VARIATIONAL ITERATION SOLUTIONS FOR NONLINEAR DIFFERENTIAL EQUATIONS OF DIFFERENT DAMPING MODELS IN OSCILLATIONS AND VIBRATIONS

M. H. Hojjati (Iran) and Hafez Tari (Iran)

Received October 26, 2007

References:

[1] G. Adomian, Solving Frontier Problems on Physics: The Decomposition Method, Kluwer Academic, Dordrecht, 1994. [2] J. D. Cole, Perturbation Methods in Applied Mathematics, Blaisdell Waltham, MA, 1968. [3] H. T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover Publications, Inc., New York, 1962. [4] G. E. Draganescu, Application of a variational iteration method to linear and nonlinear viscoelastic models with fractional derivatives, J. Math. Phys. 47(3) (2006), 082902. [5] G. E. Draganescu and V. Capalnasan, Nonlinear relaxation phenomena in polycrystalline solids, Int. J. Nonlinear Sci. Num. Sim. 4(3) (2003), 219-225. [6] G. E. Draganescu, N. Cofan and D. L. Rujan, Nonlinear vibrations of a nonzero sized sensor with fractional damping, J. Optoelect. Adv. Mater. 7(2) (2005), 877-884. [7] J. H. He, Homotopy perturbation method for bifurcation of nonlinear problems, Int. J. Non-linear Sci. Num. Sim. 6(2) (2005), 207-208. [8] J. H. He, The homotopy perturbation method for nonlinear oscillators with discontinuities, App. Math. Comp. 151 (2004), 287-292. [9] J. H. He, Variational iteration method for autonomous ordinary differential systems, App. Math. Comp. 114 (2000), 115-123. [10] J. H. He, Variational iteration method – a kind of non-linear analytical technique: some examples, Int. J. Non-Linear Mechanics 34(4) (1999), 699-708. [11] J. H. He, Variational iteration method – some recent results and new interpretations, J. Comp. Appl. Math. 207(1) (2007), 3-17. [12] J. H. He, Non-perturbative Methods for Strongly Nonlinear Problems, Dissertation, De-Verlag im Internet GmbH, Berlin, 2006. [13] J. H. He, Some asymptotic methods for strongly nonlinear equations, Int. J. Modern Physics B 20(10) (2006), 1141-1199. [14] R. Iwankiewicz and S. R. K. Nielsen, Advanced Methods in Stochastic Dynamics of Non-linear Systems, Vibration Theory, Vol. 4, Aalborg University Press, Denmark, 1999. [15] A. V. Karmishin, A. I. Zhukov and V. G. Kolosov, Methods of Dynamics Calculation and Testing for Thin-walled Structures, Mashinostroyenie, Moscow, 1990. [16] A. M. Lyapunov, General Problem on Stability of Motion, Taylor & Francis, London, 1992 (English Translation). [17] V. Marinca, An approximate solution for one-dimensional weakly nonlinear oscillators, Int. J. Nonlinear Sci. Num. Sim. 3(2) (2002), 107-120. [18] A. H. Nayfeh, Perturbation Methods, Wiley, New York, 2000. [19] C. Ray Wylie and Louis C. Barrett, Advanced Engineering Mathematics, McGraw-Hill, Fifth ed., 1989. [20] A. B. Sotiropoulou and D. E. Panayotounakos, On the reduction of some second-order nonlinear ODEs in physics and mechanics to first-order nonlinear integrodifferential and Abel’s classes of equations, Theoretical and App. Fract. Mech. 40 (2003), 255-270. [21] L. Xu, He’s parameter-expanding methods for strongly nonlinear oscillators, J. Com. App. Math. (in press). [22] O. T. Yidirim, A study of nonlinear oscillators with u(1/3) force by He’s variational iteration method, J. Sound Vibration 306(1-2) (2007), 372-376.

Keywords and phrases: variational iteration method, vibration, oscillation, damping models, non-hysteretic dynamical systems.