Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2014 (2014), Article ID 513473, 11 pages
http://dx.doi.org/10.1155/2014/513473
Research Article

Approximate Periodic Solutions for Oscillatory Phenomena Modelled by Nonlinear Differential Equations

“Politehnica” University of Timişoara, Department of Mathematics, Piata Victoriei 2, 300006 Timişoara, Romania

Received 14 January 2014; Accepted 29 March 2014; Published 23 April 2014

Academic Editor: Baocang Ding

Copyright © 2014 Constantin Bota 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, “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 · View at Scopus
  2. M. H. Pashai, I. Khatami, 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 Scopus
  3. X. Ma, L. Wei, and Z. Guo, “He's homotopy perturbation method to periodic solutions of nonlinear Jerk equations,” Journal of Sound and Vibration, vol. 314, no. 1-2, pp. 217–227, 2008. View at Publisher · View at Google Scholar · View at Scopus
  4. S. S. Ganji, D. D. Ganji, M. G. Sfahani, and S. Karimpour, “Application of AFF and HPM to the systems of strongly nonlinear oscillation,” Current Applied Physics, vol. 10, no. 5, pp. 1317–1325, 2010. View at Publisher · View at Google Scholar · View at Scopus
  5. Y. Khan, M. Akbarzade, and A. Kargar, “Coupling of homotopy and the variational approach for a conservative oscillator with strong odd-nonlinearity,” Scientia Iranica A, vol. 19, no. 3, pp. 417–422, 2012. View at Google Scholar
  6. 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 · View at Scopus
  7. M. O. Kaya, S. Altay Demirbağ, and F. Özen Zengin, “Higher-order approximate periodic solutions of a nonlinear oscillator with discontinuity by variational approach,” Mathematical Problems in Engineering, vol. 2009, Article ID 450862, 9 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  8. H. P. W. Gottlieb, “Harmonic balance approach to periodic solutions of non-linear jerk equations,” Journal of Sound and Vibration, vol. 271, no. 3–5, pp. 671–683, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  9. H. P. W. Gottlieb, “Harmonic balance approach for a degenerate torus of a nonlinear jerk equation,” Journal of Sound and Vibration, vol. 322, no. 4-5, pp. 1005–1008, 2009. View at Publisher · View at Google Scholar · View at Scopus
  10. S. Telli and O. Kopmaz, “Free vibrations of a mass grounded by linear and nonlinear springs in series,” Journal of Sound and Vibration, vol. 289, no. 4-5, pp. 689–710, 2006. View at Publisher · View at Google Scholar · View at Scopus
  11. A. Beléndez, A. Hernández, T. Beléndez et al., “Application of the harmonic balance method to a nonlinear oscillator typified by a mass attached to a stretched wire,” Journal of Sound and Vibration, vol. 302, no. 4-5, pp. 1018–1029, 2007. View at Publisher · View at Google Scholar · View at Scopus
  12. A. Beléndez, E. Gimeno, T. Beléndez, and A. Hernández, “Rational harmonic balance based method for conservative nonlinear oscillators: application to the Duffing equation,” Mechanics Research Communications, vol. 36, no. 6, pp. 728–734, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. S. K. Lai, C. W. Lim, B. S. Wu, C. Wang, Q. C. Zeng, and X. F. He, “Newton-harmonic balancing approach for accurate solutions to nonlinear cubic-quintic Duffing oscillators,” Applied Mathematical Modelling, vol. 33, no. 2, pp. 852–866, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  14. B. Ahmad and B. S. Alghamdi, “Approximation of solutions of the nonlinear Duffing equation involving both integral and non-integral forcing terms with separated boundary conditions,” Computer Physics Communications, vol. 179, no. 6, pp. 409–416, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  15. H. Yao, “Solution of the Duffing equation involving both integral and non-integral forcing terms,” Computer Physics Communications, vol. 180, no. 9, pp. 1481–1488, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. S. Ghosh, A. Roy, and D. Roy, “An adaptation of Adomian decomposition for numeric-analytic integration of strongly nonlinear and chaotic oscillators,” Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 4–6, pp. 1133–1153, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  17. L. Xu, “Application of He's parameter-expansion method to an oscillation of a mass attached to a stretched elastic wire,” Physics Letters, Section A: General, Atomic and Solid State Physics, vol. 368, no. 3-4, pp. 259–262, 2007. View at Publisher · View at Google Scholar · View at Scopus
  18. S. Durmaz, S. Altay Demirbağ, and M. O. Kaya, “Approximate solutions for nonlinear oscillation of a mass attached to a stretched elastic wire,” Computers & Mathematics with Applications, vol. 61, no. 3, pp. 578–585, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  19. V. Marinca and N. Herişanu, “Periodic solutions of Duffing equation with strong non-linearity,” Chaos, Solitons and Fractals, vol. 37, no. 1, pp. 144–149, 2008. View at Publisher · View at Google Scholar · View at Scopus
  20. N. Jamshidi and D. D. Ganji, “Application of energy balance method and variational iteration method to an oscillation of a mass attached to a stretched elastic wire,” Current Applied Physics, vol. 10, no. 2, pp. 484–486, 2010. View at Publisher · View at Google Scholar · View at Scopus
  21. F. Geng, “A piecewise variational iteration method for treating a nonlinear oscillator of a mass attached to a stretched elastic wire,” Computers & Mathematics with Applications, vol. 62, no. 4, pp. 1641–1644, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  22. D. D. Ganji, M. Gorji, S. Soleimani, and M. Esmaeilpour, “Solution of nonlinear cubic-quintic Duffing oscillators using He's Energy Balance Method,” Journal of Zhejiang University: Science A, vol. 10, no. 9, pp. 1263–1268, 2009. View at Publisher · View at Google Scholar · View at Scopus
  23. S.-D. Feng and L.-Q. Chen, “Homotopy analysis approach to periodic solutions of a nonlinear jerk equation,” Chinese Physics Letters, vol. 26, no. 12, Article ID 124501, 2009. View at Publisher · View at Google Scholar · View at Scopus
  24. S. J. Liao, “Homotopy analysis method: a new analytic method for nonlinear problems,” Applied Mathematics and Mechanics, vol. 19, no. 10, pp. 885–890, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. Y. H. Qian, S. K. Lai, W. Zhang, and Y. Xiang, “Study on asymptotic analytical solutions using HAM for strongly nonlinear vibrations of a restrained cantilever beam with an intermediate lumped mass,” Numerical Algorithms, vol. 58, no. 3, pp. 293–314, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  26. A. Golbabai, M. Fardi, and K. Sayevand, “Application of the optimal homotopy asymptotic method for solving a strongly nonlinear oscillatory system,” Mathematical and Computer Modelling, vol. 58, no. 11-12, pp. 1837–1843, 2012. View at Publisher · View at Google Scholar · View at Scopus
  27. A. Y. T. Leung and Z. Guo, “Residue harmonic balance approach to limit cycles of non-linear jerk equations,” International Journal of Non-Linear Mechanics, vol. 46, no. 6, pp. 898–906, 2011. View at Publisher · View at Google Scholar · View at Scopus
  28. A. Elías-Zúñiga, O. Martínez-Romero, and R. K. Córdoba-Díaz, “Approximate solution for the Duffing-harmonic oscillator by the enhanced cubication method,” Mathematical Problems in Engineering, vol. 2012, Article ID 618750, 12 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. S. S. Motsa and P. Sibanda, “A note on the solutions of the Van der Pol and Duffing equations using a linearisation method,” Mathematical Problems in Engineering, vol. 2012, Article ID 693453, 10 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. J. Awrejcewicz, “Numerical investigations of the constant and periodic motions of the human vocal cords including stability and bifurcation phenomena,” Dynamics and Stability of Systems, vol. 5, no. 1, pp. 11–28, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. I. Andrianov and J. Awrejcewicz, “A role of initial conditions choice on the results obtained using different perturbation methods,” Journal of Sound and Vibration, vol. 236, no. 1, pp. 161–165, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. X. Wu and B. Wang, “Multidimensional adapted Runge-Kutta-Nyström methods for oscillatory systems,” Computer Physics Communications. An International Journal and Program Library for Computational Physics and Physical Chemistry, vol. 181, no. 12, pp. 1955–1962, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  33. A. A. Kosti, Z. A. Anastassi, and T. E. Simos, “An optimized explicit Runge-Kutta-Nyström method for the numerical solution of orbital and related periodical initial value problems,” Computer Physics Communications, vol. 183, no. 3, pp. 470–479, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  34. W. Shi and X. Wu, “On symplectic and symmetric ARKN methods,” Computer Physics Communications, vol. 183, no. 6, pp. 1250–1258, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  35. Z. Chen, X. You, W. Shi, and Z. Liu, “Symmetric and symplectic ERKN methods for oscillatory Hamiltonian systems,” Computer Physics Communications, vol. 183, no. 1, pp. 86–98, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  36. V. P. Chua and M. Porter, “Cubic-quintic duffing oscillator,” in Proceedings of the Annual Conference on Chaos and Nonlinear Dynamics, Dynamic Days, 2004.
  37. T. Kalmár-Nagy and B. Balachandran, “Forced harmonic vibration of a Duffing oscillator with linear viscous damping,” in The Duffing Equation: Nonlinear Oscillators and their Behaviour, I. Kovacic and M. J. Brennan, Eds., pp. 139–174, John Wiley and Sons, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  38. J. Awrejcewicz, “Numerical versus analytical conditions for chaos, using the example of the Duffing oscillator,” Journal of the Physical Society of Japan, vol. 60, no. 3, pp. 785–788, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  39. V. Patidar and K. K. Sud, “Bifurcation and chaos in simple jerk dynamical systems,” Pramana - Journal of Physics, vol. 64, no. 1, pp. 75–93, 2005. View at Google Scholar · View at Scopus
  40. 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 Zentralblatt MATH · View at MathSciNet · View at Scopus
  41. B. S. Wu, C. W. Lim, and W. P. Sun, “Improved harmonic balance approach to periodic solutions of non-linear jerk equations,” Physics Letters, Section A: General, Atomic and Solid State Physics, vol. 354, no. 1-2, pp. 95–100, 2006. View at Publisher · View at Google Scholar · View at Scopus
  42. H. Hu, “Perturbation method for periodic solutions of nonlinear jerk equations,” Physics Letters A, vol. 372, no. 23, pp. 4205–4209, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  43. H. Hu, M. Y. Zheng, and Y. J. Guo, “Iteration calculations of periodic solutions to nonlinear jerk equations,” Acta Mechanica, vol. 209, no. 3-4, pp. 269–274, 2010. View at Publisher · View at Google Scholar · View at Scopus
  44. J. I. Ramos, “Analytical and approximate solutions to autonomous, nonlinear, third-order ordinary differential equations,” Nonlinear Analysis: Real World Applications, vol. 11, no. 3, pp. 1613–1626, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus