Skip to main content
SpringerLink
Log in

Fourier spectral approximation for generalized time fractional Burgers equation

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

Abstract

In this paper, a linearized fully discrete scheme is presented to solve the generalized time fractional Burgers equation. The proposed method is on the basis of finite difference method in time and Fourier spectral approximation in space. Based on a temporal-spatial error splitting argument, the boundedness of the solution and convergence of the numerical scheme are proved rigorously without the time step size condition dependent on the spatial mesh size. Numerical examples are given to illustrate the theoretical results.

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. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    Book  MATH  Google Scholar 

  2. Magin, R.L.: Fractional Calculus in Bioengineering. Begell House Publishers, Danbury, CT (2006)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Liu, F., Zhuang, P., Anh, V., Turner, I., Burrage, K.: Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation. Appl. Math. Comput. 191, 12–20 (2007)

    MathSciNet  MATH  Google Scholar 

  5. Gao, G.H., Sun, Z.Z.: A compact finite difference scheme for the fractional sub-diffusion equations. J. Comput. Phys. 230, 586–595 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Liu, Q., Liu, F., Turner, I., Anh, V.: Finite element approximation for a modified anomalous subdiffusion equation. Appl. Math. Model. 35, 4103–4116 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Jin, B.T., Lazarov, R., Zhou, Z.: Error estimates for a semidiscrete finite element method for fractional order parabolic equations. SIAM J. Numer. Anal. 51, 445–466 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, H., Lü, S.J., Chen, W.P.: Spectral methods for the time fractional diffusion-wave equation in a semi-infinite channel. Comput. Math. Appl. 71, 1818–1830 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, H., Hu, X.H., Ren, J.C., Sun, T., Tang, Y.F.: \(L1\) scheme on graded mesh for the linearized time fractional KdV equation with initial singularity. Int. J. Model. Simul. Sci. Comput. 10, 1941006 (2019)

    Article  Google Scholar 

  12. Huang, J.F., Nie, N.M., Tang, Y.F.: A second order finite difference-spectral method for space fractional diffusion equation. Sci. China Math. 57, 1303–1317 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bateman, H.: Some recent researches on the motion of fluids. Mon. Weather Rev. 43, 163–170 (1915)

    Article  Google Scholar 

  14. Woyczyński, W.A.: Lévy processes in the physical sciences. Birkhäuser Boston (2001)

  15. Sugimoto, N.: Burgers equation with a fractional derivative; hereditary effects on nonlinear acoustic waves. J. Fluid Mech. 225, 631–653 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  16. Funaki, T., Woyczyński, W.A.: Interacting particle approximation for fractal Burgers equation. Stochastic Processes and Related Topics (1998)

  17. Garra, R.: Fractional-calculus model for temperature and pressure waves in fluid-saturated porous rocks. Phys. Rev. E 84, 1–6 (2011)

    Article  Google Scholar 

  18. Xu, Y.F., Agrawal, O.P.: Numerical solutions and analysis of diffusion for new generalized fractional Burgers equation. Fract. Calc. Appl. Anal. 16, 709–736 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mohebbi, A.: Analysis of a numerical method for the solution of time fractional Burgers equation. Bull. Iran. Math. Soc. 44, 457–480 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  20. Qiu, W.L., Chen, H.B., Zheng, X.: An implicit difference scheme and algorithm implementation for the one dimensional time fractional Burgers equations. Math. Comput. Simulat. 166, 298–314 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. Yang, M.M.: Numerical approximation of time-fractional Burgers-type equation. Adv. Differ. Equ. 182, 1–11 (2020)

    MathSciNet  Google Scholar 

  22. Hussein, A.J.: A weak Galerkin finite element method for solving time-fractional coupled Burgers’ equations in two dimensions. Appl. Numer. Math. 156, 265–275 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  23. Li, D.F., Zhang, C.J., Ran, M.H.: A linear finite difference scheme for generalized time fractional Burgers equation. Appl. Math. Model. 40, 6069–6081 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Asgari, Z., Hosseini, S.M.: Efficient numerical schemes for the solution of generalized time fractional Burgers type equations. Numer. Algor. 77, 763–792 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  25. Li, D.F., Wang, J.L., Zhang, J.W.: Unconditionally convergent \(L1\)-Galerkin FEMs for nonlinear time-fractional Schrödinger equations. SIAM. J. Sci. Comput. 39, A3067–A3088 (2017)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Li, D.F., Wu, C.D., Zhang, Z.M.: 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  MATH  Google Scholar 

  28. Canuto, C., Quarteroni, A.: Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comp. 38, 201–229 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  29. Maz’ja, V.G.: Sobolev Space. Springer-Verlag, New York (1985)

    Book  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work is supported by National Natural Science Foundation of China (Grant no. 11672011).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Beihang University, Beijing, 100191, China

    Li Chen & Shujuan Lü

Authors
  1. Li Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shujuan Lü
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shujuan Lü.

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

Chen, L., Lü, S. Fourier spectral approximation for generalized time fractional Burgers equation. J. Appl. Math. Comput. 68, 3979–3997 (2022). https://doi.org/10.1007/s12190-021-01686-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-021-01686-8

Keywords

Mathematics Subject Classification

Search

Navigation