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Study of strongly nonlinear oscillators using the Aboodh transform and the homotopy perturbation method

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Abstract.

A generalized equation is constructed for a class of classical oscillators with strong anharmonicity which are not exactly solvable. The Aboodh transform-based homotopy perturbation method (ATHPM) is applied to get the approximate analytical solution for the generalized equation and hence some physically relevant anharmonic oscillators are studied as the special cases of this solution. ATHPM provides the approximate analytical solution of the generalized equation in a simple way. The solution from this simple method not only shows excellent agreement with the numerical results but is also found to be of better accuracy in comparison to the solutions obtained from other established approximation methods whenever compared for physically relevant special cases.

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References

  1. R.A. Bonham, L.S. Su, J. Chem. Phys. 45, 2827 (1966)

    Article  ADS  Google Scholar 

  2. Carl M. Bender, Tai Tsun Wu, Phys. Rev. Lett. 21, 406 (1968)

    Article  ADS  Google Scholar 

  3. A. Nayfeh, Perturbation Methods (Willey-Interscience, New York, 1973)

  4. A.H. Nayfeh, D. Mook, Nonlinear Oscillations (John Willey and Sons, New York, 1979)

  5. N.N. Bogoliubov, Y.A. Mitropolsky, Asymptotic Methods in the Theory of Non-Linear Oscillations (CRC Press, 1961)

  6. V. Agrwal, H. Denman, J. Sound Vib. 99, 463 (1985)

    Article  ADS  Google Scholar 

  7. S. Chen, Y. Cheung, S. Lau, Int. J. Nonlinear Mech. 26, 125 (1991)

    Article  ADS  Google Scholar 

  8. Y. Cheung, S. Chen, S. Lau, Int. J. Nonlinear Mech. 26, 367 (1991)

    Article  ADS  Google Scholar 

  9. G. Adomian, J. Math. Anal. Appl. 135, 501 (1988)

    Article  MathSciNet  Google Scholar 

  10. G. Liu, in Conference of 7th Modern Mathematics and Mechanics, Shanghai (Scirp, 1997) pp. 47--53

  11. S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, PhD Thesis, Shanghai Jiao Tong University Shanghai (1992)

  12. Y.P. Liu, S.J. Liao, Z.B. Li, J. Symb. Comput. 55, 72 (2013)

    Article  Google Scholar 

  13. N. Anjum, J.-H. He, Appl. Math. Lett. 92, 134 (2019)

    Article  MathSciNet  Google Scholar 

  14. N. Herisanu, V. Marinca, G. Madescu, F. Dragan, Energies 12, 915 (2019)

    Article  Google Scholar 

  15. J.-H. He, Mech. Res. Commun. 29, 107 (2002)

    Article  Google Scholar 

  16. S. Ganji, D.D. Ganji, Z. Ganji, S. Karimpour, Acta Appl. Math. 106, 79 (2009)

    Article  MathSciNet  Google Scholar 

  17. J.-H. He, Phys. Rev. Lett. 90, 174301 (2003)

    Article  ADS  Google Scholar 

  18. I. Mehdipour, D.D. Ganji, M. Mozaffari, Curr. Appl. Phys. 10, 104 (2010)

    Article  ADS  Google Scholar 

  19. J.-H. He, Int. J. Nonlinear Mech. Num. Simul. 9, 211 (2008)

    Google Scholar 

  20. J. Langari, M. Akbarzade, Adv. Stud. Theor. Phys. 5, 343 (2011)

    Google Scholar 

  21. A. El-Naggar, G. Ismail, Appl. Math. Sci. 6, 2071 (2012)

    MathSciNet  Google Scholar 

  22. T.A. Nofal, G.M. Ismail, A.A.M. Mady, S. Abdel-Khalek, J. Electromagn. Anal. Appl. 5, 388 (2013)

    ADS  Google Scholar 

  23. J.-H. He, Comput. Methods Appl. Mech. Eng. 178, 257 (1999)

    Article  ADS  Google Scholar 

  24. J.-H. He, Int. J. Nonlinear Mech. 35, 37 (2000)

    Article  ADS  Google Scholar 

  25. A. Yildirim, J. Math. Phys. 50, 023510 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  26. Z. Ayati, J. Biazar, J. Egypt. Math. Soc. 23, 424 (2015)

    Article  Google Scholar 

  27. J. Biazar, M. Eslami, Comput. Math. Appl. 62, 225 (2011)

    Article  MathSciNet  Google Scholar 

  28. P. Bera, T. Sil, Appl. Math. Comput. 219, 3272 (2012)

    MathSciNet  Google Scholar 

  29. K.S. Aboodh, Global J. Pure Appl. Math. 9, 35 (2013)

    Google Scholar 

  30. A. Ghorbani, Chaos, Solitons Fractals 39, 1486 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  31. J.-H. He, Eur. J. Phys. 29, L19 (2008)

    Article  Google Scholar 

  32. A.G. Davodi, D.D. Ganji, R. Azami, H. Babazadeh, Mod. Phys. Lett. B 23, 3427 (2009)

    Article  ADS  Google Scholar 

  33. H.-L. Zhang, Comput. Math. Appl. 58, 2480 (2009)

    Article  MathSciNet  Google Scholar 

  34. V. Marinca, N. Herisanu, J. Sound Vib. 329, 1450 (2010)

    Article  ADS  Google Scholar 

  35. D.D. Ganji, M. Azimi, M. Mostofi, Indian. J. Pure Appl. Phys. 50, 670 (2012)

    Google Scholar 

  36. D.P. Billington, The Tower and the Bridge: The New Art of Structural Engineering (Princeton University Press, 1985)

  37. M. Akbarzade, Y. Khan, Math. Comput. Model. 55, 480 (2012)

    Article  Google Scholar 

  38. S.H. Hoseini, T. Pirbodaghi, M.T. Ahmadian, G.H. Farrahi, Mech. Res. Commun. 36, 892 (2009)

    Article  Google Scholar 

  39. Daniel J. Gorman, Free Vibration Analysis of Beams and Shafts (John Wiley & Sons, 1975)

  40. M.Nv. Hamdan, N.H. Shabaneh, J. Sound Vib. 199, 711 (1997)

    Article  ADS  Google Scholar 

  41. S.-S. Chen et al., Nonlinear Anal. Real World Appl. 10, 881 (2009)

    Article  MathSciNet  Google Scholar 

  42. P.M. Mathews, M. Lakshmanan, Q. Appl. Math. 32, 215 (1974)

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Physics, Indian Institute of Information Technology Design and Manufacturing Kancheepuram, 600127, Chennai, Tamil Nadu, India

    K. Manimegalai, Sagar Zephania C F & Tapas Sil

  2. Department of Physics, Dumkal College, Basantapur, Dumkal, 742303, Murshidabad, West Bengal, India

    P. K. Bera

  3. School of Electronics Engineering, VIT University, 623014, Vellore, Tamil Nadu, India

    P. Bera

  4. Department of Mechanical Engineering, IIT Ropar, 140001, Rupnagar, Punjab, India

    S. K. Das

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  1. K. Manimegalai
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  2. Sagar Zephania C F
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  3. P. K. Bera
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  4. P. Bera
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  6. Tapas Sil
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Correspondence to Tapas Sil.

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Manimegalai, K., Zephania C F, S., Bera, P.K. et al. Study of strongly nonlinear oscillators using the Aboodh transform and the homotopy perturbation method. Eur. Phys. J. Plus 134, 462 (2019). https://doi.org/10.1140/epjp/i2019-12824-6

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  • DOI: https://doi.org/10.1140/epjp/i2019-12824-6

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