Numerical Algorithms

, Volume 80, Issue 2, pp 533–555 | Cite as

Some second-order 𝜃 schemes combined with finite element method for nonlinear fractional cable equation

  • Yang Liu
  • Yanwei Du
  • Hong Li
  • Fawang LiuEmail author
  • Yajun Wang
Original Paper


In this article, some second-order time discrete schemes covering parameter 𝜃 combined with Galerkin finite element (FE) method are proposed and analyzed for looking for the numerical solution of nonlinear cable equation with time fractional derivative. At time tk𝜃, some second-order 𝜃 schemes combined with weighted and shifted Grünwald difference (WSGD) approximation of fractional derivative are considered to approximate the time direction, and the Galerkin FE method is used to discretize the space direction. The stability of second-order 𝜃 schemes is derived and the second-order time convergence rate in L2-norm is proved. Finally, some numerical calculations are implemented to indicate the feasibility and effectiveness for our schemes.


Second-order 𝜃 scheme Nonlinear fractional cable equation Finite element algorithm Stability Error estimates 

Mathematics Subject Classification (2010)

65M60 65N15 65N30 


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The authors thank the reviewers and editor very much for their insightful comments for improving our work.

Funding information

This work is supported by the National Natural Science Fund (11661058,11761053,11772046), Australian Research Council (ARC) via the Discovery Project (DP180103858) and Natural Science Fund of Inner Mongolia Autonomous Region (2016MS0102, 2017MS0107).


  1. 1.
    Zhao, Y., Zhang, Y., Shi, D., Liu, F., Turner, I.: Superconvergence analysis of nonconforming finite element method for two-dimensional time fractional diffusion equations. Appl. Math. Lett. 59, 38–47 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Liu, F., Zhuang, P., Turner, I., Burrage, K., Anh, V.: A new fractional finite volume method for solving the fractional diffusion equation. Appl. Math. Model. 38(15), 3871–3878 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Wang, H., Yang, D., Zhu, S.F.: A Petrov-Galerkin finite element method for variable-coefficient fractional diffusion equations. Comput. Methods Appl. Mech. Eng. 290, 45–56 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Yuste, S.B., Quintana-Murillo, J.: A finite difference method with non-uniform time steps for fractional diffusion equations. Comput. Phys. Comm. 183(12), 2594–2600 (2012)CrossRefzbMATHGoogle Scholar
  5. 5.
    Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172(1), 65–77 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    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, 3153–3174 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Zhang, H., Liu, F., Anh, V.: Galerkin finite element approximation of symmetric space-fractional partial differential equations. Appl. Math. Comput. 217, 2534–2545 (2010)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Jiang, Y.J., Ma, J.T.: Moving finite element methods for time fractional partial differential equations. Sci. China Math. 56, 1287–1300 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Li, C.P., Zhao, Z.G., Chen, Y.Q.: Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion. Comput. Math. Appl. 62, 855–875 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ding, H.F., Li, C.P.: A novel second-order midpoint approximation formula for Riemann-Liouville derivative: algorithm and application. arXiv:1605.02177 (2016)
  11. 11.
    Liu, Y., Du, Y.W., Li, H., He, S., Gao, W.: Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction-diffusion problem. Comput. Math. Appl. 70(4), 573–591 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Liu, Y., Du, Y.W., Li, H., Li, J.C., He, S.: A two-grid mixed finite element method for a nonlinear fourth-order reaction-diffusion problem with time-fractional derivative. Comput. Math. Appl. 70(10), 2474–2492 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Feng, L.B., Zhuang, P., Liu, F., Turner, I., Gu, Y.T.: Finite element method for space-time fractional diffusion equation. Numer. Algor. 72(3), 749–767 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Jin, B., Lazarov, R., Liu, Y.K., Zhou, Z.: The Galerkin finite element method for a multi-term time-fractional diffusion equation. J. Comput. Phys. 281, 825–843 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Tian, W.Y., Zhou, H., Deng, W.H.: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comput. 84, 1703–1727 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Wang, Z.B., Vong, S.W.: Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation. J. Comput. Phys. 277, 1–15 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Bu, W.P., Tang, Y.F., Wu, Y.C., Yang, J.Y.: Finite difference/finite element method for two-dimensional space and time fractional Bloch-Torrey equations. J. Comput. Phys. 293, 264–279 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Zeng, F., Li, C., Liu, F., Turner, I.: The use of finite difference/element approaches for solving the time-fractional subdiffusion equation. SIAM J. Sci. Comput. 35(6), A2976–A3000 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lin, Y.M., Li, X.J., Xu, C.J.: Finite difference/spectral approximations for the fractional Cable equation. Math. Comput. 80, 1369–1396 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Langlands, T.A.M., Henry, B., Wearne, S.: Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions. J. Math. Biol. 59(6), 761–808 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Bhrawy, A.H., Zaky, M.A.: Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation. Nonlinear Dyn. 80(1-2), 101–116 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Zhuang, P., Liu, F., Turner, I., Anh, V.: Galerkin finite element method and error analysis for the fractional cable equation. Numer. Algor. 72(2), 447–466 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Liu, J.C., Li, H., Liu, Y.: A new fully discrete finite difference/element approximation for fractional Cable equation. J. Appl. Math. Comput. 52(1-2), 345–361 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Yu, B., Jiang, X.Y.: Numerical identification of the fractional derivatives in the two-dimensional fractional Cable equation. J. Sci. Comput. 68(1), 252–272 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    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, 61 (2016). CrossRefGoogle Scholar
  27. 27.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Sun, H., Sun, Z.Z., Gao, G.H.: Some temporal second order difference schemes for fractional wave equations. Numer. Methods Partial Differential Eq. 32 (3), 970–1001 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Alikhanov, A.A.: A new difference scheme for the fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    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)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Ji, C.C., Sun, Z.Z.: A high-order compact finite difference scheme for the fractional sub-diffusion equation. J. Sci. Comput. 64(3), 959–985 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Li, M., Huang, C.M., Wang, P.D.: Galerkin finite element method for nonlinear fractional Schrödinger equations. Numer. Algor. 74(2), 499–525 (2017)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesInner Mongolia UniversityHohhotChina
  2. 2.School of Mathematical SciencesQueensland University of TechnologyBrisbaneAustralia
  3. 3.School of MathematicsJilin UniversityChangchunChina

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