. Am. Math. Soc. 65, 1–13 (1949)MathSciNetCrossRefMATH

20.

Li, C., Deng, W.H., Zhao, L.J.: Well-posedness and numerical algorithm for the tempered fractional differential equations. Discrete Contin. Dyn. Syst. Ser. B 24, 1989–2015 (2019) MathSciNetMATH

21.

Lubich, C.: Convolution quadrature and discretized operational calculus. I. Numer. Math. 52, 129–145 (1988)MathSciNetCrossRefMATH

22.

Lubich, C.: Convolution quadrature and discretized operational calculus. II. Numer. Math. 52, 413–425 (1988)MathSciNetCrossRefMATH

23.

Lubich, C., Sloan, I.H., Thomée, V.: Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term. Math. Comput. 65, 1–17 (1996)MathSciNetCrossRefMATH

24.

Meerschaert, M.M., Sikorskii, A.: Stochastic Models for Fractional Calculus. De Gruyter, Berlin (2012) MATH

25.

Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)MathSciNetCrossRefMATH

26.

Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York (1993) MATH

27.

Sun, J., Nie, D.X., Deng, W.H.: Algorithm implementation and numerical analysis for the two-dimensional tempered fractional Laplacian. arXiv:1802.02349 [math.NA]

28.

Tatar, N.: The decay rate for a fractional differential equation. J. Math. Anal. Appl. 295, 303–314 (2004)MathSciNetCrossRefMATH

29.

Turgeman, L., Carmi, S., Barkai, E.: Fractional Feynman–Kac equation for non-Brownian functionals. Phys. Rev. Lett. 103, 190201 (2009)MathSciNetCrossRef

30.

Wu, X.C., Deng, W.H., Barkai, E.: Tempered fractional Feynman–Kac equation: theory and examples. Phys. Rev. E 93, 032151 (2016)MathSciNetCrossRef

31.

Xing, Y.Y., Yan, Y.B.: A higher order numerical method for time fractional partial differential equations with nonsmooth data. J. Comput. Phys. 357, 305–323 (2018)MathSciNetCrossRefMATH

32.

Yan, Y.B., Khan, M., Ford, N.J.: An analysis of the modified L1 scheme for time-fractional partial differential equations with nonsmooth data. SIAM J. Numer. Anal. 56, 210–227 (2018)MathSciNetCrossRefMATH

33.

Yang, Y., Yan, Y.B., Ford, N.J.: Some time stepping methods for fractional diffusion problems with nonsmooth data. Comput. Methods Appl. Math. 18, 129–146 (2018)MathSciNetCrossRefMATH

34.

Zhang, Z.J., Deng, W.H., Fan, H.T.: Finite difference schemes for the tempered fractional Laplacian. Numer. Math. Theory Methods Appl. 12, 492–516 (2018)MathSciNetCrossRef

35.

Zhang, Z.J., Deng, W.H., Karniadakis, G.E.: A Riesz basis Galerkin method for the tempered fractional Laplacian. SIAM J. Numer. Anal. 56, 3010–3039 (2018)MathSciNetCrossRefMATH

Titel: Numerical Algorithms of the Two-dimensional Feynman–Kac Equation for Reaction and Diffusion Processes
verfasst von: Daxin Nie
Jing Sun
Weihua Deng
Publikationsdatum: 08.08.2019
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 1/2019
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-019-01027-9

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Weitere Artikel der Ausgabe 1/2019

An Improved Discrete Least-Squares/Reduced-Basis Method for Parameterized Elliptic PDEs

On a Goal-Oriented Version of the Proper Generalized Decomposition Method

Stokes Problem with Slip Boundary Conditions of Friction Type: Error Analysis of a Four-Field Mixed Variational Formulation

A Robust Multilevel Preconditioner Based on a Domain Decomposition Method for the Helmholtz Equation

A Third Order Exponential Time Differencing Numerical Scheme for No-Slope-Selection Epitaxial Thin Film Model with Energy Stability

A Second-Order Boundary Condition Capturing Method for Solving the Elliptic Interface Problems on Irregular Domains

Premium Partner