Validated solutions of initial value problems for ordinary differential equations
type: Item
instance of: scholarly article
title: Validated solutions of initial value problems for ordinary differential equations
zbMATH Open document ID: 0934.65073
DOI: 10.1016/S0096-3003(98)10083-8
published in: Applied Mathematics and Computation
publication date: 2000-04-03T00:00:00Z
review text: The purpose of the paper is to review the most significant developments in the area of validated solutions of initial value problems (IVPs) for ordinary differential equations (ODEs), especially interval methods based on Taylor series, and to identify areas for future research. After a short introduction, Section 2 introduces interval-arithmetic operations, interval-valued functions, interval integration and discuss a method for efficient generation of Taylor series coefficients. Section 3 states in more detail the IVP that is the subject of the paper. Section 4 presents an overview of Taylor series methods for validated solutions of IVPs for ODEs. Section 5 examines three techniques for computing an a priori enclosure of the solution and selecting a stepsize. Section 6 explains the wrapping effect. Section 7 describes five algorithms for computing a tight enclosure and reducing the wrapping effect. Section 8 consider some particular stepsize and order control strategies. Section 9 describes a method based on controlling the defect of the computed solution.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L50
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 65G20
Mathematics Subject Classification ID: 65G40
zbMATH DE Number: 1355447
zbMATH Keywords: initial value problems
zbMATH Keywords: Taylor series method
zbMATH Keywords: interval arithmetic
zbMATH Keywords: interval analysis
zbMATH Keywords: stepsize selection
zbMATH Keywords: validated solutions
zbMATH Keywords: interval-valued functions
zbMATH Keywords: wrapping effect
zbMATH Keywords: algorithms
zbMATH Keywords: order control
author: Nedialko S. Nedialkov
author: Kenneth R. Jackson
author: George F. Corliss
reviewed by: Dana Petcu
describes a project that uses: ATOMFT
MaRDI profile type: Publication
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full work available at URL: s0096-3003(98)10083-8
OpenAlex ID: W2012580470
description: scientific article; zbMATH DE number 1355447