Modified Lindstedt-Poincaré methods for some strongly nonlinear oscillations. I: Expansion of a constant. (English) Zbl 1116.34320

He, Ji-Huan

Int. J. Non-Linear Mech. 37, No. 2, 309-314 (2002).

Summary: A modified Lindstedt–Poincare method is proposed. In this technique, a constant, rather than the non-linear frequency, is expanded in powers of the expanding parameter to avoid the occurrence of secular terms in the perturbation series solution. Some examples are given here to illustrate its effectiveness and convenience. The results show that the obtained approximate solutions are uniformly valid on the whole solution domain, and they are suitable not only for weakly non-linear systems, but also for strongly non-linear systems.

MSC:

34C15	Nonlinear oscillations and coupled oscillators for ordinary differential equations
34E05	Asymptotic expansions of solutions to ordinary differential equations
34E10	Perturbations, asymptotics of solutions to ordinary differential equations
70K60	General perturbation schemes for nonlinear problems in mechanics

Keywords:

Perturbation method; Nonlinear equation; Duffing equation; Lindstedt-Poincaré method