Advertisement

Improvement of Analytical Solution to the Inverse Truly Nonlinear Oscillator by Extended Iterative Method

  • B. M. Ikramul HaqueEmail author
  • Md. Asifuzzaman
  • M. Kamrul Hasan
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 655)

Abstract

A new approach to the Mickens extended iteration method has been presented to obtain approximate analytic solutions for nonlinear oscillatory differential equation. To illustrate the accuracy of the approximate solution of the inverse nonlinear oscillator “\( \ddot{x} + x^{ - 1} = 0 \)”, we have used the Fourier series and utilized indispensable truncated terms in each iterative step. In this article the solution gives more accurate result significantly than other existing methods and shows a good agreement with its exact solution. The percentage of error between exact frequency and our third approximate frequency is very low. We have compared all the results to exact results and other existing results and the method is convergent as well as consistent. Finally, an example is given to show the effectiveness of the approximate solution.

Keywords

Extended iterative method Inverse truly nonlinear oscillator Analytical solution 

AMS Subject Classification

34A34 34B99 

Notes

Acknowledgement

The authors are grateful to the honorable reviewers’ for their constructive suggestions/comments to improve the quality of this article. The authors are also grateful to Md. Shahinur Alam Sarker, Assistant Professor, Department of Humanities, Khulna University of Engineering & Technology, Khulna-9203, for his assistance to prepare the revised manuscript.

References

  1. 1.
    Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillation. Wiley, New York (1979)zbMATHGoogle Scholar
  2. 2.
    Alam, A., Rahman, H., Haque, B.M.I., Ali Akbar, M.: Perturbation technique for analytic solutions of fourth order near critically damped nonlinear systems. Int. J. Basic Appl. Sci. 11, 131–138 (2011)Google Scholar
  3. 3.
    He, J.H.: Modified Lindstedt-Poincare methods for some non-linear oscillations. Part III: double series expansion. Int. J. Nonlinear Sci. Numer. Simul. 2, 317–320 (2001)zbMATHGoogle Scholar
  4. 4.
    Ramos, J.I.: Approximate methods based on order reduction for the periodic solutions of nonlinear third-order ordinary differential equations. Appl. Math. Comput. 215, 4304–4319 (2010)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Xu, H., Cang, J.: Analysis of a time fractional wave-like equation with the homotopy analysis method. Phys. Lett. A 372, 1250–1255 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Belendez, A., Pascual, C., Ortuno, M., Belendez, T., Gallego, S.: Application of a modified He’s homotopy perturbation method to obtain higher-order approximations to a nonlinear oscillator with discontinuities. Real World Appl. 10, 601–610 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Mickens, R.E.: Comments on the method of harmonic balance. J. Sound Vib. 94, 456–460 (1984)CrossRefGoogle Scholar
  8. 8.
    Wu, B.S., Sun, W.P., Lim, C.W.: An analytical approximate technique for a class of strongly nonlinear oscillator. Int. J. Nonlinear Mech. 41, 766–774 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hosen, M.A.: Accurate approximate analytical solutions to an anti-symmetric quadratic nonlinear oscillator. Afr. J. Math. Comput. Sci. Res. 6, 77–81 (2013)Google Scholar
  10. 10.
    Mickens, R.E.: Iteration Procedure for determining approximate solutions to nonlinear oscillator equation. J. Sound Vib. 116, 185–188 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lim, C.W., Wu, B.S.: A modified procedure for certain non-linear oscillators. J. Sound Vib. 257, 202–206 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hu, H., Tang, J.H.: A classical iteration procedure valid for certain strongly nonlinear oscillator. J. Sound Vib. 299, 397–402 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mickens, R.E.: Harmonic balance and iteration calculations of periodic solutions to ÿ + y−1 = 0. J. Sound Vib. 306, 968–972 (2007)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chen, Y.M., Liu, J.K.: A modified Mickens iteration procedure for nonlinear oscillators. J. Sound Vib. 314, 465–473 (2008)CrossRefGoogle Scholar
  15. 15.
    Mickens, R.E.: Truly Nonlinear Oscillations. World Scientific, Singapore (2010)CrossRefzbMATHGoogle Scholar
  16. 16.
    Alquran, M., Al-Khaled, K.: Effective approximate methods for strongly nonlinear differential equations with oscillations. Math. Sci. 6, 32 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Turkyilmazoglu, M.: An effective approach for approximate analytical solutions of the damped Duffing equation. Phys. Scr. 86, 01530 (2012)zbMATHGoogle Scholar
  18. 18.
    Haque, B.M.I.: Modified solutions of some oscillators by iteration procedure. J. Egypt. Math. Soc. 21, 68–73 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Haque, B.M.I., Hossain, M.R.: An analytic investigation of the quadratic nonlinear oscillator by an iteration method. BJMCS 13, 1–8 (2015)Google Scholar
  20. 20.
    Haque, B.M.I., Bostami, M.B., Hossain, M.M.A., Hossain, M.R., Rahman, M.M.: Mickens iteration like method for approximate solutions of the inverse cubic truly nonlinear oscillator. BJMCS 13, 1–9 (2015). Article no. 22823Google Scholar
  21. 21.
    Haque, B.M.I., Hossain, M.M.A., Bostami, M.B., Hossain, M.R.: Analytical approximate solutions to the nonlinear singular oscillator: an iteration procedure. BJMCS 14, 1–7 (2016). Article no. 23263CrossRefGoogle Scholar
  22. 22.
    Taylor, A.E., Mann, W.R.: Advance Calculus. Wiley, New York (1983)Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  • B. M. Ikramul Haque
    • 1
    Email author
  • Md. Asifuzzaman
    • 1
  • M. Kamrul Hasan
    • 2
  1. 1.Department of MathematicsKhulna University of Engineering & TechnologyKhulnaBangladesh
  2. 2.Department of MathematicsRajshahi University of Engineering & TechnologyRajshahiBangladesh

Personalised recommendations