Analytical solution of nonlinear vibrating systems

American Journal of Applied Sciences

Volumn 5, Issue 9, 2008, Pages 1219-1224

  Tolou, N ^a   Khatam, I ^b   Jafari, B ^b   Ganji, D D ^b  

^a ISLAMIC AZAD UNIVERSITY   (Iran)

^b UNIVERSITY OF MAZANDARAN   (Iran)

Author keywords

Homotopy Perturbation Method (HPM); Nonlinear dynamic equations; Nonlinear vibrating systems

Indexed keywords

EID: 41149132737     PISSN: 15469239     EISSN: 15543641     Source Type: Journal    
DOI: 10.3844/ajassp.2008.1219.1224     Document Type: Article

References (23)

1. Livija, C. and K. Ivana, 2007. Parametrically excited vibrations of an oscillator with strong cubic negative nonlinearity. J. Sound Vib., 304: 201-212.
2. Mond, M. and G. Cederbaum, 1993. Stability analysis of the non-linear Mathieu equation. J. Sound Vib., 167 (1): 77-89.
3. Younesian, D., E. Esmailzadeh and R. Sedaghati, 2005. Existence of periodic solutions for the generalized form of Mathieu Equation. Nonl. Dyn., 39: 335-348.
4. Bellman, R., 1964. Perturbation Techniques in Mathematic, Physics and Engineering, Holt, Rinehart and Winston, New York.
5. Cole, J.D., 1968. Perturbation Methods in Applied Mathematics, Blaisedell, Waltham, MA.
6. Liu, G.L., 1997. New research directions in singular perturbation theory: Artificial parameter approach and inverse-perturbation technique. Conference of 7th Modern Mathematics and Mechanics, Shanghai.
7. He, J.H., 1998. Approximate analytical solution for seepage flow with fractional derivatives in porous media. J. Comput. Math. Appl. Mech. Eng., 167: 57-68.
8. He, J.H., 1998. Approximate solution for nonlinear differential equations with convolution product nonlinearities. Comput. Math. Appl. Mech. Eng., 167: 69-73.
9. He, J.H., 1999. Variational iteration method: A kind of nonlinear analytical technique: Some examples. Int. J. Non-L. Mech., 344: 699-708.
10. Ganji, D.D., 2006. The application of He's homotopy perturbation method to nonlinear equations arising in heat transfer. Phys. Lett. A, 355: 337-341.
11. Tolou, N., D.D. Ganji, M.J. Hosseini and Z.Z. Ganji, 2007. Applications of Homotopy Perturbation Method in Nonlinear Heat Diffusion-Convection-Reaction Equations. The Open Mechanic J., 1: 20-25.
12. Ganji, D.D. and A. Sadighi, 2007. Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations. J. Comput. Appl. Math., 207: 24-34.
13. Tari, H. and D.D. Ganji, 2007. Approximate explicit solutions of nonlinear BBMB equations by He's methods and comparison with the exact solution, Phy. Lett., 367: 95-101.
14. Rafei, M.and D.D. Ganji, 2006. Explicit solutions of Helmholtz equation and fifth-order KdV equation using homotopy perturbation method, Int. J. Nonl. Sci. Numer. Simulation, 7 (3): 321-328.
15. Ganji, D.D. and A. Sadighi, 2006. Application of He's homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations. Int. J. Nonl. Sci. Numer. Simulation, 7 (4): 411-418.
16. Suying, Z., Z. Minzhen, D. Zichen and L. Wencheng, 2006. Solution of nonlinear dynamic differential equations based on numerical Laplace transform inversion. Applied Math. Comput., 189: 79-86.
17. Cveticanin, L., 2005. Homotopy-perturbation method for pure nonlinear differential equation. Chaos Solitons Fractals, 30: 1221-1230.
18. Belendez, A., A. Hernandez and T. Belendez et al., 2007. Application of He's homotopy perturbation method to the Duffing-harmonic oscillator. Int. J. Nonl. Sci. Numer. Simulation, 8 (1): 79-88.
19. He, J.H., 1999. Homotopy perturbation technique, J. Comput. Math. Appl. Mech. Eng., 178: 257-262.
20. He, J.H., 2000. A coupling method of a homotopy technique and a perturbation technique for non-linear problems. Int. J. Non-L. Mech., 35: 35-43.
21. He, J.H., 2003. Homotopy perturbation method: a new nonlinear analytical technique. J. Appl. Math. Comput., 135: 73-79.
22. He, J.H., 2004. The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl. Math. Comput., 151: 287-292.
23. Shou D.H. and J.H. He, 2007. Application of parameter-expanding method to strongly nonlinear oscillators. J. Comput. Int. J. Nonl. Sci. Num. Simulation, 8 (1): 121-124.