Exponential Fourier Collocation Methods for First-Order Differential Equations

Abstract

Commencing from the variation-of-constants formula and incorporating a local Fourier expansion of the underlying problem with collocation methods, this chapter presents a novel class of exponential Fourier collocation methods (EFCMs) for solving systems of first-order ordinary differential equations. We discuss in detail the connections of EFCMs with trigonometric Fourier collocation methods (TFCMs), the well-known Hamiltonian Boundary Value Methods (HBVMs), Gauss methods and Radau IIA methods. It turns out that the novel EFCMs are an extension, in a strict mathematical sense, of these existing methods in the literature.