Obrechkoff one-step method fitted with Fourier spectrum for undamped Duffing equation

Using the Fourier-fitting technique, which was put forward by the author recently, we construct a new kind of Obrechkoff one-step method, which combines the Fourier spectrum and the high-order derivatives. We compare the relationship between the local truncation error to the number of fitted coefficients and the error in the analytical approximations to the containing terms of the Fourier spectrum for two types of undamped Duffing equations and find that it exists a one-to-one relation for the error of the two kinds of solutions. The numerical results demonstrate the new method has great advantages in the accuracy, stability and efficiency; it not only can be used as a high-accurate integrator with any large step size but also as a frequency filter to detect the fundamental frequency of a unknown periodic dynamic system.