On implicit systems of differential equations.
type: Item
instance of: scholarly article
title: On implicit systems of differential equations.
zbMATH Open document ID: 1037.34001
DOI: 10.1016/S0022-0396(03)00191-8
author: F. Leon Pritchard
published in: Journal of Differential Equations
publication date: 2003-11-19T00:00:00Z
review text: The author studies implicit autonomous systems of ordinary differential equations with polynomial dependency on both the unknown functions and their derivatives. This allows him to make heavy use of methods from commutative algebra and algebraic geometry in his analysis. After a brief review of the relevant methods, the main part of the article consists of an extensive discussion of the completion of such a system in ideal theoretic terms leading to explicit algorithms. The author also offers a geometric interpretation of completion in terms of the tangent variety of the constraint variety along the lines usually used in differential-geometric approaches to differential algebraic equations. Then he provides a field-theoretic formulation of the conditions under which initial data uniquely determine the initial derivatives and shows that this defines then in fact a birational equivalence. This result forms the basis for proving that systems satisfying these conditions possess (locally) unique analytic solutions. Finally, some concrete examples are discussed in detail.
Mathematics Subject Classification ID: 34A09
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 34A12
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 2005882
zbMATH Keywords: differential-algebraic equation
zbMATH Keywords: completion
zbMATH Keywords: existence and uniqueness of solutions
reviewed by: Werner M. Seiler
describes a project that uses: SINGULAR
MaRDI profile type: Publication
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full work available at URL: s0022-0396(03)00191-8
OpenAlex ID: W2073439437
description: scientific article; zbMATH DE number 2005882