Abstract and Applied Analysis

Volume 2013 (2013), Article ID 579731, 2 pages

http://dx.doi.org/10.1155/2013/579731

Letter to the Editor

He's Max-Min Approach to a Nonlinear Oscillator with Discontinuous Terms

Hui-Li Zhang and Fang Xie

Department of Mathematics, Kunming University, No. 2 Puxin Road, Kunming, Yunnan 650214, China

Received 25 December 2012; Accepted 29 December 2012

Copyright © 2013 Hui-Li Zhang and Fang Xie. 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.

Recently, the max-min approach was systematically studied in the review article (Ji-Huan, 2012). This paper concludes that He's max-min approach is also a very much effective method for nonlinear oscillators with discontinuous terms.

The ancient Chinese mathematics revives modern applications [1–8]; hereby, we show that He’s max-min approach [1, 9–11] is also very effective for nonlinear oscillators with discontinuous terms.

The max-min approach was first proposed in 2008 based on an ancient Chinese mathematics, and it has become a well-known method for nonlinear oscillators; see, for example, [12–14].

To illustrate the basic idea of the max-min approach [1], we consider the following nonlinear oscillator:

By a similar treatment as given in [1], we have

where is the unknown frequency.

According to an ancient Chinese inequality [1, 8, 10, 11], we have

where , and are constants.

According to He’s max-min approach, we set

or

from which the frequency can be determined approximately as

which is the same as that obtained by the homotopy perturbation method [15].

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