Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2009, Article ID 450862, 9 pages
http://dx.doi.org/10.1155/2009/450862
Research Article

Higher-Order Approximate Periodic Solutions of a Nonlinear Oscillator with Discontinuity by Variational Approach

1Faculty of Aeronautics and Astronautics, Istanbul Technical University, Maslak 34469, Istanbul, Turkey
2Faculty of Science and Letters, Istanbul Technical University, Maslak 34469, Istanbul, Turkey

Received 16 October 2008; Revised 12 March 2009; Accepted 14 May 2009

Academic Editor: Ekaterina Pavlovskaia

Copyright © 2009 M. Orhan Kaya 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.

Linked References

  1. J.-H. He and X.-H. Wu, “Variational iteration method: new development and applications,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 881–894, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. M. Rafei, D. D. Ganji, H. Daniali, and H. Pashaei, “The variational iteration method for nonlinear oscillators with discontinuities,” Journal of Sound and Vibration, vol. 305, no. 4-5, pp. 614–620, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  3. L.-N. Zhang and J.-H. He, “Resonance in Sirospun yarn spinning using a variational iteration method,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 1064–1066, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. T. Öziş and A. Yıldırım, “A study of nonlinear oscillators with u1/3 force by He's variational iteration method,” Journal of Sound and Vibration, vol. 306, no. 1-2, pp. 372–376, 2007. View at Publisher · View at Google Scholar
  5. J.-H. He, “Variational iteration method-some recent results and new interpretations,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 3–17, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. J.-H. He, “Variational iteration method—a kind of non-linear analytical technique: some examples,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 699–708, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. H. Ozer, “Application of the variational iteration method to the boundary value problems with jump discontinuities arising in solid mechanics,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 4, pp. 513–518, 2007. View at Google Scholar
  8. A. Beléndez, C. Pascual, M. Ortuño, T. Beléndez, and S. Gallego, “Application of a modified He's homotopy perturbation method to obtain higher-order approximations to a nonlinear oscillator with discontinuities,” Nonlinear Analysis: Real World Applications, vol. 10, no. 2, pp. 601–610, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. A. Beléndez, A. Hernandez, T. Beléndez, C. Neipp, and A. Marquez, “Higher accuracy analytical approximations to a nonlinear oscillator with discontinuity by He's homotopy perturbation method,” Physics Letters A, vol. 372, no. 12, pp. 2010–2016, 2008. View at Publisher · View at Google Scholar
  10. A. Beléndez, C. Pascual, S. Gallego, M. Ortuño, and C. Neipp, “Application of a modified He's homotopy perturbation method to obtain higher-order approximations of an x1/3 force nonlinear oscillator,” Physics Letters A, vol. 371, no. 5-6, pp. 421–426, 2007. View at Publisher · View at Google Scholar
  11. A. Beléndez, C. Pascual, T. Beléndez, and A. Hernández, “Solution for an anti-symmetric quadratic nonlinear oscillator by a modified He's homotopy perturbation method,” Nonlinear Analysis: Real World Applications, vol. 10, no. 1, pp. 416–427, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. A. Beléndez, T. Beléndez, A. Marquez, and C. Neipp, “Application of He's homotopy perturbation method to conservative truly nonlinear oscillators,” Chaos, Solitons & Fractals, vol. 37, no. 3, pp. 770–780, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. J.-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
  14. A. Beléndez, A. Hernández, T. Beléndez, E. Fernández, M. L. Álvarez, and C. Neipp, “Application of He's homotopy perturbation method to the duffin-harmonic oscillator,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 1, pp. 79–88, 2007. View at Google Scholar
  15. T. Özis and A. Yıldırım, “A comparative study of He's homotopy perturbation method for determining frequency-amplitude relation of a nonlinear oscillator with discontinuities,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 2, pp. 243–248, 2007. View at Google Scholar
  16. H.-M. Liu, “Approximate period of nonlinear oscillators with discontinuities by modified Lindstedt-Poincare method,” Chaos, Solitons & Fractals, vol. 23, no. 2, pp. 577–579, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. S.-Q. Wang and J.-H. He, “Nonlinear oscillator with discontinuity by parameter-expansion method,” Chaos, Solitons & Fractals, vol. 35, no. 4, pp. 688–691, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. F. Ö. Zengin, M. O. Kaya, and S. A. Demirbağ, “Application of parameter-expansion method to nonlinear oscillators with discontinuities,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 3, pp. 267–270, 2008. View at Google Scholar
  19. S. Momani and S. Abuasad, “Application of He's variational iteration method to Helmholtz equation,” Chaos, Solitons & Fractals, vol. 27, no. 5, pp. 1119–1123, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. J.-H. He, “Variational approach for nonlinear oscillators,” Chaos, Solitons & Fractals, vol. 34, no. 5, pp. 1430–1439, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. J.-H. He, “Variational principles for some nonlinear partial differential equations with variable coefficients,” Chaos, Solitons & Fractals, vol. 19, no. 4, pp. 847–851, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. J.-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 Zentralblatt MATH
  23. D.-H. Shou, “Variational approach to the nonlinear oscillator of a mass attached to a stretched wire,” Physica Scripta, vol. 77, no. 4, Article ID 045006, 4 pages, 2008. View at Publisher · View at Google Scholar
  24. Z.-L. Tao, “The frequency-amplitude relationship for some nonlinear oscillators with discontinuity by He's variational method,” Physica Scripta for Experimental and Theoretical Physics, vol. 78, no. 1, Article ID 015004, 2 pages, 2008. View at Google Scholar · View at MathSciNet
  25. B. S. Wu, W. P. Sun, and C. W. Lim, “An analytical approximate technique for a class of strongly non-linear oscillators,” International Journal of Non-Linear Mechanics, vol. 41, no. 6-7, pp. 766–774, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet