Computational and Applied Mathematics

, Volume 36, Issue 2, pp 915–927 | Cite as

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

  • Akbar Mohebbi
  • Zahra Faraz


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.


Benjamin–Bona–Mahony–Burgers equation Finite difference scheme Solvability Unconditional stability Convergence Solitary waves 

Mathematics Subject Classification

65N06 65M12 


  1. Al-Khaled K, Momani S, Alawneh A (2005) Approximate wave solutions for generalized Benjamin-Bona-Mahony-Burgers equations. Appl Math Comput 171:281–292MathSciNetMATHGoogle Scholar
  2. Alquran M, K. Al-Khaled (2011) Sinc and solitary wave solutions to the generalized BBMB equations. Physica Scripta 83:6 (065010) Google Scholar
  3. 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 6:127–131Google Scholar
  4. 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 24:897–910MathSciNetCrossRefMATHGoogle Scholar
  5. Fardi M, Sayevand K (2012) Homotopy analysis method: a fresh view on Benjamin-Bona-Mahony-Burgers equation. J Math Comput Sci 4:494–501Google Scholar
  6. Ganji ZZ, Ganji DD, Bararnia H (2009) Approximate general and explicit solutions of nonlinear BBMB equations by Exp-Function method. Appl Math Model 33:1836–1841MathSciNetCrossRefMATHGoogle Scholar
  7. Guo C, Fang S (2012) Optimal decay rates of solutions for a multidimensional generalized Benjamin-Bona-Mahony equation. Nonlin Anal Theory Methods Appl 75:3385–3392MathSciNetCrossRefMATHGoogle Scholar
  8. Hu J, Hu B, Xu Y (2011) Average implicit linear difference scheme for generalized Rosenau-Burgers equation. Appl Math Comput 217:7557–7563MathSciNetMATHGoogle Scholar
  9. Kaya D (2004) A numerical simulation of solitary-wave solutions of the generalized regularized long-wave equation. Appl Math Comput 149:833–841MathSciNetMATHGoogle Scholar
  10. Kaya D, Inan IE (2004) Exact and numerical traveling wave solutions for nonlinear coupled equations using symbolic computation. Appl Math Comput 151:775–787MathSciNetMATHGoogle Scholar
  11. 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 3:461–471MATHGoogle Scholar
  12. Korpusov MO (2012) On the blow-up of solutions of the Benjamin-Bona-Mahony-Burgers and Rosenau-Burgers equations. Nonlin Anal Theory Methods Appl 75:1737–1743MathSciNetCrossRefMATHGoogle Scholar
  13. Mei M (1998) Large-time behavior of solution for generalized Benjamin-Bona-Mahony-Burgers equations. Nonlin Anal Theory Methods Appl 33:699–714MathSciNetCrossRefMATHGoogle Scholar
  14. Omrani Kh, Ayadi M (2008) Finite difference discretization of the Benjamin-Bona-Mahony-Burgers equation. Num Methods Part Diff Eq 24:239–248MathSciNetCrossRefMATHGoogle Scholar
  15. Tari H, Ganji DD (2007) Approximate explicit solutions of nonlinear BBMB equations by Hes methods and comparison with the exact solution. Phys Lett A 367:95–101CrossRefMATHGoogle Scholar
  16. 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 73:1729–1738MathSciNetCrossRefMATHGoogle Scholar
  17. Zhou YL (1990) Applications of discrete functional analysis of finite difference method. International Academic Publishers, New YorkGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2015

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of Mathematical ScienceUniversity of KashanKashanIran

Personalised recommendations