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Numerical analysis of strongly nonlinear oscillation systems using He’s max-min method

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Abstract

Nonlinear functions are crucial points and terms in engineering problems. Actual and physical problems can be solved by solving and processing such functions. Thus, most scientists and engineers focus on solving these equations. This paper presents a novel method called the max-min method for presenting an accurate approximate analytical solution to strong nonlinear oscillators. It can solve many linear or nonlinear differential equations without the tangible restriction of sensitivity to the degree of the nonlinear term. It is also quite convenient due to the reduction in the size of calculations. The algorithm suggests a promising approach and is systematically illustrated step by step.

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References

  1. Bender C M, Milton K A, Pinsky S S, Simmons L M. A new perturbative approach to nonlinear problems. Journal of Mathematical Physics, 1989, 30(7): 1447–1455

    Article  MATH  MathSciNet  Google Scholar 

  2. He J H. A note on delta-perturbation expansion method. Applied Mathematics and Mechanics, 2002, 23(6): 634–638

    Article  MATH  Google Scholar 

  3. Ganji D D. The application of He’s homotopy perturbation method to nonlinear equations arising in heat transfer. Physics Letters. [Part A], 2006, 355(4–5): 337–341

    MATH  MathSciNet  Google Scholar 

  4. Bararnia H, Ghasemi E, Soleimani S, Barari A, Ganji D D. HPMPade’s method on natural convection of darcian fluid about a vertical full cone embedded in porous media. Journal of Porous Media, 2011, 14(6): 545–553

    Article  Google Scholar 

  5. Barari A, Omidvar M, Ghotbi A R, Ganji D D. Application of homotopy perturbation method and variational iteration method to nonlinear oscillator differential equations. Acta Applicandae Mathematicae, 2008, 104(2): 161–171

    Article  MATH  MathSciNet  Google Scholar 

  6. Barari A, Ghotbi A R, Farrokhzad F, Ganji D D. Variational iteration method and Homotopy-perturbation method for solving different types of wave equations. Journal of Applied Sciences, 2008, 8(1): 120–126

    Article  Google Scholar 

  7. Miansari M O, Miansari M E, Barari A, Domairry G. Analysis of blasius equation for flat-plate flow with infinite boundary value. International Journal for Computational Methods in Engineering Science and Mechanics, 2010, 11(2): 79–84

    Article  MATH  MathSciNet  Google Scholar 

  8. He J H. The homotopy perturbation method for nonlinear oscillators with discontinuities. Applied Mathematics and Computation, 2004, 151(1): 287–292

    Article  MATH  MathSciNet  Google Scholar 

  9. He J H. A coupling method of a homotopy technique and a perturbation technique for non-linear problems. International Journal of Non-linear Mechanics, 2000, 35(1): 37–43

    Article  MATH  MathSciNet  Google Scholar 

  10. Ozis T, Yildirim A. A comparative study of He’s homotopy perturbation method for determining frequency-amplitude relation of a nonlinear oscillator with discontinuities. International Journal of Nonlinear Sciences and Numerical Simulation, 2007, 8(2): 243–248

    Article  Google Scholar 

  11. He J H. Homotopy perturbation method: a new nonlinear analytical technique. Applied Mathematics and Computation, 2003, 135(1): 73–79

    Article  MATH  MathSciNet  Google Scholar 

  12. He J H. Variational iteration method: a kind of nonlinear analytical technique: some examples. International Journal of Non-linear Mechanics, 1999, 34(4): 699–708

    Article  MATH  Google Scholar 

  13. Fouladi F, Hosseinzadeh E, Barari A, Domairry G. Highly nonlinear temperature dependent fin analysis by variational iteration method. Journal of Heat Transfer Research, 2010, 41(2): 155–165

    Article  Google Scholar 

  14. Ganji D D, Babazadeh H, Jalaei M H, Tashakkorian H. Application of He’s variational iteration methods for solving nonlinear BBMB equations and free vibrations of systems. Acta Applicandae Mathematicae, 2009, 106(3): 359–367

    Article  MATH  MathSciNet  Google Scholar 

  15. Barari A, Omidvar M, Ganji D D, Poor A T. An approximate solution for boundary value problems in structural engineering and fluid mechanics. Journal of Mathematical Problems in Engineering, 2008, 1–13

  16. Xu L. Variational principles for coupled nonlinear Schrödinger equations. Physics Letters. [Part A], 2006, 359(6): 627–629

    MATH  Google Scholar 

  17. He J H. Bookkeeping parameter in perturbation methods. International Journal of Non-Linear Sciences and Numerical Simulation, 2001, 2(3): 257–264

    Article  MATH  Google Scholar 

  18. Ganji S S, Ganji D D, Ganji Z Z, Karimpour S. Periodic solution for strongly nonlinear vibration systems by energy balance method. Acta Applicandae Mathematicae, 2009, 106(1): 79–92

    Article  MATH  MathSciNet  Google Scholar 

  19. Momeni M, Jamshidi N, Barari A, Ganji D D. Application of He’s energy balance method to Duffing harmonic oscillators. International Journal of Computer Mathematics, 2011, 88(1): 135–144

    Article  MATH  MathSciNet  Google Scholar 

  20. He J H. A max-min approach to nonlinear oscillator. International Journal of Nonlinear Sciences and Numerical Simulation, 2008, 9(2): 207–210

    Article  Google Scholar 

  21. Arnold M D. An efficient solution for scattering by a perfectly conducting strip grating. Journal of Electromagnetic Waves and Applications, 2006, 20(7): 891–900

    Article  Google Scholar 

  22. He J H. A review on some new recently developed nonlinear analytical techniques. International Journal of Nonlinear Sciences and Numerical Simulation, 2000, 1(1): 51–70

    Article  MATH  MathSciNet  Google Scholar 

  23. Zhao J X. Numerical and analytical formulations of the extended MIE theory for solving the sphere scattering problem. Journal of Electromagnetic Waves and Applications, 2006, 20(7): 967–983

    Article  MathSciNet  Google Scholar 

  24. Rui P L, Chen R S. Implicitly restarted gmres fast Fourier transform method for electromagnetic scattering. Journal of Electromagnetic Waves and Applications, 2007, 21(7): 973–986

    Article  Google Scholar 

  25. Wang M Y, Xu J, Wu J, Yan Y, Li H L. FDTD study on scattering of metallic column covered by double negative meta-material. Journal of Electromagnetic Waves and Applications, 2007, 21(14): 1905–1914

    Article  Google Scholar 

  26. Liu X F, Wang B Z, Lai S J. Element-free Galerkin method in electromagnetic scattering field computation. Journal of Electromagnetic Waves and Applications, 2007, 21(14): 1915–1923

    Article  MATH  Google Scholar 

  27. Hosein Nia S H, Ranjbar A, Soltani H, Ghasemi J. Effect of the initial approximation on stability and convergence in homotopy perturbation method. International Journal of Nonlinear Dynamics in Engineering and Sciences, 2008, 1: 79–90

    Google Scholar 

  28. Omidvar M, Barari A, Momeni M, Ganji D D. New class of solutions for water infiltration problems in unsaturated soils. Geomechanics and Geoengineering. International Journal (Toronto, Ont.), 2010, 5(2): 127–135

    Article  Google Scholar 

  29. Mirmoradi H, Mazaheripour H, Ghanbarpour S, Barari A. Homotopy perturbation method for solving twelfth order boundary value problems. International Journal of Research and Reviews in Applied Sciences, 2009, 1(2): 163–173

    MATH  Google Scholar 

  30. Qian B C. History of Chinese Mathematics. Beijing: Science Publisher, 1992 (in Chinese)

    Google Scholar 

  31. Ji Z. Mathematics in Northern-Southern, Sui and Tang Dynasties. Shijiazhuang: Hebei Science and Technology Publishing House, 1999 (in Chinese)

    Google Scholar 

  32. He J H. Some asymptotic methods for strongly nonlinear equations. International Journal of Modern Physics B, 2006, 20(10): 1141–1199

    Article  MATH  MathSciNet  Google Scholar 

  33. He J H, Tang H. Rebuild of King Fang 40 BC musical scales by He’s inequality. Applied Mathematics and Computation, 2005, 168(2): 909–914

    Article  MATH  MathSciNet  Google Scholar 

  34. He J H. Mysterious pi and a possible link to DNA sequencing. International Journal of Nonlinear Sciences, 2004, 5(3): 263–274

    Article  Google Scholar 

  35. He J H. Solution of nonlinear equations by an ancient Chinese algorithm. Applied Mathematics and Computation, 2004, 151(1): 293–297

    Article  MATH  MathSciNet  Google Scholar 

  36. He J H. He Chengtian’s inequality and its applications. Applied Mathematics and Computation, 2004, 151(3): 887–891

    Article  MATH  MathSciNet  Google Scholar 

  37. Nayfeh A H. Introduction to Perturbation Techniques. New York: John Wiley & Sons, 1981

    MATH  Google Scholar 

  38. He J H. Non-pertubative methods for strongly nonlinear problems, dissertation. De-Verlag IM Internet GmbH 2006

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Authors and Affiliations

  1. Department of Electrical Engineering, California State University, Los Angeles, CA, 90840, USA

    H. Babazadeh

  2. Departments of Mechanical, Electrical and Civil Engineering, Babol University of Technology, Babol, Iran

    G. Domairry & R. Azami

  3. Department of Civil Engineering, Aalborg University, Aalborg, Denmark

    A. Barari

  4. Department of Civil Engineering, Shahrood University of Technology, Shahrood, Iran

    A. G. Davodi

Authors
  1. H. Babazadeh
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  2. G. Domairry
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  3. A. Barari
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  4. R. Azami
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  5. A. G. Davodi
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Correspondence to A. Barari.

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Babazadeh, H., Domairry, G., Barari, A. et al. Numerical analysis of strongly nonlinear oscillation systems using He’s max-min method. Front. Mech. Eng. 6, 435–441 (2011). https://doi.org/10.1007/s11465-011-0243-x

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  • DOI: https://doi.org/10.1007/s11465-011-0243-x

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