Analytical solution of sub-harmonic nonlinear oscillation

Abstract

The study of nonlinear oscillator is important issue in the development of theory of dynamical system. One of the newest analytical methods to solve nonlinear equations is the application of both homotopy perturbation and variational iteration techniques. Homotopy perturbation method (HPM), which does not need small parameters is compared with variational iteration method (VIM) and both of them compare with numerical solution in the field of sub-harmonic nonlinear oscillation. The justification for using this method is the difficulties and limitations of perturbation and homotopy when used individually. In this paper, homotopy perturbation method and varational iteration method are used to solve for periodic method for sub-harmonic of nonlinear oscillation. After solving the equations, we found effect of each parameter and the best value for solving equations was   ε=0.1,λ=1,α=0.1,β=1.

Keywords

• HPM,VIM,analytical solution,nonlinear oscillation

References

1. J. H. He, dissertation . de-verlag im internet GmbH,2006.
2. J. H. He, Int. J. Mod. Phys. B 20 (10) (2006) 1141.
3. A. Rajabi , D.D. Ganji, H. Taherian , Phys. Lett.360 (2) (2007) 131.
4. S. J. Liao, Int. J. Non-Linear Mech. 303 (1995) 371.
5. S. J. Liao, Eng. Anal. Bound. Elm. 202 (1997) 91.
6. J. H. He, J. Comput. Math. Appl. Mech. Eng. 167 (1998) 57.
7. J. H. He, Comput. Math. Appl. Mech. Eng. 167 (1998) 69.
8. J. H. He, Int. J. Non-Linear Mech. 344 (1999) 699.

9. J. H. He, J. Comput. Math. Appl. Mech. Eng. 17 (8) (1999) 257.
10. J. H. He, Int. J. Non-Linear Mech. 351 (2000) 37.
11. J. H. He, Phys. Lett. A 350 (1–2) (2006) 87.
12. J. H. He, X.H. Wu, Chaos Solitons Fractals 29 (1) (2006) 108.
13. J. H. He, Phys. Lett. A 347 (4–6) (2005) 228.
14. J. H. He, Chaos Solitons Fractals 26 (3) (2005) 827.
15. J. H. He, Chaos Solitons Fractals 26 (3) (2005) 695.
16. J. H. He, Int. J. Non-Linear Sci. Numer. Simul. 6 (2) (2005) 207.
17. X. C. Cai, W.Y. Wu, M.S. Li, Int. J. Non-Linear. Sci. Numer. Simul. 7 (1) (2006) 109.
18. L. Cveticanin, Chaos Solitons Fractals 30 (5) (2006) 1221.
19. M. El-Shahed, Int. J. Non-Linear Sci. Numer. Simul. 6 (2) (2005) 163.
20. S. Abbasbandy, Chaos Solitons Fractals 30 (5) (2006) 1206.
21. A. M. Siddiqui, R. Mahmood, Q. K. Ghori, Int. J. Non-Linear Sci. Numer.Simul. 7 (1) (2006) 7.
22. A. M. Siddiqui, M. Ahmed, Q. K. Ghori, Int. J. Non-Linear Sci. Numer. Simul. 7 (1) (2006) 15.
23. D. D. Ganji, M. Rafei, Phys. Lett. A 356 (2) (2006) 131.
24. D. D. Ganji, A. Rajabi, Int. Commun. Heat Mass Transfer 33 (3) (2006) 391.
25. Mohsen Sheikholeslami, Magnetic field influence on nanofluid thermal radiation in a cavity with tilted elliptic inner cylinder, Journal of Molecular Liquids, Journal of Molecular Liquids 229 (2017) 137–147.
26. M. Sheikholeslami, D. D. Ganji, Transportation of MHD nanofluid free convection in a porous semi annulus using numerical approach, Chemical Physics Letters 669 (2017) 202–210.
27. M. Sheikholeslami, K. Vajravelu, Nanofluid flow and heat transfer in a cavity with variable magnetic field, Applied Mathematics and Computation, http://www.sciencedirect.com/science/journal/00963003/298/supp/C298 (2017) 272–282.
28. Mohsen Sheikholeslami, Influence of Lorentz forces on nanofluid flow in a porous cylinder considering Darcy model, Journal of Molecular Liquids 225 (2017) 903–912.
29. Mohsen Sheikholeslami, CVFEM for magnetic nanofluid convective heat transfer in a porous curved enclosure, Eur. Phys. J. Plus (2016) 131: 413, DOI: 10.1140/epjp/i2016-16413-y.
30. M. Sheikholeslami, Houman B. Rokni, Nanofluid two phase model analysis in existence of induced magnetic field, International Journal of Heat and Mass Transfer http://www.sciencedirect.com/science/journal/00179310/107/supp/C107 (2017) 288–299.
31. M. Sheikholeslami, S. A. Shehzad, Magnetohydrodynamic nanofluid convection in a porous enclosure considering heat flux boundary condition, International Journal of Heat and Mass Transfer 106 (2017) 1261–1269.