Periodic wave solution of a second order nonlinear ordinary differential equation by Homotopy analysis method

Hao Song, Longbin Tao


The periodic wave solution of a second order nonlinear ordinary differential equation is obtained by the homotopy analysis method, an analytical, totally explicit mathematical technique. By choosing a proper auxiliary parameter, the new series solution converges very fast. The method provides us with a simple way to adjust the convergence region. Furthermore, a significant improvement of the convergence rate and region is achieved by applying Homotopy-Pade approximants. Three examples demonstrate the excellent computation accuracy and efficiency of the present HAM approach. The present method could be extended for more complicated wave equations.

  • Abbasbandy, S., Homotopy analysis method for generalized Benjamin--Bona--Mahony equation, Z. angew. Math. Phys., 59, 2008, 51--62. doi:10.1007/s00033-007-6115-x
  • Liao, S. J., An approximate solution technique not depending on small parameters: a special example, Int. J. Nonlinear Mech., 30, 1995, 371--380. doi:10.1016/0020-7462(94)00054-E
  • Liao, S. J., Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman and Hall/CRC, Florida, 2004.
  • Liao, S. J. and Cheung, K. F., Homotopy analysis of nonlinear progressive waves in deep water, J. Eng. Math., 45, 2003, 105--116. doi:10.1023/A:1022189509293
  • Tao, L., Song, H. and Chakrabarti, S., Nonlinear progressive waves in water of finite depth--óan analytic approximation, Coast. Eng., 54, 2007, 825--834. doi:10.1016/j.coastaleng.2007.05.008
  • Wang, C., Wu, Y. and Wu, W., Solving the nonlinear periodic wave problems with the Homotopy Analysis Method, Wave Motion, 41, 2005, 329--337. doi:10.1016/j.wavemoti.2004.08.002


Homotopy analysis method, nonlinear wave equation, periodic wave solution

Full Text:



Remember, for most actions you have to record/upload into this online system
and then inform the editor/author via clicking on an email icon or Completion button.
ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.