Spectral fixed point method for nonlinear oscillation equation with periodic solution

Based on the fixed point concept in functional analysis, an improvement on the traditional spectral method is proposed for nonlinear oscillation equations with periodic solution. The key idea of this new approach (namely, the spectral fixed point method, SFPM) is to construct a contractive map to replace the nonlinear oscillation equation into a series of linear oscillation equations. Usually the series of linear oscillation equations can be solved relatively easily. Different from other existing numerical methods, such as the well-known Runge-Kutta method, SFPM can directly obtain the Fourier series solution of the nonlinear oscillation without resorting to the Fast Fourier Transform (FFT) algorithm. In the meanwhile, the steepest descent seeking algorithm is proposed in the framework of SFPM to improve the computational efficiency. Finally, some typical cases are investigated by SFPM and the comparison with the Runge-Kutta method shows that the present method is of high accuracy and efficiency.