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In this study, an optimally scaled Fourier expansion method with half-order technique, referred to as OSFEM(H), is developed based on the combination of the optimal scaling and the half-order techniques. The optimal scaling technique is employed to reduce the ill-posedness which may arise from using high order Fourier expansion series to approximate periodic solutions. The half-order technique is a powerful tool with which one can use an m -order Fourier expansion to interpolate as many as up to 4 m + 1 points. In this paper, the best scales in the multi-scale Fourier expansion interpolation are derived based on the idea of equating the norm of each column of the interpolated matrix, such that the condition number of the preconditioned matrix is minimized. Then, the present half-order OSFEM, i.e. OSFEM(H), is used to solve the Duffing equation. It is shown that excellent results are achieved by comparing with the differential transformation method and the harmonic balance method. Numerical simulations verify the accuracy and effectiveness of the presently proposed method. [ABSTRACT FROM AUTHOR]

Recently, a simple time domain collocation (TDC) method was proposed by researchers including the first author of the present paper, and has been successfully applied to obtain the harmonic, subharmonic, and superharmonic responses of the nonlinear Duffing oscillator. The TDC method is based on the point-collocation method performing within an appropriate period of the periodic solution, wherein the approximate solution is assumed as a Fourier series. Upon using the TDC method, the ordinary differential equation is transformed into a system of non-linear algebraic equations (NAEs), which can be readily solved by an NAE solver. In this study, using the Duffing oscillator as the prototype, we develop a multiple scale time domain collocation (MSTDC) method, by introducing a series of optimal multiple scales to the Fourier series of the approximate solution, to alleviate the ill-posedness of the system of collocation-resulting NAEs due to the inclusion of very high order harmonics in the approximate solution. Besides the MSTDC method, a multiple scale differential transformation (MSDT) method is proposed by introducing the multiple scales to the classical differential transformation method. Finally, numerical experiments are carried out to verify the accuracy and efficiency of the MSTDC and the MSDT methods. [ABSTRACT FROM AUTHOR]