Skip to main content
SpringerLink
Log in

A High-Order Method with a Temporal Nonuniform Mesh for a Time-Fractional Benjamin–Bona–Mahony Equation

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

Abstract

The solution of a time-fractional differential equation often exhibits a weak singularity near the initial time. It makes classical numerical methods with uniform mesh usually lose their accuracy. Technique of nonuniform mesh was found to be a very efficient approach in the literatures to recover the full accuracy based on reasonable regularity of the solution. In this paper, we study finite difference scheme with temporal nonuniform mesh for time-fractional Benjamin–Bona–Mahony equations with non-smooth solutions. Our approximation bases on an integral equation equivalent to the nonlinear problem under consideration. We employ high-order interpolation formulas to obtain a linearized scheme on a nonuniform mesh and, by using a modified Grönwall inequality established recently, we show that the proposed scheme with a temporal graded mesh is unconditionally third-order convergent in time with respect to discrete \(H^1\)-norm. Besides high order convergence the proposed scheme has the advantage that only linear systems are needed to be solved for obtaining approximated solutions. Numerical examples are provided to justify the accuracy.

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

Similar content being viewed by others

References

  1. 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 

  2. Benjamin, T.B., Bona, J.L., Mahony, J.J.: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. A 272, 47–78 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, H., Stynes, M.: Error analysis of a second-order method on fitted meshes for a time-fractional diffusion problem. J. Sci. Comput. (2019). https://doi.org/10.1007/s10915-018-0863-y

    MathSciNet  MATH  Google Scholar 

  4. Dehghan, M., Abbaszadeh, M., Deng, W.: Fourth-order numerical method for the space-time tempered fractional diffusion-wave equation. Appl. Math. Lett. 73, 120–127 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dehghan, M., Abbaszadeh, M., Mohebbi, A.: The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on non-rectangular domains with error estimate. J. Comput. Appl. Math. 286, 211–231 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dehghan, M., Abbaszadeh, M., Mohebbi, A.: The numerical solution of nonlinear high dimensional generalized Benjamin–Bona–Mahony–Burgers equation via the meshless method of radial basis functions. Comput. Math. Appl. 68, 212–237 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dehghan, M., Manafian, J., Saadatmandi, A.: Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numer. Methods Part. Differ. Equ. 26, 448–479 (2010)

    MathSciNet  MATH  Google Scholar 

  8. Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gracia, J.L., O’Riordan, E., Stynes, M.: A fitted scheme for a Caputo initial-boundary value problem. J. Sci. Comput. 76, 583–609 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Guner, O., Bekir, A.: Bright and dark soliton solutions for some nonlinear fractional differential equations. Chin. Phys. B 25, 030203 (2016)

    Article  Google Scholar 

  11. Jin, B., Lazarov, R., Pasciak, J., Zhou, Z.: Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion. IMA J. Numer. Anal. 35, 561–582 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jin, B., Lazarov, R., Zhou, Z.: An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data. IMA J. Numer. Anal. 36, 197–221 (2016)

    MathSciNet  MATH  Google Scholar 

  13. Jin, B., Lazarov, R., Zhou, Z.: Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data. SIAM J. Sci. Comput. 38, A146–A170 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Application of Fractional Differential Equations. Elsevier, Amersterdam (2006)

    MATH  Google Scholar 

  15. Li, C.: Linearized difference schemes for a BBM equation with a fractional nonlocal viscous term. Appl. Math. Comput. 311, 240–250 (2017)

    MathSciNet  MATH  Google Scholar 

  16. Li, C.P., Yi, Q., Chen, A.: Finite difference methods with non-uniform meshes for nonlinear fractional differential equations. J. Comput. Phys. 316, 614–631 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Liao, H.L., McLean, W., Zhang, J.: A second-order scheme with nonuniform time steps for a linear reaction-subdiffusion problem. arXiv:1803.09873v2 [math.NA]

  19. Liao, H.L., Sun, Z.Z.: Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations. Numer. Methods Part. Differ. Equ. 26, 37–60 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Liu, Y., Roberts, J., Yan, Y.: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. Int. J. Comput. Math. 95, 1151–1169 (2018)

    Article  MathSciNet  Google Scholar 

  21. Lyu, P., Vong, S.: A linearized and second-order unconditionally convergent scheme for coupled time fractional Klein–Gordon–Schrödinger equation. Numer. Methods Part. Differ. Equ. 34, 2153–2179 (2018)

    Article  MATH  Google Scholar 

  22. Lyu, P., Vong, S.: A linearized second-order finite difference scheme for time fractional generalized BBM equation. Appl. Math. Lett. 78, 16–23 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lyu, P., Vong, S.: A linearized second-order scheme for nonlinear time fractional Klein–Gordon type equations. Numer. Algorithms 78, 485–511 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ma, J., Tang, T.: Error analysis for a fast numerical method to a boundary integral equation of the first kind. J. Comput. Math. 26, 56–68 (2008)

    MathSciNet  MATH  Google Scholar 

  25. McLean, W.: Regularity of solutions to a time-fractional diffusion equation. ANZIAM J. 52, 123–138 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Medeiros, L.A., Menzala, G.P.: Existence and uniqueness for periodic solutions of the Benjamin–Bona–Mahony equation. SIAM J. Math. Anal. 8, 792–799 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Morgado, M.L., Rebelo, M.: Numerical approximation of distributed order reaction diffusion equations. J. Comput. Appl. Math. 275, 216–227 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mustapha, K., McLean, W.: Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations. SIAM J. Numer. Anal. 51, 491–515 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mustapha, K., Schötzau, D.: Well-posedness of hp-version discontinuous Galerkin methods for fractional diffusion wave equations. IMA J. Numer. Anal. 34, 1426–1446 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  32. Omrani, K., Ayadi, M.: Finite difference discretization of the Benjamin–Bona–Mahony–Burgers equation. Numer. Methods Part. Differ. Equ. 24, 239–248 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  34. Rosiera, L., Zhang, B.Y.: Unique continuation property and control for the Benjamin–Bona–Mahony equation on a periodic domain. J. Differ. Equ. 254, 141–178 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. 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  MATH  Google Scholar 

  36. Song, L., Zhang, H.: Solving the fractional BBM-Burgers equation using the homotopy analysis method. Chaos Solitons Fractals 40, 1616–1622 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  38. 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  MATH  Google Scholar 

  39. Sun, Z.Z.: Numerical Methods of Partial Differential Equations, 2nd edn. Science Press, Beijing (2012)

    Google Scholar 

  40. Tang, T.: A note on collocation methods for Volterra integro-differential equations with weakly singular kernels. IMA J. Numer. Anal. 13, 93–99 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  41. Tang, T.: Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations. Numer. Math. 61, 373–382 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  42. Vong, S., Lyu, P.: Unconditional convergence in maximum-norm of a second-order linearized scheme for a time-fractional Burgers-type equation. J. Sci. Comput. 76, 1252–1273 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zhang, J., Xu, C.: Finite difference/spectral approximations to a water wave model with a nonlocal viscous term. Appl. Math. Model. 38, 4912–4925 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank anonymous reviewers for their constructive comments and suggestions, which helped them to improve significantly the quality of the paper.

Author information

Authors and Affiliations

  1. School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, China

    Pin Lyu

  2. Department of Mathematics, University of Macau, Macao, China

    Pin Lyu & Seakweng Vong

Authors
  1. Pin Lyu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Seakweng Vong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Seakweng Vong.

Additional information

Publisher's Note

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

This research is supported by the Macao Science and Technology Development Fund 050/2017/A, the Grant MYRG2017-00098-FST and MYRG2018-00047-FST from University of Macau.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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). https://doi.org/10.1007/s10915-019-00991-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-019-00991-6

Keywords

Mathematics Subject Classification

Search

Navigation