Application of modified homotopy perturbation method and amplitude frequency formulation to strongly nonlinear oscillators

Abstract

In this paper, two powerful analytical methods known as modified homotopy perturbation method and Amplitude Frequency Formulation called respectively MHPM and AFF, are introduced to derive approximate solutions of a system of ordinary differential equations appear in mechanical applications. These methods convert a difficult problem into a simple one, which can be easily handled. The obtained solutions are compared with numerical fourth order runge-kutta method to show the applicability and accuracy of both MHPM and AFF in solving this sample problem. The results attained in this paper confirm the idea that MHPM and AFF are powerful mathematical tools and they can be applied to linear and nonlinear problems.

Keywords

• Nonlinear oscillation,modified homotopy perturbation method,amplitude frequency formulation

References

1. L.Xu, Determination of limit cycle by He’s parameter-expanding method for strongly nonlinear oscillators, Journal of Sound and Vibration 302, 178_184 (2007).
2. L.Xu, Application of He’s parameter-expansion method to an oscillation of a mass attached to a stretched elastic wire, Physics Letters A 368, 259-262 (2007).
3. A. Kimiaeifar, A.R. Saidi, A.R. Sohouli, D.D. Ganji, Analysis of modified Van der Pol’s oscillator using He’s parameter-expanding methods, Current Applied Physics 10, 279_283 (2010).
4. Ji-Huan He, Xu-Hong Wu, Variational iteration method, New development and applications, Computers and Mathematics with Applications 54, 881_894 (2007).
5. D.D. Ganji, G.A. Afrouzi, R.A. Talarposhti, Application of variational iteration method and homotopy perturbation method for nonlinear heat diffusion and heat transfer equations, Physics Letters A 368, 450_457 (2007).
6. L. Xu, Variational approach to solution of nonlinear dispersive K (m,n) equation, Chaos, Solitons and Fractals 37 (1), 137_143 (2008).
7. L. Xu, Variational principles for coupled nonlinear Schrödinger equations, Physics Letters A 359, 627_629 (2006).
8. D.D. Ganji, A. Sadighi, I. Khatami, Assessment of two analytical approaches in some nonlinear problems arising in engineering sciences, Physics Letters A 372, 4399_4406 (2008).

9. Xumei Chen, Linjun Wang, The variational iteration method for solving a neutral functional-differential equation with proportional delays, Computers and Mathematics with applications 59, 2696_2702 (2010).
10. Saeed Rastegar , Bahram Azizollah Ganji , Mojtaba Varedi , Mehran Erza, Application of He’s variational iteration method to the estimation of diaphragm deflection in MEMS capacitive microphone, Measurement 44, 113_120 (2011).
11. Ji-Huan He, An elementary introduction to the homotopy perturbation method, Computers and Mathematics with Applications 57, 410_412 (2009).
12. A Belendez,T Belendez, A Marquez, C Neipp, Application of He’s homotopy perturbation method to conservative truly nonlinear oscillators. Chaos, Solitons and Fractals 37, 770_80 (2008).
13. A. Beléndez, A. Hernández, T.Beléndez, C. Neipp, A. Márquez, Higher accuracy analytical approximations to a nonlinear oscillator with discontinuity by He’s homotopy perturbation method, Physics Letters A 372, 2010-2016 (2008).
14. Augusto Beléndez, Homotopy perturbation method for a conservative x ^(1/3) force nonlinear oscillator, Computers & Mathematics with Applications 58, 2267-2273 (2009).
15. M. Gorji-Bandpy, M. Azimi, and M. Mostofi, 2011, Analytical Methods to a Generalized Duffing Oscillator, Australian Journal of Basic and Applied Science, Vol. 5, No. 11, pp. 788-796.
16. S.M. Moghimi, D.D. Ganji, H. Bararnia, M. Hosseini, M. Jalaal, Homotopy perturbation method for nonlinear MHD Jeffery–Hamel problem, Computers and Mathematics with Applications 61, 2213_2216 (2011).
17. Songxin Liang, David J.Jeffrey, Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation, Communications in Nonlinear Science and Numerical Simulation, 4057_4064 (2009).
18. J.H.He, An improved amplitude frequency formulation for nonlinear oscillators, International Journal of Nonlinear Sciences and Numerical Simulation 9, 211_212 (2008).
19. L. Geng and X. C. Cai, He’s frequency formulation for nonlinear oscillators, European Journal of Physics 28, 923_931 (2007).
20. Ling Zhao, He’s frequency_amplitude formulation for nonlinear oscillators with an irrational force, Computers and Mathematics with Applications 58, 2473_2476, (2009).
21. S.S. Ganji, D.D.Ganji, M.G.Sfahani, S.Karimpour, Application of AFF and HPM to the systems of strongly nonlinear oscillation, Current Applied Physics 10, 1317-1325 (2010).
22. J. H. He, Max-Min Approach to Nonlinear Oscillators, International Journal of Nonlinear Sciences and Numerical Simulation 9, 207_210 (2008).
23. De-Qiang Zeng, Nonlinear oscillator with discontinuity by the max_min approach, Chaos, Solitons and Fractals 42 (2009) 2885-2889.
24. S.S. Ganji, A. Barari, D.D. Ganji, Approximate analysis of two-mass–spring systems and buckling of a column, Computers and Mathematics with Applications 61,1088_1095 (2011).
25. Xinlong Feng, Yinnian He, Modified homotopy perturbation method for solving the Stokes equations, Computers and Mathematics with Applications 61, 2262_2266 (2011).
26. A. Beléndez , C. Pascual, S. Gallego, M. Ortuño, C. Neipp, Application of a modified He’s homotopy perturbation method to obtain higher-order approximations of an x ^(1/3) force nonlinear oscillator, Physics Letters A 371, 421-426 (2007).
27. A. Beléndez, C. Pascual, T. Beléndez, A. Hernández, Solution for an anti-symmetric quadratic nonlinear oscillator by a modified He’s homotopy perturbation method, Nonlinear Analysis: Real World Applications 10, 416-427 (2009).
28. Junfeng Lu, An analytical approach to the sine Gordon equation using the modified homotopy perturbation method, Computers and Mathematics with Applications 58, 2313_2319 (2009).
29. Davood D Ganji, Mohammadreza Azimi, Mehdi Mostofi, Energy Balance Method and amplitude frequency formulation based of strongly nonlinear oscillator, indian journal of pure & applied physics, Vol. 50, 2012, pp. 670-675.
30. D. D. Ganji, M. Sheikholeslami, and H. R. Ashorynejad, Analytical Approximate Solution of Nonlinear Differential Equation Governing Jeffery-Hamel Flow with High Magnetic Field by Adomian Decomposition Method, ISRN Mathematical AnalysisVolume 2011 (2011).
31. A. Sadighi, D.D. Ganji, Analytic treatment of linear and nonlinear Schrödinger equations: A study with homotopy-perturbation and Adomian decomposition methods, Physics Letters A 372, 465_469 (2008).
32. D.D.Ganji, M.Azimi, Application of DTM on MHD Jeffery Hamel Problem with Nanoparticle, 2013, The http://www.scientificbulletin.upb.ro/?lang=english&page=mainScientific Bulletin Series D, vol. 75, pp.223-230.
33. M. Azimi, D. D. Ganji, F. Abbassi, Study on MHD Viscous Flow over a Stretching Sheet Using DTM-Pade’ Technique, Modern Mechanical Engineering, Vol. 2, No. 4, 2012, pp. 126-129.
34. D.D.Ganji, M.Azimi, Application of Max min Approach and Amplitude Frequency Formulation to the nonlinear oscillation systems, 2012, The http://www.scientificbulletin.upb.ro/?lang=english&page=mainScientific Bulletin Series A, Issue.3, pp.131-140.