Skip to main content
SpringerLink
Log in

Correction of High-Order BDF Convolution Quadrature for Fractional Feynman–Kac Equation with Lévy Flight

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this work, we present the correction schemes of the k-step BDF convolution quadrature at the starting \(k-1\) steps for the fractional Feynman–Kac equation with Lévy flight. Based on the idea of Jin et al. (SIAM J Sci Comput 39:A3129–A3152, 2017), we provide a detailed kth-order convergence analysis for the correction BDFk with nonsmooth data. The numerical experiments with spectral method are given to illustrate theoretical results. Moreover, some simulations and corresponding theoretical for the correction BDFk of the multi-term time fractional model are extended.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Chen, C.-M., Liu, F., Turner, I., Anh, V.: A Fourier method for the fractional diffusion equation describing sub-diffusion. J. Comput. Phys. 227, 886–897 (2007)

    Article  MathSciNet  Google Scholar 

  5. Chen, M.H., Deng, W.H.: Discretized fractional substantial calculus. ESAIM Math. Model. Numer. Anal. (M2AN) 49, 373–394 (2015)

    MathSciNet  MATH  Google Scholar 

  6. Chen, M.H., Deng, W.H.: High order algorithms for the fractional substantial diffusion equation with truncated Lévy flights. SIAM J. Sci. Comput. 37, A890–A917 (2015)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Chen, M.H., Deng, W.H., Wu, Y.J.: Superlinearly convergent algorithms for the two-dimensional space-time Caputo–Riesz fractional diffusion equation. Appl. Numer. Math. 70, 22–41 (2013)

    Article  MathSciNet  Google Scholar 

  9. Cuesta, E., Lubich, Ch., Palencia, C.: Convolution quadrature time discretization of fractional diffusion-wave equations. Math. Comput. 75, 673–696 (2006)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  11. Dungey, N.: Asymptotic type for sectorial operators and integral of fractional powers. J. Funct. Anal. 256, 1387–1407 (2009)

    Article  MathSciNet  Google Scholar 

  12. 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)

    Article  MathSciNet  Google Scholar 

  13. Friedrich, R., Jenko, F., Baule, A., Eule, S.: Anomalous diffusion of inertial, weakly damped particles. Phys. Rev. Lett. 96, 230601 (2006)

    Article  Google Scholar 

  14. Gao, G.H., Sun, H.H., Sun, Z.Z.: Stability and convergence of finite difference schemes for a class of time-fractional sub-diffusion equation based on certain superconvergence. J. Comput. Phys. 280, 510–528 (2015)

    Article  MathSciNet  Google Scholar 

  15. Hao, Z.P., Cao, W.R., Lin, G.: A second-order difference scheme for the time fractional substantial diffusion equation. J. Comput. Appl. Math. 313, 54–69 (2017)

    Article  MathSciNet  Google Scholar 

  16. Huang, C., Zhang, Z.M., Song, Q.S.: Spectral methods for substantial fractional differential equations. J. Sci. Comput. 74, 1554–1574 (2018)

    Article  MathSciNet  Google Scholar 

  17. Ji, C.C., Sun, Z.Z.: A high-order compact finite difference schemes for the fractional sub-diffusion equation. J. Sci. Comput. 64, 959–985 (2015)

    Article  MathSciNet  Google Scholar 

  18. Jin, B., 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)

    Article  MathSciNet  Google Scholar 

  19. Jin, B., Li, B.Y., Zhou, Z.: Correction of high-order BDF convolution quadrature for fractional evolution equations. SIAM J. Sci. Comput. 39, A3129–A3152 (2017)

    Article  MathSciNet  Google Scholar 

  20. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, New York (2006)

    MATH  Google Scholar 

  21. Li, C.P., Ding, H.F.: Higher order finite difference method for the reaction and anomalous-diffusion equation. Appl. Math. Model. 38, 3802–3821 (2014)

    Article  MathSciNet  Google Scholar 

  22. Lin, Y.M., Xu, C.J.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)

    Article  MathSciNet  Google Scholar 

  23. Lischke, A., Pang, G., Gulian, M., Song, F., Glusa, C., Zheng, X., Mao, Z., Cai, W., Meerschaert, M., Ainsworth, M., Karniadakis, G.: What is the fractional Laplacian? A comparative review with new results. J. Comput. Phys. 404, 109009 (2020)

    Article  MathSciNet  Google Scholar 

  24. Lubich, Ch.: Discretized fractional calculus. SIAM J. Math. Anal. 17, 704–719 (1986)

    Article  MathSciNet  Google Scholar 

  25. Lubich, Ch.: Convolution quadrature revisited. BIT 44, 503–514 (2004)

    Article  MathSciNet  Google Scholar 

  26. Lubich, Ch., 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)

    Article  MathSciNet  Google Scholar 

  27. Lv, C.H., Xu, C.J.: Error analysis of a high order method for time-fractional diffusion equations. SIAM J. Sci. Comput. 38, A2699–A2724 (2016)

    Article  MathSciNet  Google Scholar 

  28. Martínez, C., Sanz, M.: The Theory of Fractional Powers of Operators. Elsevier, New York (2001)

    MATH  Google Scholar 

  29. Meerschaert, M., Tadjeran, C.: Finite difference approximations for fractional advection–dispersion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004)

    Article  MathSciNet  Google Scholar 

  30. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)

    MATH  Google Scholar 

  31. Sakamoto, K., Yamamoto, M.: Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382, 426–447 (2011)

    Article  MathSciNet  Google Scholar 

  32. Shen, J., Tang, T., Wang, L.L.: Spectral Methods: Algorithms, Analysis and Applications. Springer, Berlin (2011)

    Book  Google Scholar 

  33. Stynes, M., O’riordan, E., Gracia, J.L.: Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal. 55, 1057–1079 (2017)

    Article  MathSciNet  Google Scholar 

  34. Sun, J., Nie, D.X., Deng, W.H.: Error estimates for backward fractional Feynman–Kac equation with non-smooth initial data. J. Sci. Comput. 84, 6 (2020)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  36. Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, New York (2006)

    MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by NSFC 11601206. The authors are grateful to Professor Martin Stynes and Dr. Zhi Zhou for them valuable comments. We would like to thank the anonymous reviewers for suggesting to analyse and simulate the multi-term time fractional model (2.5).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Jiankang Shi & Minghua Chen

Authors
  1. Jiankang Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Minghua Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Minghua Chen.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shi, J., Chen, M. Correction of High-Order BDF Convolution Quadrature for Fractional Feynman–Kac Equation with Lévy Flight. J Sci Comput 85, 28 (2020). https://doi.org/10.1007/s10915-020-01331-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-020-01331-9

Keywords

Search

Navigation