Global Residue Harmonic Balance Method for Obtaining Higher-Order Accurate Solutions to the Strongly Nonlinear Oscillator

Gamal Mohamed Ismail, Md. Alal Hosen

Abstract

In this paper, a suitable analytical technique has been introduced
based on the principles of the homotopy perturbation method and the residue
harmonic balance method. The proposed technique has been applied to obtain
approximate higher-order angular frequencies and corresponding periodic solutions of the strongly nonlinear oscillator with a cubic and harmonic restoring
force. Dissimilar other harmonic balance methods, all the earlier residual errors
are presented in the approximate solutions to enhance the accuracy. The expressions of the frequency-amplitude relationship are obtained in a novel analytical way. It is highly remarkable that the second-order approximate solutions produce better than previously existing results and almost similar as compared with the corresponding numerical solutions (considered to be exact). The high accuracy
and simple solution procedure are the merits of the proposed technique which
could also be applied to other nonlinear oscillatory problems arising in science and
engineering