Topological Methods in Nonlinear Analysis

Chinese mathematics for nonlinear oscillators

Ling Zhao

Full-text: Open access

Abstract

Ancient Chinese mathematicians made dramatic progress toward answering one of the oldest, most fundamental problem of how to solve approximately a real root of a nonlinear algebra equation in about 2nd century BC. The idea was further extended to nonlinear differential equations by J. H. He in 2002. In this paper, J. H. He's frequency-amplitude formation is used to find periodic solution of a pure nonlinear oscillator (without a linear term). The obtained result is of remarkable accuracy.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 31, Number 2 (2008), 383-387.

Dates
First available in Project Euclid: 13 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1463150283

Mathematical Reviews number (MathSciNet)
MR2432097

Zentralblatt MATH identifier
1146.01303

Citation

Zhao, Ling. Chinese mathematics for nonlinear oscillators. Topol. Methods Nonlinear Anal. 31 (2008), no. 2, 383--387. https://projecteuclid.org/euclid.tmna/1463150283


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References

  • A. Beléndez, A. Hernández, T. Beléndez, C. Neipp and A. Márquez, Application of the homotopy perturbation method to the nonlinear pendulum , European J. Phys., 28 (2007), 93–104 \ref\key 2
  • A. Beléndez, A. Hernández, T. Beléndez, et al., Application of the homotopy perturbation method to the Duffing-harmonic oscillator , Internat. J. Nonlinear Sci., 8 (2007), 79–88 \ref\key 3
  • A. Beléndez, C. Pascual, A. Márquez and D. I. Méndez, Application of He's homotopy perturbation method to the relativistic (an)harmonic oscillator. \romI. Comparison between approximate and exact frequencies, Internat. J. Nonlinear Sci., 8 (2007), 483–492
  • \ref\key 4A. Beléndez, C. Pascual, D. I. Méndez, M. L. Álvarez and C. Neipp, Application of He's homotopy perturbation method to the relativistic (an)harmonic oscillator. \romII. A More accurate approximate aolution, Internat. J. Nonlinear Sci., 8 (2007), 493–504 \ref\key 5
  • L. Geng and X.-C. Cai, He's frequency formulation for nonlinear oscillators , European J. Phys., 28 , 923–931 (2007) \ref\key 6
  • J. H. He, Variational iteration method –- a kind of non-linear analytical technique: Some examples , Internat. J. Nonlinear Mech., 34 , 699–708 (1999) \ref\key 7 ––––, Ancient Chinese algorithm: The Ying Buzu Shu \rom(method of surplus and deficiency\rom) vs Newton iteration method, Appl. Math Mech., 23 , 1407–1412 (2002) \ref\key 8 ––––, Solution of nonlinear equations by an ancient Chinese algorithm , Applied Mathematics and Computation, 151 , no. (2004), 293–297 \ref\key 9 ––––, Homotopy perturbation method for bifurcation of nonlinear problems , Internat. J. Nonlinear Sci., 6 , 207–208 (2005) \ref\key 10 ––––, New interpretation of homotopy perturbation method , Internat. J. Mod. Phys. B, 20 , 2561–2568 (2006) \ref\key 11 ––––, Some asymptotic methods for strongly nonlinear equations , Internat. J. Mod. Phys. B, 20 , 1141–1199 (2006) \ref\key 12 ––––, Nonperturbative methods for strongly nonlinear problems , dissertation.de -Verlag im Internet GmbH (2006) \ref\key 13
  • J. H. He and X.H. Wu, Exp-function method for nonlinear wave equations , Chaos Solitons Fractals, 30 (2006), 700–708 \ref\key 14
  • T. Öziş and A. Y\ild\ir\im, A comparative study of He's homotopy perturbation method for determining frequency-amplitude relation of a nonlinear oscillator with discontinuities , Internat. J. Nonlinear Sci., 8 , 243–248 (2007) \ref\key 15
  • D. H. Shou et al., Application of parameter-expanding method to strongly nonlinear oscillators , Internat. J. Nonlinear Sci., 8 , 121–124 (2007) \ref\key 16
  • E. Yusufoglu, Variational iteration method for construction of some compact and noncompact structures of Klein–Gordon equations , Internat. J. Nonlinear Sci., 8 , 152–158 (2007) \ref\key 17 ––––, Homotopy perturbation mMethod for solving a nonlinear system of second order boundary value problems , Internat. J. Nonlinear Sci., 8 (2007), 353–358 \ref\key 18
  • S.-D. Zhu, Exp-function method for the hybrid-lattice system , Internat. J. Nonlinear Sci., 8 (2007), 461–464 \ref\key 19 ––––, Exp-function method for the discrete mKdV lattice , Internat. J. Nonlinear Sci., 8 (2007), 465–468