Finite Element Methods Based on Two Families of Second-Order Numerical Formulas for the Fractional Cable Model with Smooth Solutions

Abstract

We apply two families of novel fractional \(\theta \)-methods, the FBT-\(\theta \) and FBN-\(\theta \) methods developed by the authors in previous work, to the fractional Cable model, in which the time direction is approximated by the fractional \(\theta \)-methods, and the space direction is approximated by the finite element method. Some positivity properties of the coefficients for both of these methods are derived, which are crucial for the proof of the stability estimates. We analyse the stability of the scheme and derive an optimal convergence result with \(O(\tau ^2+h^{r+1})\) for smooth solutions, where \(\tau \) is the time mesh size and h is the spatial mesh size. Some numerical experiments with smooth and nonsmooth solutions are conducted to confirm our theoretical analysis. To overcome the singularity at initial value, the starting part is added to restore the second-order convergence rate in time.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

References

  1. 1.

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

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Al-Maskari, M., Karaa, S.: The lumped mass FEM for a time-fractional cable equation. Appl. Numer. Math. 132, 73–90 (2018)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Banjai, L., López-Fernández, M.: Efficient high order algorithms for fractional integrals and fractional differential equations. Numer. Math. 141(2), 289–317 (2019)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    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)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Diethelm, K., Ford, N.J., Freed, A.D.: Detailed error analysis for a fractional Adams method. Numer. Algorithms 36(1), 31–52 (2004)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Ding, H.F., Li, C.P., Yi, Q.: A new second-order midpoint approximation formula for Riemann–Liouville derivative: algorithm and its application. IMA J. Appl. Math. 82(5), 909–944 (2017)

    MathSciNet  Google Scholar 

  7. 7.

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

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Feng, L.B., Zhuang, P., Liu, F.W., Turner, I., Li, J.: High-order numerical methods for the Riesz space fractional advection–dispersion equations. Comput. Math. Appl. (2016). https://doi.org/10.1016/j.camwa.2016.01.015

    Google Scholar 

  9. 9.

    Fisher, M.E., Robert, E.H.: Toeplitz determinants: some applications, theorems, and conjectures. Advances in Chemical Physics: Stochastic processes in chemical physics 333–353, (1969)

  10. 10.

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

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Gatto, P., Hesthaven, J.S.: Numerical approximation of the fractional laplacian via \(hp\)-finite elements, with an application to image denoising. J. Sci. Comput. 65(1), 249–270 (2015)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Grenander, U., Szeg\(\ddot{\text{o}}\), G.: Toeplitz forms and their applications. Univ of California Press, (1984)

  13. 13.

    Hassani, H., Avazzadeh, Z., Machado, J.A.T.: Solving two-dimensional variable-order fractional optimal control problems with transcendental bernstein series. J. Comput. Nonlinear Dyn. 14(6), 061001 (2019)

    Google Scholar 

  14. 14.

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

    Google Scholar 

  15. 15.

    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(1), A146–A170 (2016)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Jin, B.T., Li, B.Y., Zhou, Z.: Correction of high-order BDF convolution quadrature for fractional evolution equations. SIAM J. Sci. Comput. 39(6), A3129–A3152 (2017)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Jin, B.T., Li, B.Y., Zhou, Z.: An analysis of the Crank–Nicolson method for subdiffusion. IMA J. Numer. Anal. 38(1), 518–541 (2017)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Langlands, T.A.M., Henry, B.I., Wearne, S.L.: Fractional cable equation models for anomalous electrodiffusion in nerve cells: finite domain solutions. SIAM J. Appl. Math. 71(4), 1168–1203 (2011)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Li, C.P., Zeng, F.H.: Numerical Methods for Fractional Calculus. Chapman and Hall/CRC, Boca Raton (2015)

    Google Scholar 

  20. 20.

    Li, J.C., Huang, Y.Q., Lin, Y.P.: Developing finite element methods for Maxwell’s equations in a Cole–Cole dispersive medium. SIAM J. Sci. Comput. 33(6), 3153–3174 (2011)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Li, D.F., Zhang, J.W., Zhang, Z.M.: Unconditionally optimal error estimates of a linearized galerkin method for nonlinear time fractional reaction-subdiffusion equations. J. Sci. Comput. 76(2), 848–866 (2018)

    MathSciNet  MATH  Google Scholar 

  22. 22.

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

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Liu, F.W., Yang, Q.Q., Turner, I.: Two new implicit numerical methods for the fractional Cable equation. J. Comput. Nonlinear Dyn. 6(1), 011009 (2011)

    Google Scholar 

  24. 24.

    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, 2535–2548 (2016)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    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)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Liu, Z.G., Cheng, A.J., Li, X.L.: A fast-high order compact difference method for the fractional cable equation. Numer. Meth. Part Differ. Equ. 34(6), 2237–2266 (2018)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Liu, Y., Du, Y.W., Li, H., Liu, F.W., Wang, Y.J.: Some second-order \(\theta \) schemes combined with finite element method for nonlinear fractional cable equation. Numer. Algorithms 80(2), 533–555 (2019). https://doi.org/10.1007/s11075-018-0496-0

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Lubich, C.: Discretized fractional calculus. SIAM J. Math. Anal. 17(3), 704–719 (1986)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Magin, R.L.: Fractional Calculus in Bioengineering. Begell House, Redding (2006)

    Google Scholar 

  30. 30.

    McLean, W., Mustapha, K.: A second-order accurate numerical method for a fractional wave equation. Numer. Math. 105, 481–510 (2007)

    MathSciNet  MATH  Google Scholar 

  31. 31.

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

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    Google Scholar 

  33. 33.

    Stynes, M.: Too much regularity may force too much uniqueness. Fract. Calc. Appl. Anal. 19(6), 1554–1562 (2016)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    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(2), 1057–1079 (2017)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Thomée, Vidar: Galerkin Finite Element Methods for Parabolic Problems, 2nd edn. Springer, Berlin (2006)

    Google Scholar 

  36. 36.

    Wang, Y.J., Liu, Y., Li, H., Wang, J.F.: Finite element method combined with second-order time discrete scheme for nonlinear fractional cable equation. Eur. Phys. J. Plus. 131(3), 61 (2016)

    Google Scholar 

  37. 37.

    Yan, Y.B., Pal, K., Ford, N.J.: Higher order numerical methods for solving fractional differential equations. BIT Numer. Math. 54(2), 555–584 (2014)

    MathSciNet  MATH  Google Scholar 

  38. 38.

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

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Yin, B.L., Liu, Y., Li, H., Zhang, Z.M.: Two families of novel second-order fractional numerical formulas and their applications to fractional differential equations. arXiv preprint arXiv:1906.01242v2 (2019)

  40. 40.

    Yin, B.L., Liu, Y., Li, H., He, S.: Fast algorithm based on TT-M FE system for space fractional Allen–Cahn equations with smooth and non-smooth solutions. J. Comput. Phys. 379, 351–372 (2019)

    MathSciNet  Google Scholar 

  41. 41.

    Zeng, F.H., Zhang, Z., Karniadakis, G.E.: Second-order numerical methods for multi-term fractional differential equations: smooth and non-smooth solutions. Comput. Methods Appl. Mech. Eng. 327, 478–502 (2017)

    MathSciNet  MATH  Google Scholar 

  42. 42.

    Zhang, H., Jiang, X.Y., Fan, W.P.: Parameter estimation for the fractional Schrödinger equation using Bayesian method. J. Math. Phys. 57(8), 082104 (2016)

    MathSciNet  MATH  Google Scholar 

  43. 43.

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

    MathSciNet  MATH  Google Scholar 

  44. 44.

    Zhu, P., Xie, S.L., Wang, X.S.: Nonsmooth data error estimates for FEM approximations of the time fractional cable equation. Appl. Numer. Math. 121, 170–184 (2017)

    MathSciNet  MATH  Google Scholar 

  45. 45.

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

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to Professor Buyang Li, two anonymous referees and editors for their valuable suggestions which improve the presentation of this work. The work of the second author was supported in part by the NSFC grant 11661058. The work of the third author was supported in part by the NSFC grant 11761053, the NSF of Inner Mongolia 2017MS0107, and the program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region NJYT-17-A07. The work of the fourth author was supported in part by grants NSFC 11871092 and U1930402.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Yang 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

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yin, B., Liu, Y., Li, H. et al. Finite Element Methods Based on Two Families of Second-Order Numerical Formulas for the Fractional Cable Model with Smooth Solutions. J Sci Comput 84, 2 (2020). https://doi.org/10.1007/s10915-020-01258-1

Download citation

Keywords

  • FBT-\(\theta \) method
  • FBN-\(\theta \) method
  • Fractional cable model
  • Finite element method