A novel perturbation method to approximate the solution of nonlinear ordinary differential equation after being linearized to the Mathieu equation

Abstract

To save the manipulation cost for seeking a higher order approximation to enhance the accuracy of analytic solution, the present paper develops a novel perturbation method by linearizing the nonlinear ordinary differential equation (ODE) with respect to a zeroth order solution in advance, where a weight factor splits the nonlinear terms into two sides of the ODE. Consequently, a series of linear ODEs are solved sequentially to obtain higher order approximate analytic solutions, and meanwhile the frequency can be determined explicitly by solving a frequency equation. When the nonlinear problems are linearized to the Mathieu equations endowing with periodic forcing terms, we develop a novel homotopy perturbation method to determine their solutions, and then provide accurate formulas for nonlinear oscillators. For Duffing oscillator as an example, the accuracy of frequency obtained by the linearized homotopy perturbation method can be raised to 10(-8), and even for a huge value of nonlinear coefficient, the error is of the order 10(-5). A numerical procedu r e is developed to implement the proposed method, where the computed order of convergence reveals a linear convergence that the accuracy of nth order approximate solution is better than 10(-(n+1)). The super-and sub-harmonic periodic solutions are exhibited for the forced Duffing equation.