On the choice of the frequency in trigonometrically-fitted methods for periodic problems

In this paper the use of a trigonometrically fitted method to obtain the approximate solutions of some nonlinear periodic oscillators is presented. A great number of different approaches have been considered to obtain analytical approximations for this kind of problems: a generalized decomposition method (GDM), a linearized harmonic balance procedure (LHB), the homotopy perturbation method (HPM), the harmonic balanced method (HBM), the Adomian decomposition method, etc. From those approaches, analytical approximations to the frequency of oscillation and periodic solutions are obtained, which are valid for a large range of amplitudes of oscillation. However, these techniques have been limited to obtain only one or two iterates because of the great amount of algebra involved. We use a trigonometrically adapted method to obtain numerical approximations to the solutions, yielding very acceptable results, on the basis that the approximation of the frequency of the method is done with great accuracy. There are a lot of trigonometrically fitted methods in the literature, but there is not a definite way to obtain the optimal value of the frequency. We present a strategy for the choice of the parameter value in the adapted method based on the minimization of the total energy. Some examples solved numerically confirm the good performance of the adopted strategy.