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

Abstract

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. References 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