In this paper a new approach combining the features of the homotopy concept with variational approach is proposed to find accurate analytical solutions for nonlinear oscillators with and without a fractional power restoring force. Since the first-order approximation leads to very accurate results, comparisons with other results are presented to show the effectiveness of this method. The validity of the method is independent of whether or not there exist small or large parameters in the considered nonlinear equations; the obtained results prove the validity and efficiency of the method, which can be easily extended to other strongly nonlinear problems. At the end we compare our procedure with the optimal homotopy perturbation method.

In this paper, two novel and different methods are applied to nonlinear oscillators. It has been found that the coupled method of homotopy perturbation method and variational formulation and amplitude-frequency formulation work very well for the whole range of initial amplitudes. The analytical approximate frequency and the corresponding periodic solution are valid for small as well as large amplitudes of oscillation. Contrary to the conventional methods, only one iteration leads to high accuracy of the solutions. Some examples are given to illustrate the accuracy and effectiveness of these methods.

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