Skip to main content
SpringerLink
Log in

Solitary wave solution of nonlinear Benjamin–Bona–Mahony–Burgers equation using a high-order difference scheme

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this work, we investigate the solitary wave solution of nonlinear Benjamin–Bona–Mahony–Burgers (BBMB) equation using a high-order linear finite difference scheme. We prove that this scheme is stable and convergent with the order of \(O(\tau ^2+h^4)\). Furthermore, we discuss the existence and uniqueness of numerical solutions. Numerical results obtained from propagation of a single solitary and interaction of two and three solitary waves confirm the efficiency and high accuracy of proposed method.

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

Similar content being viewed by others

References

  • Al-Khaled K, Momani S, Alawneh A (2005) Approximate wave solutions for generalized Benjamin-Bona-Mahony-Burgers equations. Appl Math Comput 171:281–292

    MathSciNet  MATH  Google Scholar 

  • Alquran M, K. Al-Khaled (2011) Sinc and solitary wave solutions to the generalized BBMB equations. Physica Scripta 83:6 (065010)

  • Darvishi MT, Najafi M, Najafi M (2010) Exact three-wave solutions for high nonlinear form of Benjamin-Bona-Mahony-Burgers equations. Int J Math Comput Sci 6:127–131

    Google Scholar 

  • Dehghan M, Mohebbi A (2008) The combination of collocation, finite difference, and multigrid methods for solution of the two-dimensional wave equation. Num Methods Part Diff Equ 24:897–910

    Article  MathSciNet  MATH  Google Scholar 

  • Fardi M, Sayevand K (2012) Homotopy analysis method: a fresh view on Benjamin-Bona-Mahony-Burgers equation. J Math Comput Sci 4:494–501

    Google Scholar 

  • Ganji ZZ, Ganji DD, Bararnia H (2009) Approximate general and explicit solutions of nonlinear BBMB equations by Exp-Function method. Appl Math Model 33:1836–1841

    Article  MathSciNet  MATH  Google Scholar 

  • Guo C, Fang S (2012) Optimal decay rates of solutions for a multidimensional generalized Benjamin-Bona-Mahony equation. Nonlin Anal Theory Methods Appl 75:3385–3392

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Kaya D (2004) A numerical simulation of solitary-wave solutions of the generalized regularized long-wave equation. Appl Math Comput 149:833–841

    MathSciNet  MATH  Google Scholar 

  • Kaya D, Inan IE (2004) Exact and numerical traveling wave solutions for nonlinear coupled equations using symbolic computation. Appl Math Comput 151:775–787

    MathSciNet  MATH  Google Scholar 

  • Kazeminia M, Tolou P, Mahmoudi J, Khatami I, Tolou N (2009) Solitary and periodic solutions of BBMB equation via exp-function method. Adv Studies Theor Phys 3:461–471

    MATH  Google Scholar 

  • Korpusov MO (2012) On the blow-up of solutions of the Benjamin-Bona-Mahony-Burgers and Rosenau-Burgers equations. Nonlin Anal Theory Methods Appl 75:1737–1743

    Article  MathSciNet  MATH  Google Scholar 

  • Mei M (1998) Large-time behavior of solution for generalized Benjamin-Bona-Mahony-Burgers equations. Nonlin Anal Theory Methods Appl 33:699–714

    Article  MathSciNet  MATH  Google Scholar 

  • Omrani Kh, Ayadi M (2008) Finite difference discretization of the Benjamin-Bona-Mahony-Burgers equation. Num Methods Part Diff Eq 24:239–248

    Article  MathSciNet  MATH  Google Scholar 

  • Tari H, Ganji DD (2007) Approximate explicit solutions of nonlinear BBMB equations by Hes methods and comparison with the exact solution. Phys Lett A 367:95–101

    Article  MATH  Google Scholar 

  • Yina H, Hu J (2010) Exponential decay rate of solutions toward traveling waves for the Cauchy problem of generalized Benjamin-Bona-Mahony-Burgers equations. Nonlin Anal Theory Methods Appl 73:1729–1738

    Article  MathSciNet  MATH  Google Scholar 

  • Zhou YL (1990) Applications of discrete functional analysis of finite difference method. International Academic Publishers, New York

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematical Science, University of Kashan, Kashan, Iran

    Akbar Mohebbi & Zahra Faraz

Authors
  1. Akbar Mohebbi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zahra Faraz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Akbar Mohebbi.

Additional information

Communicated by Jorge Zubelli.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mohebbi, A., Faraz, Z. Solitary wave solution of nonlinear Benjamin–Bona–Mahony–Burgers equation using a high-order difference scheme. Comp. Appl. Math. 36, 915–927 (2017). https://doi.org/10.1007/s40314-015-0272-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40314-015-0272-x

Keywords

Mathematics Subject Classification

Search

Navigation