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Mathematical Problems in Engineering
Volume 2015, Article ID 132651, 7 pages
http://dx.doi.org/10.1155/2015/132651
Research Article

Homotopy Analysis Method for Nonlinear Periodic Oscillating Equations with Absolute Value Term

State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

Received 25 March 2015; Revised 23 June 2015; Accepted 5 July 2015

Academic Editor: Raffaele Solimene

Copyright © 2015 Jifeng Cui et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Based on the homotopy analysis method (HAM), an analytic approach is proposed for highly nonlinear periodic oscillating equations with absolute value terms. The nonsmoothness of absolute value terms is handled by means of Fourier expansion, and the convergence is accelerated by using the iteration method. Two typical examples which can not be solved by the method of averaging of perturbation technique are employed to illustrate the validity and flexibility of this approach. Rather, accurate approximations are obtained using the HAM-based approach. The proposed approach has general meanings and thus can be used to solve many highly nonlinear periodic oscillating systems with this type of nonsmoothness of absolute value term.