An adaptive optimized Nyström method for second‐order IVPs

Abstract

This research work deals with the development, analysis, and implementation of an adaptive optimized one-step Nyström method for solving second-order initial value problems of ODEs and time-dependent partial differential equations. The new method is developed through a collocation technique with a new approach for selecting the collocation points. An embedding-like procedure is used to estimate the error of the proposed optimized method. The current approach has been used to compute efficiently approximate solutions to general second-order IVPs. The numerical experiments demonstrate that the introduced error estimation and step-size control strategy presented in this manuscript have produced a good performance compared to some of the other existing numerical methods. KEYWORDS collocation method, error estimation and control, ordinary and time-dependent partial differential equations, optimized Nyström method, variable stepsize formulation MSC CLASSIFICATION 65L05, 65L20

A new block method with variable stepsize implementation for solving third‐order differential systems

“…Additional numerical techniques for solving () and related problems can be found in the literature. Examples of these methods include the harmonic balance method, spline method, energy conservation method, finite element method, boundary value method, multi‐derivative method, Trigonometric fitted method, embedded explicit method, direct integration Obrechkoff method, mesh‐free technique, Nyström method, and block method; see [13–33] for further details.…”