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

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.

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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: 𝑢^′′+𝛽𝑢³+𝜀𝑢|𝑢|=0,𝑢(0)=𝐴,𝑢^′(0)=0. (1) By a similar treatment as given in [1], we have 0<𝜔²<𝛽𝐴²+𝜀𝐴, (2) where 𝜔 is the unknown frequency.

According to an ancient Chinese inequality [1, 8, 10, 11], we have 𝜔²=𝑛𝛽𝐴²+𝜀𝐴𝑚+𝑛=𝑘𝛽𝐴²𝑛+𝜀𝐴,𝑘=(),𝑚+𝑛 (3) where 𝑚,𝑛, and 𝑘 are constants.

According to He’s max-min approach, we set ₀^𝑇/4𝑘𝛽𝐴²+𝜀𝐴𝑢−𝛽𝑢³−𝜀𝑢|𝑢|cos𝜔𝑡𝑑𝑡=0,𝑇=2𝜋𝜔, (4) or ₀^𝑇/4𝑘𝛽𝐴²+𝜀𝐴𝐴cos𝜔𝑡−𝛽𝐴³cos³||𝜔𝑡−𝜀𝐴cos𝜔𝑡𝐴cos𝜔𝑡cos𝜔𝑡𝑑𝑡=0, (5) from which the frequency 𝜔 can be determined approximately as 𝜔=34𝛽𝐴²+83𝜋𝜀𝐴, (6) which is the same as that obtained by the homotopy perturbation method [15].