nach oben

Journal of Scientific Computing

Erschienen in:

01.07.2020

Error Estimates for Backward Fractional Feynman–Kac Equation with Non-Smooth Initial Data

verfasst von: Jing Sun, Daxin Nie, Weihua Deng

Erschienen in: Journal of Scientific Computing | Ausgabe 1/2020

Einloggen

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

search-config

KI-gestützte Suche

Aus

Abstract

In this paper, we are concerned with the numerical solution for the backward fractional Feynman–Kac equation with non-smooth initial data. Here we first provide the regularity estimate of the solution. And then we use the backward Euler and second-order backward difference convolution quadratures to approximate the Riemann–Liouville fractional substantial derivative and get the first- and second-order convergence in time. The finite element method is used to discretize the Laplace operator with the optimal convergence rates. Compared with the previous works for the backward fractional Feynman–Kac equation, the main advantage of the current discretization is that we don’t need the assumption on the regularity of the solution in temporal and spatial directions. Moreover, the error estimates of the time semi-discrete schemes and the fully discrete schemes are also provided. Finally, we perform the numerical experiments to verify the effectiveness of the presented algorithms.

Vorheriger Artikel Barrier Function Local and Global Analysis of an L1 Finite Element Method for a Multiterm Time-Fractional Initial-Boundary Value Problem

Nächster Artikel Enriched Finite Volume Approximations of the Plane-Parallel Flow at a Small Viscosity

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

Nur mit Berechtigung zugänglich

Acosta, G., Bersetche, F.M., Borthagaray, J.P.: Finite element approximations for fractional evolution problems. Fract. Calc. Appl. Anal. 22, 767–794 (2019)MathSciNetCrossRef

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

Bazhlekova, E., Jin, B.T., Lazarov, R., Zhou, Z.: An analysis of the Rayleigh-Stokes problem for a generalized second-grade fluid. Numer. Math. 131, 1–31 (2015)MathSciNetCrossRef

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

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

Carmi, S., Barkai, E.: Fractional Feynman-Kac equation for weak ergodicity breaking. Phys. Rev. E 84, 061104 (2011)CrossRef

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

Chen, S., Liu, F., Zhuang, P., Anh, V.: Finite difference approximation for the fractional Fokker-Planck equation. Appl. Math. Model. 33, 256–273 (2009)MathSciNetCrossRef

Cheng, A.J., Wang, H., Wang, K.X.: A Eulerian-Lagrangian control volume method for solute transport with anomalous diffusion. Numer. Methods Partial Differ. Equ. 31, 253–267 (2015)MathSciNetCrossRef

10.

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)MathSciNetCrossRef

11.

Deng, W.H., Li, B.Y., Qian, Z., Wang, H.: Time discretization of a tempered fractional Feynman-Kac equation with measure data. SIAM J. Numer. Anal. 56, 3249–3275 (2018)MathSciNetCrossRef

12.

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

13.

Ervin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Equ. 22, 558–576 (2006)MathSciNetCrossRef

14.

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)MathSciNetCrossRef

15.

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

16.

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

17.

Li, C.P., Zeng, F.H., Liu, F.: Spectral approximations to the fractional integral and derivative. Frac. Calcu. Appl. Anal. 15, 383–406 (2012)MathSciNetMATH

18.

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

19.

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

20.

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

21.

Nie, D.X., Sun, J., Deng, W.H.: Numerical algorithms of the two-dimensional Feynman-Kac equation for reaction and diffusion processes. J. Sci. Comput. 81, 537–568 (2019)MathSciNetCrossRef

22.

Podlubny, I.: Fractional Differential Equations. Academic Press, Cambridge (1999)MATH

23.

Sun, Z.Z., Wu, X.N.: A fully discrete scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193–209 (2006)MathSciNetCrossRef

24.

Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (2006)MATH

25.

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

26.

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)MathSciNetCrossRef

Titel: Error Estimates for Backward Fractional Feynman–Kac Equation with Non-Smooth Initial Data
verfasst von: Jing Sun
Daxin Nie
Weihua Deng
Publikationsdatum: 01.07.2020
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 1/2020
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-020-01256-3

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 1/2020

A Mixed Method for Time-Transient Acoustic Wave Propagation in Metamaterials

Barrier Function Local and Global Analysis of an L1 Finite Element Method for a Multiterm Time-Fractional Initial-Boundary Value Problem

The Fine Error Estimation of Collocation Methods on Uniform Meshes for Weakly Singular Volterra Integral Equations

Enriched Finite Volume Approximations of the Plane-Parallel Flow at a Small Viscosity

A Posteriori Error Estimates for Fully Discrete Finite Element Method for Generalized Diffusion Equation with Delay

Fast and High-Order Accuracy Numerical Methods for Time-Dependent Nonlocal Problems in

Premium Partner