Nonlinear Dynamics

, Volume 51, Issue 1–2, pp 83–87 | Cite as

Soliton solutions for the fifth-order KdV equation with the homotopy analysis method

  • S. Abbasbandy
  • F. Samadian Zakaria
Original Paper


An analytic technique, the homotopy analysis method (HAM), is applied to obtain the soliton solution of the fifth-order KdV equation. The homotopy analysis method (HAM) provides us with a new way to obtain series solutions of such problems. HAM contains the auxiliary parameter ℏ, which provides us with a simple way to adjust and control the convergence region of series solution.


KdV equation Homotopy analysis method Fifth-order KdV Soliton solution 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abbasbandy, S.: The application of homotopy analysis method to nonlinear equations arising in heat transfer. Phys. Lett. A 360, 109–113 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Abbasbandy, S.: The application of homotopy analysis method to solve a generalized Hirota–Satsuma coupled KdV equation. Phys. Lett. A 361, 478–483 (2007)Google Scholar
  3. 3.
    Abbasbandy, S.: Homotopy analysis method for heat radiation equations. Int. Commun. Heat Mass (in Press)Google Scholar
  4. 4.
    Ayub, M., Rasheed, A., Hayat, T.: Exact flow of a third grade fluid past a porous plate using homotopy analysis method. Int. J. Eng. Sci. 41, 2091–2103 (2003)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Hayat, T., Khan, M.: Homotopy solutions for a generalized second-grade fluid past a porous plate. Nonlinear Dyn. 42, 395–405 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Hayat, T., Khan, M., Ayub, M.: On non-linear flows with slip boundary condition. Z. Angew. Math. Phys. 56, 1012–1029 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Lax, P.D.: Periodic solutions of the Korteweg—de Vries equation. Commun. Pure Appl. Math. 28, 141–188 (1975)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Liao, S.J.: The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems. Ph.D. Thesis, Shanghai Jiao Tong University (1992)Google Scholar
  9. 9.
    Liao, S.J.: Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman and Hall/CRC, Boca Raton, FL (2003)Google Scholar
  10. 10.
    Liao, S.J.: On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet. J. Fluid Mech. 488, 189–212 (2003)Google Scholar
  11. 11.
    Liao, S.J.: A new branch of solutions of boundary-layer flows over an impermeable stretched plate. Int. J. Heat Mass Transfer 48, 2529–2539 (2005)CrossRefzbMATHGoogle Scholar
  12. 12.
    Liao, S.J., Su, J., Chwang, A.T.: Series solutions for a nonlinear model of combined convective and radiative cooling of a spherical body. Int. J. Heat Mass Transfer 49, 2437–2445 (2006)CrossRefzbMATHGoogle Scholar
  13. 13.
    Liao, S.J., Magyari, E.: Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones. Z. Angew. Math. Phys. 57, 777–792 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Liao, S.J.: Series solutions of unsteady boundary-layer flows over a stretching flat plate. Stud. Appl. Math. 117, 239–264 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Sajid, M., Hayat, T., Asghar, S.: On the analytic solution of the steady flow of a fourth grade fluid. Phys. Lett. A 355, 18–26 (2006)CrossRefGoogle Scholar
  16. 16.
    Tan, Y., Abbasbandy, S.: Homotopy analysis method for quadratic Riccati differential equation. Commun. Nonlinear Sci. Numer. Simul. (in press)Google Scholar
  17. 17.
    Wazwaz, A.: Solitons and periodic solutions for the fifth-order KdV equations. Appl. Math. Lett. 19, 1162–1167 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Wang, C., Wu, Y., Wu, W.: Solving the nonlinear periodic wave problems with the homotopy analysis method. Wave Motion 41, 329–337 (2005)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Department of MathematicsImam Khomeini International UniversityGhazvinIran
  2. 2.College of EtesamTehranIran

Personalised recommendations