An optimally scaled polynomial-Fourier-series method for the numerical solution of the Duffing oscillator

In this paper, a novel multiple-scale polynomial-Fourier-series method (PFSM) is developed to be used in the data interpolation, in which the multiple-scale Rk can be determined exactly and optimally in terms of the data nodes. For solving the nonlinear Duffing equation, an optimally scaled harmonic balance method (OSHB) is derived, which is better than the classic harmonic balance method (HB). In terms of the OSHB, the periodic solutions of the Duffing oscillator, and reconstruction of the frequency response curves, which exhibit a hysteresis within which the multiple solutions can happen in an interval of frequency near to the resonant frequency can be precisely solved. The PFSM is further adapted to solve the initial value problem of the Duffing equation, and the periodic solution can be obtained more accurately than the HB. The proposed method has also been tested to solve the boundary value problem and the initial value problem of some nonlinear ordinary differential equations (ODEs). The conclusion can be drawn that the present OSHB and PFSM are effective to solve nonlinear ODEs, including the nonlinear Duffing equation as a demonstrative example.