Abstract
The most common and most widely studied methods for determining analytical approximate solutions of a nonlinear dynamical system are iteration methods.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
A.N. Nayfeh, D. Mook, Nonlinear Oscillations. (Wiley, 1979)
B.S. Wu, C.W. Lim, Y.F. Ma, Analytical approximation to large – amplitude oscillation of a nonlinear conservative system. Int. J. Non-linear Mech. 38, 1037–1043 (2003)
S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method (Chapman and Hall/CRC Press, Boca Raton, 2003)
V. Marinca, N. Herişanu, D. Bălă, Some optimal approximate methods with application to thin film flow. WSEAS Trans. Syst. 7(9), 744–753 (2010)
R.E. Mickens, A generalized iteration procedure for calculating approximations to periodic solutions of “truly nonlinear oscillators”. J. Sound Vib. 287, 1045–1051 (2005)
C.W. Lim, B.S. Wu, A modified procedure for certain non-linear oscillators. J. Sound Vib. 257, 202–206 (2002)
H. Hu, Solutions of a quadratic nonlinear oscillator: Iteration procedure. J. Sound Vib. 298, 1159–1165 (2006)
Y.M. Chen, J.K. Liu, A modified Mickens iteration procedure for nonlinear oscillators. J. Sound Vib. 314, 465–473 (2008)
H. Hu, A classical iteration procedure valid for certain strongly nonlinear oscillators. J. Sound Vib. 29, 397–402 (2007)
J.H. He, Some asymptotic methods for strongly nonlinear equations. Int. J. Mod. Phys. B 20(10), 1141–1189 (2006)
V. Marinca, N. Herişanu, A modified iteration perturbation method for some nonlinear oscillation problems. Acta Mech. 184, 231–242 (2006)
J.H. He, Variational iteration method – a kind of non-linear analytical technique: Some examples. Int. J. Non-linear Mech. 34, 699–708 (1999)
N. Herişanu, V. Marinca, An iteration procedure with application to Van der Pol oscillator. Int. J. Nonlin. Sci. Numer. Simul. 10(3), 353–361 (2009)
R.E. Mickens, Perturbation procedure for the Van der Pol oscillator based on Hopf bifurcation theorem. J. Sound Vib. 127, 187–193 (1988)
S.H. Chen, Y.K. Cheung, S.L. Lau, On perturbation procedure for limit cycle analysis. Int. J. Non-linear Mech. 26, 237–250 (1993)
E.M.A. Elbashbeshy, M.F. Dimian, Effect of radiation on the flow and heat transfer over a wedge with variable viscosity. Appl. Math. Comput. 132, 445–454 (2002)
M.E.M. Ouaf, Exact solution of thermal radiation on MHD over a stretching porous sheet. Appl. Math. Comput. 170, 1117–1125 (2003)
V. Marinca, N. Herişanu, An optimal iteration method for a class of strongly nonlinear oscillators. Sci. Bul. Politehnica Univ. Timişoara Trans. Mech. 54(68), 1–8 (2009)
V. Marinca, G. Drăgănescu, Construction of approximate periodic solutions to a modified Van der Pol oscillator. Nonlin. Anal. Real World Appl. 11(5), 4355–4362 (2010)
M.N. D’Acunto, Determination of the limit cycle for a modified Van der Pol oscillator. Mech. Res. Comm. 33, 93–98 (2000)
F.M. Scudo, Volterra and theoretical ecology. Theor. Popul. Biol. 2, 1–23 (1971)
R.D. Small, Population Growth in a Closed Model, in Mathematical Modelling. Classroom Notes in Applied Mathematics, ed. by M.S. Klamkin (SIAM, Philadelphia, 1989)
S. Kobayashi, T. Matsukuma, S. Nagai, K. Umeda, Some coefficients of the TFD functions. J. Phys. Soc. Jpn. 10, 759–765 (1955)
S.J. Liao, An explicit analytic solution to the Thomas-Fermi equation. Appl. Math. Comput. 144, 495–506 (2003)
S.R. Resiga, G.D. Ciocan, I. Anton, F. Avelan, Analysis of the swirling flow downstream a Francis turbine runner. J. Fluid Eng. 128, 177–189 (2006)
A.J. Lotka, Elements of Physical Biology (William and Wilkins, Baltimore, 1925)
V. Volterra, Variazioni e fluttuazioni del numero d’individui in specie animali conniventi. Mem. R. Acad. Naz. Lincei 2–3, 30–111 (1926)
A. Repaci, Nonlinear dynamical systems: On the accuracy of Adomian decomposition method. Appl. Math. Lett. 3, 35–39 (1990)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Marinca, V., Herisanu, N. (2012). Optimal Parametric Iteration Method. In: Nonlinear Dynamical Systems in Engineering. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22735-6_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-22735-6_9
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22734-9
Online ISBN: 978-3-642-22735-6
eBook Packages: EngineeringEngineering (R0)