Optimal scale polynomial interpolation technique for obtaining periodic solutions to the Duffing oscillator

Abstract

A novel optimal scale polynomial interpolation method (OSPIM) is proposed to attack the Duffing oscillator. This method is based on the ideas of multi-scaling and equilibrated matrix, such that the condition number of the coefficient matrix of the polynomial interpolation is minimized. The OSPIM can eliminate the Runge phenomenon, which occurs in the conventional polynomial interpolation, and is well suited for solving nonlinear oscillatory systems. In addition, we further alleviate the ill-posedness of polynomial interpolation by proposing a half-order technique, with which one can use a \(m\)-order polynomial to interpolate as many as \(2m+1\) points. We then employ the half-order OSPIM, i.e., OSPIM(H), as a trial function in conjunction with the simple point-collocation method, to solve the nonlinear Duffing equation. Moreover, the differential transformation method is used for the first time to solve a forced Duffing oscillator to compare with the present method. Finally, illustrative examples verify the accuracy and efficiency of the present method.