Movatterモバイル変換

[0]ホーム
URL:

Skip to main content
Springer Nature Link
Log in

Local discontinuous Galerkin method combined with theL2 formula for the time fractional Cable model

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

In this article, we develop the local discontinuous Galerkin (LDG) method combined with theL2 formula to solve a fractional Cable model, where the BDF2 with theL2 formula for the fractional derivative is used to discretize the temporal direction, and the LDG method is used to approximate the spatial direction. The stability of the fully discrete LDG scheme is analyzed, and the rigorous optimal error estimate in\(L^2\)-norm is derived. Further, numerical examples with different boundary conditions are provided to validate our theory results.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Henry, B., Langlands, T.A.M.: Fractional cable models for spiny neuronal dendrites. Phys. Rev. Lett.100(12), 128103 (2008)

    Article  Google Scholar 

  2. Langlands, T.A.M., Henry, B., Wearne, S.: Fractional cable equation models for anomalous electro diffusion in nerve cells: infinite domain solutions. J. Math. Biol.59(6), 761–808 (2009)

  3. Bisquert, J.: Fractional diffusion in the multipletrapping regime and revision of the equivalence with the continuous-time random walk. Phys. Rev. Lett.91(1), 010602 (2003)

    Article  Google Scholar 

  4. Liu, F.W., Yang, Q.Q., Turner, I.: Two new implicit numerical methods for the fractional cable equation. J. Comput. Nonlinear Dyn.6, 011009 (2011)

    Article  Google Scholar 

  5. Dehghan, M., Abbaszadeh, M.: Analysis of the element free Galerkin (EFG) method for solving fractional cable equation with Dirichlet boundary condition. Appl. Numer. Math.109, 208–234 (2016)

    Article MathSciNet MATH  Google Scholar 

  6. Yang, X., Jiang, X.Y., Zhang, H.: A time-space spectral tau method for the time fractional cable equation and its inverse problem. Appl. Numer. Math.130, 95–111 (2018)

    Article MathSciNet MATH  Google Scholar 

  7. Lin, Y., Li, X., Xu, C.: Finite difference/spectral approximations for the fractional cable equation. Math. Comput.80(275), 1369–1396 (2011)

    Article MathSciNet MATH  Google Scholar 

  8. Liu, J., Li, H., Liu, Y.: A new fully discrete finite difference/element approximation for fractional cable equation. J. Appl. Math. Comput.52(1), 345–361 (2016)

    Article MathSciNet MATH  Google Scholar 

  9. Yin, B., Liu, Y., Li, H., Zhang, Z.: Finite element methods based on two families of second-order numerical formulas for the fractional Cable model with smooth solutions. J. Sci. Comput.84(1), 2 (2020)

    Article MathSciNet MATH  Google Scholar 

  10. Liu, Y., Du, Y.W., Li, H., Wang, J.F.: A two-grid finite element approximation for a nonlinear time-fractional Cable equation. Nonlinear Dyn.85(4), 2535–2548 (2016)

    Article MathSciNet MATH  Google Scholar 

  11. Zheng, Y., Zhao, Z.: The discontinuous Galerkin finite element method for fractional cable equation. Appl. Numer. Math.115, 32–41 (2017)

    Article MathSciNet MATH  Google Scholar 

  12. Zhuang, P., Liu, F.W., Turner, I., Anh, V.: Galerkin finite element method and error analysis for the fractional cable equation. Numer. Algor.72(2), 447–466 (2016)

    Article MathSciNet MATH  Google Scholar 

  13. Cockburn, B., Shu, C.W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal.35(6), 2440–2463 (1998)

    Article MathSciNet MATH  Google Scholar 

  14. Deng, W., Hesthaven, J.S.: Local discontinuous Galerkin methods for fractional ordinary differential equations. BIT Numer. Math.55(4), 967–985 (2015)

    Article MathSciNet MATH  Google Scholar 

  15. Zhang, M., Liu, Y., Li, H.: High-order local discontinuous Galerkin method for a fractal mobile/immobile transport equation with the Caputo-Fabrizio fractional derivative. Numer. Methods Part. Differ. Equ.35(4), 1588–1612 (2019)

    Article MATH  Google Scholar 

  16. Liu, Y., Zhang, M., Li, H., Li, J.C.: High-order local discontinuous Galerkin method combined with WSGD-approximation for a fractional subdiffusion equation. Comput. Math. Appl.73(6), 1298–1314 (2017)

    Article MathSciNet MATH  Google Scholar 

  17. Li, C., Wang, Z.: The local discontinuous Galerkin finite element methods for Caputo-type partial differential equations: numerical analysis. Appl. Numer. Math.140, 1–22 (2019)

    Article MathSciNet MATH  Google Scholar 

  18. Dehghan, M., Abbaszadeh, M.: Variational multiscale element free Galerkin (VMEFG) and local discontinuous Galerkin (LDG) methods for solving two-dimensional Brusselator reaction-diffusion system with and without cross-diffusion. Comput. Meth. Appl. Mech. Eng.300, 770–797 (2016)

    Article MathSciNet MATH  Google Scholar 

  19. Guo, L., Wang, Z., Vong, S.: Fully discrete local discontinuous Galerkin methods for some time-fractional fourth-order problems. Int. J. Comput. Math.93(10), 1665–1682 (2016)

    Article MathSciNet MATH  Google Scholar 

  20. Du, Y., Liu, Y., Li, H., Fang, Z.: Local discontinuous Galerkin method for a nonlinear time-fractional fourth-order partial differential equation. J. Comput. Phys.344, 108–126 (2017)

    Article MathSciNet MATH  Google Scholar 

  21. Sun, X.R., Li, C., Zhao, F.Q.: Local discontinuous Galerkin methods for the time tempered fractional diffusion equation. Appl. Math. Comput.365, 124725 (2020)

    MathSciNet MATH  Google Scholar 

  22. Li, C., Sun, X.R., Zhao, F.Q.: LDG schemes with second order implicit time discretization for a fractional sub-diffusion equation. Results Appl. Math.4, 100079 (2019)

    Article MathSciNet MATH  Google Scholar 

  23. Li, C.P., Li, Z.Q., Wang, Z.: Mathematical analysis and the local discontinuous Galerkin method for Caputo-Hadamard fractional partial differential equation. J. Sci. Comput.85(2), 1–27 (2020)

    Article MathSciNet MATH  Google Scholar 

  24. Li, C.P., Li, D.X., Wang, Z.: L1/LDG method for the generalized time-fractional Burgers equation. Math. Comput. Simul.187, 357–378 (2021)

    Article MathSciNet MATH  Google Scholar 

  25. Yuan, W.P., Chen, Y.P., Huang, Y.Q.: A local discontinuous Galerkin method for time-fractional Burgers equations. East Asian J. Appl. Math.10, 818–837 (2020)

    Article MathSciNet MATH  Google Scholar 

  26. Huang, C., An, N., Yu, X.: A local discontinuous Galerkin method for time-fractional diffusion equation with discontinuous coefficient. Appl. Numer. Math.151, 367–379 (2020)

    Article MathSciNet MATH  Google Scholar 

  27. Wei, L.L., Liu, L.J., Sun, H.X.: Stability and convergence of a local discontinuous Galerkin method for the fractional diffusion equation with distributed order. J. Appl. Math. Comput.59, 323–341 (2019)

    Article MathSciNet MATH  Google Scholar 

  28. Niu, Y.X., Wang, J.F., Liu, Y., Li, H., Fang, Z.C.: Local discontinuous Galerkin method based on a family of second-order time approximation schemes for fractional mobile/immobile convection-diffusion equations (submitted to Journal) (2021)

  29. Alikhanov, A.A., Huang, C.: A high-order\(L\)2 type difference scheme for the time-fractional diffusion equation. Appl. Math. Comput.411, 126545 (2021)

    MathSciNet MATH  Google Scholar 

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

    Article MathSciNet MATH  Google Scholar 

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

    Article MathSciNet MATH  Google Scholar 

  32. Xu, Y., Shu, C.W.: Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection-diffusion and KdV equations. Comput. Meth. Appl. Mech. Eng.196(37–40), 3805–3822 (2007)

    Article MathSciNet MATH  Google Scholar 

  33. Wang, H., Shu, C.W., Zhang, Q.: Stability and error estimates of local discontinuous Galerkin method with implicit-explicit time-marching for advection-diffusion problems. SIAM J. Numer. Anal.53, 206–227 (2015)

    Article MathSciNet MATH  Google Scholar 

  34. Li, X., Xing, Y., Chou, C.: Optimal energy conserving and energy dissipative local discontinuous Galerkin methods for the Benjamin-Bona-Mahony equation. J. Sci. Comput.83(1), 1–48 (2020)

    Article MathSciNet MATH  Google Scholar 

  35. Liu, Y., Du, Y., Li, H., Liu, F., Wang, Y.: Some second-order\(\theta \) schemes combined with finite element method for nonlinear fractional Cable equation. Numer. Algor.80(2), 533–555 (2019)

    Article MathSciNet MATH  Google Scholar 

  36. Castillo, P., Cockburn, B., Schötzau, D., Schwab, C.: Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection-diffusion problems. Math. Comput.71(238), 455–478 (2002)

    Article MathSciNet MATH  Google Scholar 

Download references

Acknowledgements

Authors thank three anonymous reviewers and Editor-in-Chief very much for their valuable comments and suggestions for improving our work. This work is supported by the National Natural Science Foundation of China (12061053, 12161063), Natural Science Foundation of Inner Mongolia (2020MS01003, 2021MS01018), Young Innovative Talents Project of Grassland Talents Project, Program for Innovative Research Team in Universities of Inner Mongolia Autonomous Region (NMGIRT2207) and Scientific Research Projection of Higher Schools of Inner Mongolia (NJZY21266).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010021, China

    Minghui Song, Yang Liu & Hong Li

  2. School of Statistics and Mathematics, Inner Mongolia University of Finance and Economics, Hohhot, 010070, China

    Jinfeng Wang

Authors
  1. Minghui Song

    You can also search for this author inPubMed Google Scholar

  2. Jinfeng Wang

    You can also search for this author inPubMed Google Scholar

  3. Yang Liu

    You can also search for this author inPubMed Google Scholar

  4. Hong Li

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence toYang Liu.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Song, M., Wang, J., Liu, Y.et al. Local discontinuous Galerkin method combined with theL2 formula for the time fractional Cable model.J. Appl. Math. Comput.68, 4457–4478 (2022). https://doi.org/10.1007/s12190-022-01711-4

Download citation

Keywords

Access this article

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Advertisement

[8]ページ先頭
©2009-2025 Movatter.jp