Abstract
In this paper, the homotopy analysis method is applied to deduce the periodic solutions of a conservative nonlinear oscillator for which the elastic force term is proportional to u1/3. By introducing the auxiliary linear operator and the initial guess of solution, the homotopy analysis solving is set up. By choosing the suitable convergence-control parameter, the accurate high-order approximations of solution and frequency for the whole range of initial amplitudes can easily be obtained. Comparison of the results obtained using this method with those obtained by different methods reveals that the former is more accurate, effective and convenient for these types of nonlinear oscillators.
Highlights
Classical perturbation methods including the Lindstedt-Poincaré method, the Krylov-Bogoliubov-Mitropolski method and the multiple scales method, as described by Nayfeh [1] [2], Kevorkian and Cole [3] and Verhulst [4], are limited to the weakly nonlinear systems
The homotopy analysis method is applied to deduce the periodic solutions of a conservative nonlinear oscillator for which the elastic force term is proportional to u1/3
The results presented in this paper reveal that the HAM is very effective and convenient for conservative nonlinear oscillators with non-polynomial elastic terms
Summary
Classical perturbation methods including the Lindstedt-Poincaré method, the Krylov-Bogoliubov-Mitropolski method and the multiple scales method, as described by Nayfeh [1] [2], Kevorkian and Cole [3] and Verhulst [4], are limited to the weakly nonlinear systems. During the past few decades, based on the classical ones, many improved or innovative methods applicable to the strongly nonlinear systems have been developed in open literature Such as the modified Lindstedt-Poincaré method [5], the hyperbolic Lindstedt-Poincaré method [6], the incremental harmonic balance method [7], the perturbation-incremental method [8], the homotopy anal-. By introducing the non-zero convergence-control parameter h and the nonzero auxiliary function H(t), this method provides a simple way to control and to ensure the convergence of approximation series Another advantage of the HAM is that one can construct a continuous mapping of the initial guess approximation to the exact solution of the given problem through an auxiliary linear operator. On the basic ideas and the applications of HAM, one can refer to [9] [10] [11]
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