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Mathematical Problems in Engineering
Volume 2009, Article ID 327462, 20 pages
http://dx.doi.org/10.1155/2009/327462
Research Article

Higher-Order Solutions of Coupled Systems Using the Parameter Expansion Method

1Department of Civil and Transportation Engineering, Islamic Azad University, Science and Research Branch Campus, 4716695814 Tehran, Iran
2Department of Civil and Mechanical Engineering, Babol University of Technology, P.O. Box 484, 4714871167 Babol, Iran
3Department of Civil Engineering, Khajeh Nasir University of Technology, 1996715433 Tehran, Iran

Received 29 January 2009; Revised 27 April 2009; Accepted 7 June 2009

Academic Editor: Jerzy Warminski

Copyright © 2009 S. S. Ganji 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. R. E. Mickens, Oscillations in Planar Dynamic Systems, Scientific, Singapore, 1966.
  2. A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations, Wiley-Interscience, New York, NY, USA, 1979. View at MathSciNet
  3. R. P. Agarwal, S. R. Grace, and D. O'Regan, Oscillation Theory for Second Order Dynamic Equations, vol. 5 of Series in Mathematical Analysis and Applications, Taylor & Francis, London, UK, 2003. View at MathSciNet
  4. J.-H. He, Non-Perturbative Methods for Strongly Nonlinear Problems, Dissertation. de-Verlag im Internet GmbH, Berlin, Germany, 2006.
  5. A. H. Nayfeh, Perturbation Methods, John Wiley & Sons, New York, NY, USA, 1973. View at Zentralblatt MATH · View at MathSciNet
  6. J.-H. He, “Homotopy perturbation method for bifurcation on nonlinear problems,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 6, p. 207, 2005. View at Google Scholar
  7. 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
  8. A. R. Ghotbi, A. Barari, and D. D. Ganji, “Solving ratio-dependent predator-prey system with constant effort harvesting using homotopy perturbation method,” Mathematical Problems in Engineering, vol. 2008, Article ID 945420, 8 pages, 2008. View at Google Scholar · View at MathSciNet
  9. S. S. Ganji, D. D. Ganji, S. Karimpour, and H. Babazadeh, “Applications of He's homotopy perturbation method to obtain second-order approximations of the coupled two-degree-of-freedom systems,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 3, pp. 305–314, 2009. View at Google Scholar
  10. S. T. Mohyud-Din and M. A. Noor, “Homotopy perturbation method for solving fourth-order boundary value problems,” Mathematical Problems in Engineering, vol. 2007, Article ID 98602, 15 pages, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. L.-N. Zhang and J.-H. He, “Homotopy perturbation method for the solution of the electrostatic potential differential equation,” Mathematical Problems in Engineering, vol. 2006, Article ID 83878, 6 pages, 2006. View at Google Scholar · View at MathSciNet
  12. I. Khatami, M. H. Pashai, and N. Tolou, “Comparative vibration analysis of a parametrically nonlinear excited oscillator using HPM and numerical method,” Mathematical Problems in Engineering, vol. 2008, Article ID 956170, 11 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. Y.-X. Wang, H.-Y. Si, and L.-F. Mo, “Homotopy perturbation method for solving reaction-diffusion equations,” Mathematical Problems in Engineering, vol. 2008, Article ID 795838, 5 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. J.-H. He, “Modified Lindstedt-Poincaré methods for some strongly non-linear oscillations—II: a new transformation,” International Journal of Nonlinear Mechanics, vol. 37, no. 2, pp. 315–320, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  15. D.-H. Shou and J.-H. He, “Application of parameter-expanding method to strongly nonlinear oscillators,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 1, pp. 121–124, 2007. View at Google Scholar
  16. N. H. Sweilam and R. F. Al-Bar, “Implementation of the parameter-expansion method for the coupled van der pol oscillators,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 2, pp. 259–264, 2009. View at Google Scholar
  17. L.-N. Zhang and L. Xu, “Determination of the limit cycle by He's parameter-expansion for oscillators in a u3/(1+u2) potential,” Zeitschrift für Naturforschung A, vol. 62, no. 7-8, pp. 396–398, 2007. View at Google Scholar
  18. L. Xu, “Application of He's parameter-expansion method to an oscillation of a mass attached to a stretched elastic wire,” Physics Letters A, vol. 368, no. 3-4, pp. 259–262, 2007. View at Publisher · View at Google Scholar
  19. J.-H. He, “A review on some new recently developed nonlinear analytical techniques,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 1, no. 1, pp. 51–70, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. A. Marathe and A. Chatterjee, “Wave attenuation in nonlinear periodic structures using harmonic balance and multiple scales,” Journal of Sound and Vibration, vol. 289, no. 4-5, pp. 871–888, 2006. View at Publisher · View at Google Scholar
  21. H. P. W. Gottlieb, “Harmonic balance approach to limit cycles for nonlinear jerk equations,” Journal of Sound and Vibration, vol. 297, no. 1-2, pp. 243–250, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  22. J.-H. He, “Linearized perturbation technique and its applications to strongly nonlinear oscillators,” Computers & Mathematics with Applications, vol. 45, no. 1–3, pp. 1–8, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. J.-H. He, “Preliminary report on the energy balance for nonlinear oscillations,” Mechanics Research Communications, vol. 29, no. 2-3, pp. 107–111, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. S. S. Ganji, D. D. Ganji, Z. Z. Ganji, and S. Karimpour, “Periodic solution for strongly nonlinear vibration systems by He's energy balance method,” Acta Applicandae Mathematicae, vol. 106, no. 1, pp. 79–92, 2009. View at Publisher · View at Google Scholar
  25. S. S. Ganji, D. D. Ganji, and S. Karimpour, “He's energy balance and He's variational methods for nonlinear oscillations in engineering,” International Journal of Modern Physics B, vol. 23, no. 3, pp. 461–471, 2009. View at Google Scholar
  26. 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
  27. J.-H. He and X.-H. Wu, “Construction of solitary solution and compacton-like solution by variational iteration method,” Chaos, Solitons & Fractals, vol. 29, no. 1, pp. 108–113, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. 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
  29. L. Xu, “Variational approach to solitons of nonlinear dispersive K(m,n) equations,” Chaos, Solitons & Fractals, vol. 37, no. 1, pp. 137–143, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. J.-H. He, “Iteration perturbation method for strongly nonlinear oscillations,” Journal of Vibration and Control, vol. 7, no. 5, pp. 631–642, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. M. A. Noor and S. T. Mohyud-Din, “Variational homotopy perturbation method for solving higher dimensional initial boundary value problems,” Mathematical Problems in Engineering, vol. 2008, Article ID 696734, 11 pages, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. D. D. Ganji, M. Rafei, A. Sadighi, and Z. Z. Ganji, “A comparative comparison of He's method with perturbation and numerical methods for nonlinear vibrations equations,” International Journal of Nonlinear Dynamics in Engineering and Sciences, vol. 1, no. 1, pp. 1–20, 2009. View at Google Scholar
  33. L. Cveticanin, “Vibrations of a coupled two-degree-of-freedom system,” Journal of Sound and Vibration, vol. 247, no. 2, pp. 279–292, 2001. View at Publisher · View at Google Scholar
  34. L. Cveticanin, “The motion of a two-mass system with non-linear connection,” Journal of Sound and Vibration, vol. 252, no. 2, pp. 361–369, 2002. View at Publisher · View at Google Scholar
  35. R. Vito, “On the stability of vibrations of particle in a plane constrained by identifical non-linear springs,” International Journal of Nonlinear Mechanics, vol. 9, no. 84, p. 325, 1974. View at Publisher · View at Google Scholar
  36. E. A. Jackson, Perspectives of Nonlinear Dynamics. Vol. 1, Cambridge University Press, Cambridge, UK, 1989. View at MathSciNet