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Solving nonlinear differential equations of Vanderpol, Rayleigh and Duffing by AGM


In the present paper, three complicated nonlinear differential equations in the field of vibration, which are Vanderpol, Rayleigh and Duffing equations, have been analyzed and solved completely by Algebraic Method (AGM). Investigating this kind of equations is a very hard task to do and the obtained solution is not accurate and reliable. This issue will be emerged after comparing the achieved solutions by numerical method (Runge-Kutte 4th). Based on the comparisons which have been made between the gained solutions by AGM and numerical method, it is possible to indicate that AGM can be successfully applied for various differential equations particularly for difficult ones. The results reveal that this method is not only very effective and simple, but also reliable, and can be applied for other complicated nonlinear problems.

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  1. Chopra A K. Dynamic of Structures, Theory and Applications to Earthquake Engineering. New Jersey: Prentice-Hall, 1995

    Google Scholar 

  2. Zhou J K. Differential Transformation and Its Applications for Electrical Circuits. Wuhan: Huazhong University Press, 1986, 363–364

    Google Scholar 

  3. Acton J R, Squire P T. Solving Equations with Physical Understanding. Boston: Adam Hilger Ltd, 1985

    Google Scholar 

  4. Bellman R. Perturbation Techniques in Mathematics, Physics and Engineering. New York: Holt, Rinehart and Winston, Inc, 1964, 254–267

    Google Scholar 

  5. Sfahani M G, Barari A, Omidvar M, Ganji S S, Domairry G. Dynamic response of inextensible beams by improved energy balance method. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of multi-body Dynamics, 2011, 225(1): 66–73

    Google Scholar 

  6. 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 

  7. Ganji Z Z, Ganji D D, Bararnia H. Approximate general and explicit solutions of nonlinear BBMB equations exp-function method. Applied Mathematical Modelling, 2009, 33(4): 1836–1841

    Article  MATH  MathSciNet  Google Scholar 

  8. Pirbodaghi T, Ahmadian M, Fesanghary M. On the homotopy analysis method for non-linear vibration of beams. Mechanics Research Communications, 2009, 36(2): 143–148

    Article  MATH  Google Scholar 

  9. He J H. An improved amplitude-frequency formulation for nonlinear oscillators. International Journal of Nonlinear Sciences and Numerical Simulation, 2008, 9(2): 211–223

    Google Scholar 

  10. Ganji S S, Ganji D D, Sfahani M G, Karimpour S. Application of HAFF and HPM to the systems of strongly nonlinear oscillation. Current Applied Physics, 2010, 10(5): 1317–1325

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  12. He J H, Wu X H. Exp-function method for nonlinear wave equations. Chaos. Solitos & Fractals, 2006, 30(3): 700–708

    Article  MATH  MathSciNet  Google Scholar 

  13. Chang J R. The exp-function method and generalized solitary solutions. Computers & Mathematics with Applications, 2011, 61(8): 2081–2084

    Article  MATH  MathSciNet  Google Scholar 

  14. Ganji S S, Ganji D D, Babazadeh H, Sadoughi N. Application of amplitude-frequency formulation to nonlinear oscillation system of the motion of a rigid rod rocking back. Mathematical Methods in the Applied Sciences, 2010, 33(2): 157–166

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  16. He J H. Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 1999, 178(3): 257–262

    Article  MATH  MathSciNet  Google Scholar 

  17. Ghosh S, Roy A, Roy D. An adaptation of adomian decomposition for numeric-analytic integration of strongly nonlinear and chaotic oscillators. Computer Methods in Applied Mechanics and Engineering, 2007, 196(4–6): 1133–1153

    Article  MATH  MathSciNet  Google Scholar 

  18. He J H. Approximate analytical solution for seepage flow with fractional derivatives in porous media. Computer Methods in Applied Mechanics and Engineering, 1998, 167(1–2): 57–68

    Article  MATH  MathSciNet  Google Scholar 

  19. He J H. Approximate analytical solution of Blasiu’s equation. Communications in Nonlinear Science and Numerical Simulation, 1998, 3(4): 260–263

    Article  MATH  Google Scholar 

  20. Abbasbandy S. A numerical solution of Blasius equation by Adomian’s decomposition method and comparison with homotopy perturbation method. Chaos, Solitons, and Fractals, 2007, 31(1): 257–260

    Article  MathSciNet  Google Scholar 

  21. Adomian G. Solving Frontier Problems of Physics: The Decomposition Method. Boston, MA: Kluwer Academic Publishers, 1994

    Book  MATH  Google Scholar 

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Correspondence to D. D. Ganji.

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Akbari, M.R., Ganji, D.D., Majidian, A. et al. Solving nonlinear differential equations of Vanderpol, Rayleigh and Duffing by AGM. Front. Mech. Eng. 9, 177–190 (2014).

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  • Algebraic Method (AGM)
  • Angular Frequency
  • Vanderpol
  • Rayleigh
  • Duffing