Skip to main content
SpringerLink
Log in

A fast algorithm for multi-term time-space fractional diffusion equation with fractional boundary condition

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this paper, the multi-term time-space fractional diffusion equation with fractional boundary conditions is considered. The fractional derivative in space is approximated by the standard and shifted \(Gr\ddot{u}nwald-Letnikov\) formula, and the fractional derivative in time is approximated by L2-1\(_{\sigma }\) formula. The direct scheme for the equation on the nonuniform mesh is established to deal with the weak singularity near the initial time. The sum-of-exponentials algorithm is used to accelerate the L2-1\(_{\sigma }\) formula by approximating the power function \(t^{-\alpha }\). The fast scheme for the equation on the nonuniform mesh is established, which retains the accuracy. In addition, the theoretical analysis of the schemes is made; both the direct scheme and the fast scheme are stable and convergent under certain conditions, and the temporal convergence order is obtained. Finally, numerical examples are given to verify the correctness of the theoretical analysis and the efficiency of the fast scheme.

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

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Data availability

No datasets were generated or analyzed during the current study.

References

  1. Sokolov, I.M., Klafter, J., Blumen, A.: Fractional kinetics. Phys. Today 55(11), 48–54 (2002)

    Article  Google Scholar 

  2. Ionescu, C., Lopes, A., Copot, D., Machado, J.T., Bates, J.H.: The role of fractional calculus in modeling biological phenomena: a review. Commun. Nonlinear Sci. Numer. Simul. 51, 141–159 (2017)

    Article  MathSciNet  Google Scholar 

  3. Sohail, A., Bég, O.A., Li, Z., Celik, S.: Physics of fractional imaging in biomedicine. Prog. Biophys. Mol. Biol. 140, 13–20 (2018)

    Article  Google Scholar 

  4. Wang, H., Yang, D., Zhu, S.: Inhomogeneous Dirichlet boundary-value problems of space-fractional diffusion equations and their finite element approximations. SIAM J. Numer. Anal. 52(3), 1292–1310 (2014)

    Article  MathSciNet  Google Scholar 

  5. Wang, H., Yang, D.: Wellposedness of Neumann boundary-value problems of space-fractional differential equations. Fract. Calc. Appl. Anal. 20(6), 1356–1381 (2017)

    Article  MathSciNet  Google Scholar 

  6. Huang, C., Liu, X., Meng, X., Stynes, M.: Error analysis of a finite difference method on graded meshes for a multiterm time-fractional initial-boundary value problem. Comput. Methods Appl. Math. 20(4), 815–825 (2020)

    Article  MathSciNet  Google Scholar 

  7. Zhang, B., Bu, W., Xiao, A.: Efficient difference method for time-space fractional diffusion equation with robin fractional derivative boundary condition. Numer. Algorithms 88, 1965–1988 (2021)

    Article  MathSciNet  Google Scholar 

  8. Tian, W., Zhou, H., Deng, W.: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comput. 84(294), 1703–1727 (2015)

    Article  MathSciNet  Google Scholar 

  9. Tuan, V.K., Gorenflo, R.: Extrapolation to the limit for numerical fractional differentiation. ZAMM-J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik 75(8), 646–648 (1995)

    Article  MathSciNet  Google Scholar 

  10. Sun, Z., Wu, X.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56(2), 193–209 (2006)

    Article  MathSciNet  Google Scholar 

  11. Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225(2), 1533–1552 (2007)

    Article  MathSciNet  Google Scholar 

  12. Gao, G., Sun, Z., Zhang, H.: A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J. Comput. Phys. 259, 33–50 (2014)

    Article  MathSciNet  Google Scholar 

  13. Jin, B., Lazarov, R., Liu, Y., Zhou, Z.: The Galerkin finite element method for a multi-term time-fractional diffusion equation. J. Comput. Phys. 281, 825–843 (2015)

    Article  MathSciNet  Google Scholar 

  14. Huang, C., Stynes, M.: Superconvergence of a finite element method for the multi-term time-fractional diffusion problem. J. Sci. Comput. 82(1), 10 (2020)

    Article  MathSciNet  Google Scholar 

  15. Li, X., Xu, C.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47(3), 2108–2131 (2009)

    Article  MathSciNet  Google Scholar 

  16. Zheng, M., Liu, F., Anh, V., Turner, I.: A high-order spectral method for the multi-term time-fractional diffusion equations. Appl. Math. Model. 40(7–8), 4970–4985 (2016)

    Article  MathSciNet  Google Scholar 

  17. Feng, L., Zhuang, P., Liu, F., Turner, I., Gu, Y.: Finite element method for space-time fractional diffusion equation. Numer. Algorithms 72, 749–767 (2016)

    Article  MathSciNet  Google Scholar 

  18. Bu, W., Shu, S., Yue, X., Xiao, A., Zeng, W.: Space-time finite element method for the multi-term time-space fractional diffusion equation on a two-dimensional domain. Comput. Math. Appl. 78(5), 1367–1379 (2019)

    Article  MathSciNet  Google Scholar 

  19. Liao, H., McLean, W., Zhang, J.: A second-order scheme with nonuniform time steps for a linear reaction-sudiffusion problem. Preprint arXiv:1803.09873 (2018)

  20. Liao, H., Tang, T., Zhou, T.: A second-order and nonuniform time-stepping maximum-principle preserving scheme for time-fractional Allen-Cahn equations. J. Comput. Phys. 414, 109473 (2020)

    Article  MathSciNet  Google Scholar 

  21. Liao, H., Liu, N., Lyu, P.: Discrete gradient structure of a second-order variable-step method for nonlinear integro-differential models. Preprint arXiv:2301.12474 (2023)

  22. Yan, Y., Sun, Z., Zhang, J.: Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations: a second-order scheme. Commun. Comput. Phys. 22(4), 1028–1048 (2017)

    Article  MathSciNet  Google Scholar 

  23. Li, X., Liao, H., Zhang, L.: A second-order fast compact scheme with unequal time-steps for subdiffusion problems. Numer. Algorithms 86, 1011–1039 (2021)

    Article  MathSciNet  Google Scholar 

  24. Liu, N., Chen, Y., Zhang, J., Zhao, Y.: Unconditionally optimal \(H^{1}\)-error estimate of a fast nonuniform l2–1 \(\sigma \) scheme for nonlinear subdiffusion equations. Numer. Algorithms 92(3), 1655–1677 (2023)

    Article  MathSciNet  Google Scholar 

  25. Du, R., Sun, Z.: Temporal second-order difference methods for solving multi-term time fractional mixed diffusion and wave equations. Numer. Algorithms 88, 191–226 (2021)

    Article  MathSciNet  Google Scholar 

  26. Gao, G., Alikhanov, A.A., Sun, Z.: The temporal second order difference schemes based on the interpolation approximation for solving the time multi-term and distributed-order fractional sub-diffusion equations. J. Sci. Comput. 73, 93–121 (2017)

    Article  MathSciNet  Google Scholar 

  27. Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015)

    Article  MathSciNet  Google Scholar 

  28. Liu, H., Lü, S.: A high-order numerical scheme for solving nonlinear time fractional reaction-diffusion equations with initial singularity. Appl. Numer. Math. 169, 32–43 (2021)

    Article  MathSciNet  Google Scholar 

  29. Chen, H., Stynes, M.: Error analysis of a second-order method on fitted meshes for a time-fractional diffusion problem. J. Sci. Comput. 79, 624–647 (2019)

    Article  MathSciNet  Google Scholar 

  30. Chen, H., Stynes, M.: A high order method on graded meshes for a time-fractional diffusion problem. In: International Conference on Finite Difference Methods, pp. 15–27 (2018). Springer

  31. Kilbas, A.A., Marichev, O.I., Samko, S.G.: Fractional integrals and derivatives (theory and applications). Gordon and Breach, Switzerland (1993)

    Google Scholar 

  32. Liao, H., McLean, W., Zhang, J.: A discrete Gronwall inequality with applications to numerical schemes for subdiffusion problems. SIAM J. Numer. Anal. 57(1), 218–237 (2019)

    Article  MathSciNet  Google Scholar 

  33. Royden, H.L., Fitzpatrick, P.: Real analysis. Macmillan Publishing, New York (1968)

    Google Scholar 

  34. Liao, H., McLean, W., Zhang, J.: A second-order scheme with nonuniform time steps for a linear reaction-sudiffusion problem. Preprint arXiv:1803.09873 (2018)

  35. Ren, J., Liao, H., Zhang, J., Zhang, Z.: Sharp \(H^{1}\)-norm error estimates of two time-stepping schemes for reaction-subdiffusion problems. J. Comput. Appl. Math. 389, 113352 (2021)

    Article  Google Scholar 

Download references

Acknowledgements

Initially, I would like to thank the editor for reading this article. Then, I would like to thank the reviewers for their efforts.

Funding

This work was supported by the National Natural Science Foundation of China (Grant No. 11801221) and the Natural Science Foundation of Jiangsu Province (Grant No. BK20180586).

Author information

Authors and Affiliations

  1. School of Science, Jiangnan University, Wuxi, 214122, China

    Zhenhao Lu & Wenping Fan

Authors
  1. Zhenhao Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wenping Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Zhenhao Lu wrote the main text and prepared the data. Wenping Fan guided, checked, and revised this paper. All authors reviewed the paper.

Corresponding author

Correspondence to Wenping Fan.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lu, Z., Fan, W. A fast algorithm for multi-term time-space fractional diffusion equation with fractional boundary condition. Numer Algor (2024). https://doi.org/10.1007/s11075-024-01830-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11075-024-01830-y

Keywords

Search

Navigation