Trigonometric collocation methods based on Lagrange basis polynomials for multi-frequency oscillatory second-order differential equations

Abstract

In the present work, a kind of trigonometric collocation methods based on Lagrange basis polynomials is developed for effectively solving multi-frequency oscillatory second-order differential equations $q^{″}\left(t\right)+Mq\left(t\right)=f\left(q\left(t\right)\right)$. The properties of the obtained methods are investigated. It is shown that the convergent condition of these methods is independent of $\Vert M\Vert$, which is very crucial for solving oscillatory systems. A fourth-order scheme of the methods is presented. Numerical experiments are implemented to show the remarkable efficiency of the methods proposed in this paper.