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

Abstract and Applied Analysis, Jan 2013

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

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]. References H. Ji-Huan, “Asymptotic methods for solitary solutions and compactons,” Abstract and Applied Analysis, vol. 2012, Article ID 916793, 130 pages, 2012. View at Publisher · View at Google ScholarH.-L. Zhang and L.-J. Qin, “An ancient Chinese mathematical algorithm and its application to nonlinear oscillators,” Computers & Mathematics with Applications, vol. 61, no. 8, pp. 2071–2075, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNetL. Xu, “Estimation of the length constant of a long cooling fin by an ancient Chinese algorithm,” Thermal Science, vol. 15, Supplement 1, pp. S149–S152, 2011. View at Publisher · View at Google Scholar · View at ScopusJ.-H. He and Q. Yang, “Solitary wavenumber-frequency formulation using an ancient Chinese arithmetic,” International Journal of Modern Physics B, vol. 24, no. 24, pp. 4747–4751, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNetL.-H. Zhou and J. H. He, “The variational approach coupled with an ancient Chinese mathematical method to the relativistic oscillator,” Mathematical & Computational Applications, vol. 15, no. 5, pp. 930–935, 2010. View at Google Scholar · View at MathSciNetT. Zhong, “Ancient Chinese musical scales: best approximations, but why?” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 2, pp. 161–166, 2009. View at Google Scholar · View at ScopusJ. C. Lan and Z. Yang, “Continued fraction method for an ancient Chinese musical equation,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 2, pp. 167–169, 2009. View at Google Scholar · View at ScopusJ.-H. He, “Solution of nonlinear equations by an ancient Chinese algorithm,” Applied Mathematics and Computation, vol. 151, no. 1, pp. 293–297, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNetJ. H. He, “Max-min approach to nonlinear oscillators,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 2, pp. 207–210, 2008. View at Google Scholar · View at ScopusJ. H. He, “An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering,” International Journal of Modern Physics B, vol. 22, no. 21, pp. 3487–3578, 2008. View at Publisher · View at Google Scholar · View at ScopusJ.-H. He, “Some asymptotic methods for strongly nonlinear equations,” International Journal of Modern Physics B, vol. 20, no. 10, pp. 1141–1199, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNetD. D. Gan and M. Azimi, “Application of max min approach and amplitude frequency formulation to nonlinear oscillation systems,” University Politehnica Of Bucharest Scientific Bulletin A, vol. 74, no. 3, pp. 131–140, 2012. View at Google ScholarS. A. Demirbağ and M. O. Kaya, “Application of he's max-min approach to a generalized nonlinear discontinuity equation,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 11, no. 4, pp. 269–272, 2010. View at Google Scholar · View at ScopusD. Q. Zeng, “Nonlinear oscillator with discontinuity by the max-min approach,” Chaos, Solitons & Fractals, vol. 42, no. 5, pp. 2885–2889, 2009. View at Publisher · View at Google Scholar · View at ScopusJ.-H. He, “The homotopy perturbation method nonlinear oscillators with discontinuities,” Applied Mathematics and Computation, vol. 151, no. 1, pp. 287–292, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet


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Hui-Li Zhang, Fang Xie. He's Max-Min Approach to a Nonlinear Oscillator with Discontinuous Terms, Abstract and Applied Analysis, 2013, DOI: 10.1155/2013/579731