On choice of initial guess in the variational iteration method and its applications to nonlinear oscillator

Abstract

This paper uses the variational iteration method to study nonlinear oscillator, and He’s amplitude–frequency formulation is adopted here as a good initial guess. In general, the ability of amplitude–frequency formulation to present reasonable and precision results makes it a reliable method, especially in oscillation systems. In addition, simplicity in the determination of the frequency of the system is one of the distinctive merits in this method. On the other hand, it is difficult to attain higher accurate solutions or higher order solutions in amplitude–frequency formulation. Thus, to overcome this hardship, one can select amplitude–frequency formulation as an initial guess in variational iteration method; this not only noticeably improves the accuracy and efficiency of variational iteration method (improved variational iteration method) but also accomplishing higher order solutions is feasible. Moreover, the more precise the frequency of the initial guess of variational iteration method, the more dominant the final results of variational iteration method. To show the ability and precision of this choice, some examples are presented and their results are compared to variational iteration method, amplitude–frequency formulation, energy balance method, and fourth Runge-Kutta’s numerical method. The resultant graphs and charts show an excellent agreement to this choice. In fact, the choice of amplitude–frequency formulation as an initial guess not only improves various aspects of the variational iteration method but also it distinguishes decline the relatively complex trend of calculating of initial guess compared to other ways.