Mathematical Problems in Engineering

Volume 2015, Article ID 132651, 7 pages

http://dx.doi.org/10.1155/2015/132651

## 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.

#### 1. Introduction

When a periodic oscillating system is governed by a nonlinear differential equation with terms of absolute value, it is hard to gain its analytic approximation of period by perturbation methods [1, 2]. This study is important because such kinds of systems are always encountered in the engineering problems. In 1982, the method of averaging is used to study the monitoring system named Lewis regulator by Hagedorn [3]. In 1995, a new asymptotic method is applied to study the large amplitude rolling using only roll extinction data by Chan et al. [4]. In 1999, the main theoretical and experimental aspects connected with the jumps of amplitude of rolling motion in the presence of a bifurcation have been outlined by Francescutto and Contento [5]. All of the above oscillating systems contain the absolute value terms. In this paper, let us consider a type of periodic oscillating systems with absolute value term [6, 7]subject to initial conditionswith being the amplitude of , being a continuous operator, and dot being the differentiation with respect to .

For example, consider the periodic oscillating system governed bywhere is the natural frequency and is a physical parameter. The absolute term brings great difficulty, because the nonlinear term becomes nonsmooth. The method of averaging [8, 9] in perturbation theory is frequently used to obtain analytical approximations of (3). With the method of averaging, the st-order approximate frequency of (3) reads [10] as follows:Unfortunately, the above perturbation result is valid only for small amplitude of oscillation in the case of , as shown in Figure 1. It is a common knowledge that in most cases perturbation approximations are valid only for weakly nonlinear problems. To overcome the restrictions of perturbation methods, Liao [11, 12] developed an analytic approximation technique for highly nonlinear problems, namely, the homotopy analysis method (HAM) [13, 14]. The HAM, based on a basic concept in topology, transfers nonlinear problems into an infinite number of linear subproblems which are independent of any small/large physical parameters. So, this technique works efficiently for many nonlinear problems in science and engineering. In particular, the HAM provides a convenient way to guarantee the convergence of solution series. In addition, the HAM provides us with great freedom to choose the equation type of the linear sub problems so that the high-order approximations can be obtained easily in general. Due to these advantages, the HAM has been successfully applied to solve lots of different types of nonlinear problems [15–22]. These successful applications indicate the validity and potential of the HAM.