Skip to main content
SpringerLink
Log in

An Efficient Spline Collocation Method for a Nonlinear Fourth-Order Reaction Subdiffusion Equation

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

Abstract

The nonlinear fourth-order reaction–subdiffusion equation whose solutions display a typical initial weak singularity is considered. A new analytical technique is introduced to analyze orthogonal spline collocation (OSC) method based on L1 scheme on graded mesh. By introducing a discrete convolution kernel and discrete fractional Grönwall inequality, convergence of the scheme is proved rigorously. This novel analytical technique can provide new insights in analyzing other time fractional fourth-order differential equations with weakly singular solutions.

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
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Hao, Z., Cao, W.: An improved algorithm based on finite difference schemes for fractional boundary value problems with nonsmooth solution. J. Sci. Comput. 73, 395–415 (2017)

    Article  MathSciNet  Google Scholar 

  2. Liao, H.-L., Li, D., Zhang, J.: Sharp error estimate of the nonuniform L1 formula for linear reaction-subdiffusion equations. SIAM J. Numer. Anal. 56, 1112–1133 (2018)

    Article  MathSciNet  Google Scholar 

  3. Li, M., Zhao, J., Huang, C., Chen, S.: Nonconforming virtual element method for the time fractional reaction–subdiffusion equation with non-smooth data. J. Sci. Comput. 81, 1823–1859 (2019)

    Article  MathSciNet  Google Scholar 

  4. Zhai, S., Wang, D., Weng, Z., Zhao, X.: Error analysis and numerical simulations of strang splitting method for space fractional nonlinear Schrödinger equation. J. Sci. Comput. 81, 965–989 (2019)

    Article  MathSciNet  Google Scholar 

  5. Duan, B., Zheng, Z.: An exponentially convergent scheme in time for time fractional diffusion equations with non-smooth initial data. J. Sci. Comput. 80, 717–742 (2019)

    Article  MathSciNet  Google Scholar 

  6. Yan, Y., Khan, M., Ford, N.J.: An analysis of the modified L1 scheme for time fractional partial differential equations with nonsmooth data. SIAM J. Numer. Anal. 56, 210–227 (2018)

    Article  MathSciNet  Google Scholar 

  7. Xing, Y., Yan, Y.: A higher order numerical method for time fractional partial differential equations with nonsmooth data. J. Comput. Phys. 357, 305–323 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Lyu, P., Vong, S.: A high-order method with a temporal nonuniform mesh for a time-fractional Benjamin–Bona–Mahony equation. J. Sci. Comput. 80, 1607–1628 (2019)

    Article  MathSciNet  Google Scholar 

  10. Vong, S., Wang, Z.: Compact finite difference scheme for the fourth-order fractional subdiffusion system. Adv. Appl. Math. Mech. 6, 419–435 (2014)

    Article  MathSciNet  Google Scholar 

  11. Sun, H., Zhao, X., Sun, Z.: The temporal second order difference schemes based on the interpolation approximation for the time multi-term fractional wave equation. J. Sci. Comput. 78, 467–498 (2019)

    Article  MathSciNet  Google Scholar 

  12. Zhong, J., Liao, H.-L., Ji, B., Zhang, L.M.: A fourth-order compact solver for fractional-in-time fourth-order diffusion equations. arXiv:1907.01708 (2019)

  13. Shen, J., Sheng, C.: An efficient space-time method for time fractional diffusion equation. J. Sci. Comput. 81, 1088–1110 (2019)

    Article  MathSciNet  Google Scholar 

  14. Liao, H.-L., Yan, Y., Zhang, J.: Unconditional convergence of a fast two-level linearized algorithm for semilinear subdiffusion equations. J. Sci. Comput. 80, 1–25 (2019)

    Article  MathSciNet  Google Scholar 

  15. Li, D., Wu, C., Zhang, Z.: Linearized Galerkin FEMs for nonlinear time fractional parabolic problems with non-smooth solutions in time direction. J. Sci. Comput. 80, 403–419 (2019)

    Article  MathSciNet  Google Scholar 

  16. Li, M., Huang, C., Ming, W.: A relaxation-type Galerkin FEM for nonlinear fractional Schrödinger equations. Numer. Algor. 83, 99–124 (2020)

    Article  Google Scholar 

  17. Ji, C., Dai, W., Sun, Z.: Numerical schemes for solving the time-fractional Dual-Phase-Lagging heat conduction model in a double-layered nanoscale thin film. J. Sci. Comput. 81, 1767–1800 (2019)

    Article  MathSciNet  Google Scholar 

  18. Xu, D., Guo, J., Qiu, W.: Time two-grid algorithm based on finite difference method for two-dimensional nonlinear fractional evolution equations. Appl. Numer. Math. 152, 169–184 (2020)

    Article  MathSciNet  Google Scholar 

  19. Deng, B., Zhang, Z., Zhao, X.: Superconvergence points for the spectral interpolation of Riesz fractional derivatives. J. Sci. Comput. 81, 1577–1601 (2019)

    Article  MathSciNet  Google Scholar 

  20. Ji, C., Sun, Z., Hao, Z.: Numerical algorithms with high spatial accuracy for the fourth-order fractional sub-diffusion equations with the first Dirichlet boundary conditions. J. Sci. Comput. 66, 1148–1174 (2015)

    Article  MathSciNet  Google Scholar 

  21. Qiao, L., Wang, Z., Xu, D.: An alternating direction implicit orthogonal spline collocation method for the two dimensional multi-term time fractional integro-differential equation. Appl. Numer. Math. 151, 199–212 (2020)

    Article  MathSciNet  Google Scholar 

  22. Qiao, L., Xu, D.: BDF ADI orthogonal spline collocation scheme for the fractional integro-differential equation with two weakly singular kernels. Comput. Math. Appl. 78, 3807–3820 (2019)

    Article  MathSciNet  Google Scholar 

  23. Qiao, L., Xu, D., Yan, Y.: High-order ADI orthogonal spline collocation method for a new 2D fractional integro-differential problem. Math. Method Appl. Sci. 43, 5162–5178 (2020)

    Article  MathSciNet  Google Scholar 

  24. Luchko, Y.: Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation. Fract. Calc. Appl. Anal. 15, 141–160 (2012)

    Article  MathSciNet  Google Scholar 

  25. 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 

  26. An, Y., Liu, R.: Existence of nontrivial solutions of an asymptotically linear fourth-order elliptic equation. Nonlinear Anal. 68, 3325–3331 (2008)

    Article  MathSciNet  Google Scholar 

  27. 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 

  28. Podlubny, I.: Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, To Methods of Their Solution and Some of Their Applications, vol 198 of Mathematics in Science and Engineering. Academic Press, Inc, San Diego (1999)

  29. Kopteva, N.: Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions. Math. Comput. 88, 2135–2155 (2019)

    Article  MathSciNet  Google Scholar 

  30. Percell, P., Wheeler, M.F.: A \(C^1\) finite element collocation method for elliptic equations. SIAM J. Numer. Anal. 17, 605–622 (1980)

    Article  MathSciNet  Google Scholar 

  31. Douglas, Jr., Dupont, T.: Collocation Methods for Parabolic Equations in a Single Space Variable. Lecture Notes in Mathematics, Vol. 385. Springer, New York (1974)

  32. Fernandes, R.I., Fairweather, G.: Analysis of alternating direction collocation methods for parabolic and hyperbolic problems in two space variables. Numer. Methods Part. Differ. Equ. 9, 191–211 (1993)

    Article  MathSciNet  Google Scholar 

  33. Fairweather, G., Yang, X.H., Xu, D., Zhang, H.Z.: An ADI Crank–Nicolson orthogonal spline collocation method for the two-dimensional fractional diffusion-wave equation. J. Sci. Comput. 65, 1217–1239 (2015)

    Article  MathSciNet  Google Scholar 

  34. Liao, H.-L., Mclean, W., Zhang, J.: A discrete Grönwall inequality with application to numerical schemes for subdiffusion problems. SIAM J. Numer. Anal. 57, 218–237 (2019)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank the anonymous referees for their valuable comments and suggestions which helped us to improve the manuscript a lot. The authors wish to thank Professor Graeme Fairweather for stimulating discussions and for his constant encouragement and support.

Author information

Authors and Affiliations

  1. School of Science, Hunan University of Technology, Zhuzhou, 412007, China

    Haixiang Zhang & Xuehua Yang

  2. Department of Mathematics, Hunan Normal University, Changsha, 410081, China

    Da Xu

Authors
  1. Haixiang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xuehua Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Da Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xuehua Yang.

Additional information

Publisher's Note

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

The work was supported by National Natural Science Foundation of China (11701168, 11601144), Hunan Provincial Natural Science Foundation of China (2018JJ3108, 2018JJ3109, 2018JJ4062), Scientific Research Fund of Hunan Provincial Education Department (18B304, YB2016B033), and China Postdoctoral Science Foundation (2018M631403).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, H., Yang, X. & Xu, D. An Efficient Spline Collocation Method for a Nonlinear Fourth-Order Reaction Subdiffusion Equation. J Sci Comput 85, 7 (2020). https://doi.org/10.1007/s10915-020-01308-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-020-01308-8

Keywords

Mathematics Subject Classification

Search

Navigation