Skip to main content
SpringerLink
Log in

On the numerical solution of two-dimensional Rosenau–Burgers (RB) equation

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

Our purpose is to solve numerically the nonlinear evolutionary problem called Rosenau–Burgers (RB) equation. First, we have derived error estimates of semidiscrete finite element method for the approximation of the Rosenau–Burgers problem. For a second-order accuracy in time, we propose the Galerkin–Crank–Nicolson fully discrete method. Second, a linearized difference scheme for the Rosenau–Burgers equation is considered. It is proved that the proposed difference scheme is uniquely solvable, and the method is shown to be second-order convergent both in time and space in maximum norm. Finally, some numerical experiments are given to demonstrate the validity and accuracy of our linearized difference scheme.

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

Similar content being viewed by others

References

  1. Rosenau P (1986) A quasi-continuous description of a non-linear transmission line. Physica Scripta 34(827–829):42

    Google Scholar 

  2. Rosenau P (1988) Dynamics of dense discrete systems. Progr. Theor. Phys. 79:1028–1042

    Article  Google Scholar 

  3. Park M A (1993) On the Rosenau equation in multidimensional space. Nonlinear Anal., T.M.A., 21 , pp. 77-85

  4. Chung SK, Ha SN (1994) Finite element Galerkin solutions for the Rosenau equation. Appl. Anal. 54:39–56

    Article  MathSciNet  Google Scholar 

  5. Chung SK (2001) Numerical methods for the Rosenau equation. Appl. Anal. 77:351–369

    Article  MathSciNet  Google Scholar 

  6. Turgut Ak S (2016) Battal Gazi Karakoc and Anjan Biswas, Numerical Scheme to Dispersive Shallow Water Waves. Journal of Computational and Theoretical Nanoscience 13:7084–7092

    Article  Google Scholar 

  7. Turgut Ak S (2016) Battal Gazi Karakoc, Houria Triki, Numerical simulation for treatment of dispersive shallow water waves with Rosenau-KdV equation. Eur. Phys. J. Plus 131:356

    Article  Google Scholar 

  8. Abbaszadeh Mostafa, Dehghan Mehdi (2017) The two-grid interpolating element free Galerkin (TG-IEFG) method for solving Rosenau-regularized long wave (RRLW) equation with error analysis. Applicable Analysis. https://doi.org/10.1080/00036811.2017.1303137

  9. Ghiloufi A, Kadri T (2017) Analysis of new conservative difference scheme for two-dimensional Rosenau-RLW equation. Applicable Analysis 96(7):1255–1267

    Article  MathSciNet  Google Scholar 

  10. Omrani K, Abidi F, Achouri T, Khiari N (2008) A new conservative finite difference scheme for the Rosenau equation. Appl. Math. Comput. 201:35–43

    MathSciNet  MATH  Google Scholar 

  11. Atouani Noureddine, Omrani Khaled (2015) A new conservative high-order accurate difference scheme for the Rosenau equation. Applicable Analysis 94:2435–2455

    Article  MathSciNet  Google Scholar 

  12. He Dongdong (2016) Exact solitary solution and a three-level linearly implicit conservative finite difference method for the generalized Rosenau-Kawahara-RLW equation with generalized Novikov type perturbation. Nonlinear Dynamics 85(1):479–498

    Article  MathSciNet  Google Scholar 

  13. Omrani Khaled, Ghiloufi Ahlem, An efficient computational approach for two-dimensional variant of nonlinear-dispersive model of shallow water wave, 2020, Engineering with Computers, https://doi.org/10.1007/s00366-020-00967-3

  14. Ghiloufi Ahlem, Omrani Khaled (2017) New conservative difference schemes with fourth-order accuracy for some model equation for nonlinear dispersive waves. Numer. Methods Partial Differential Eq. https://doi.org/10.1002/num.22208

  15. Razborova P, Kara AH, Biswas A (2015) Additional conservation laws for Rosenau-KdV-RLW equation with power law nonlinearity by Lie symmetry. Nonlinear Dynamics. 79:743–748

    Article  MathSciNet  Google Scholar 

  16. Hu B, Xu Y, Hu J (2008) Crank-Nicolson finite difference scheme for the Rosenau-Burgers equation. Appl. Math. Comput. 204(1):311–316

    MathSciNet  MATH  Google Scholar 

  17. Hu J, Hu B, Xu Y (2011) A verage implicit linear difference scheme for generalized Rosenau-Burgers equation. Appl. Math. Comput. 217(19):7557–7563

    MathSciNet  MATH  Google Scholar 

  18. Pan X, Zhang L (2012) A new finite difference scheme for the Rosenau-Burgers equation. Appl. Math. Comput. 218(17):8917–8924

    MathSciNet  MATH  Google Scholar 

  19. Adams RA (1975) Sobolev spaces. Academics Press, New York

    MATH  Google Scholar 

  20. Boulaaras Salah (2017) Some new properties of asynchronous algorithms of theta scheme combined with finite elements methods for an evolutionary implicit 2-sided obstacle problem. Math Meth Appl Sci 40(18):7231–7239

    Article  MathSciNet  Google Scholar 

  21. Ciarlet PG (1978) The finite element method for elliptic problems. North-Holland, Amsterdam

    MATH  Google Scholar 

  22. Thomée V (1984) Galerkin Finite Element Methods for Parabolic Problems, L.N.M, vol. 1054, Springer-verlag, Berlin

  23. Browder F E (1965) Existence and uniqueness theorems for solutions of nonlinear boundary value problems, In: R. Finn, editors, Applications of Nonlinear P.D.Es. Proceedings of Symposium of Applied Mathematics, Vol. 17, A.M.S, Providence, pp. 24-49

  24. Rouatbi Asma, Rouis Moez, Omrani Khaled, Numerical scheme for a model of shallow water waves in ( 2 + 1 )-dimensions Computers and Mathematics with Applications 74(8) (2017) Pages 1871-1884

  25. Zhou Y L (1990) Applications of discrete functional analysis to the finite difference method

  26. Piao Guang-Ri, Lee June-Yub, Cai Guo-Xian (2016) Analysis and computational method based on quadratic B-spline FEM for the Rosenau-Burgers equation. Numer Methods Partial Differential Eq 32:877–895

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to thank the reviewers that their comments and suggestions have really improved the quality of the paper.

Author information

Authors and Affiliations

  1. Université de Sousse, Institut Supérieur des Sciences Appliquées et de Technologie de Sousse, Sousse Ibn Khaldoun, 4003, Sousse, Tunisia

    Khaled Omrani

  2. Université de Carthage, Ecole Polytechnique de Tunis, Rue El Khawarezmi, 2078, La Marsa, Tunisia

    Hajer Debebria

  3. Université de Sousse, École Supérieur des Sciences et de Technologie de Hammam Sousse, 4011 H. Sousse, Rue L. Abassi, Sousse, Tunisia

    Khedidja Bayarassou

Authors
  1. Khaled Omrani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hajer Debebria
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Khedidja Bayarassou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Khaled Omrani.

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Omrani, K., Debebria, H. & Bayarassou, K. On the numerical solution of two-dimensional Rosenau–Burgers (RB) equation. Engineering with Computers 38, 715–726 (2022). https://doi.org/10.1007/s00366-020-01055-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-020-01055-2

Keywords

Mathematics Subject Classification

Search

Navigation