Generalized solutions of algebraic differential equations
type: Item
instance of: scholarly article
title: Generalized solutions of algebraic differential equations
zbMATH Open document ID: 0578.12017
DOI: 10.1016/0022-0396(86)90100-2
author: Lee A. Rubel
published in: Journal of Differential Equations
publication date: 1986-01-01T00:00:00Z
review text: Two related notions are introduced of a function or distribution u being a ''solution'' of certain non-linear ordinary or partial differential equations. They are the notion of N-solution and IN-solution. The driving idea of the paper is that a certain u ''should be'' a solution except only that it is not differentiable enough to plug into the equation. But by using Ritt's differential Nullstellensatz in a new way, the lack of sufficient differentiability is circumvented. There are two main advantages of this procedure. First, it gives an intriguing new context in which a Schwartz distribution may be a solution of a non-linear equation, despite the impossibility of a completely satisfactory multiplication of distributions. Next, it gives a new context in which certain classical theorems of differential algebra (like the Ritt-\(Raudenbush\) basis theorem) are valid, despite the absence of higher derivatives. Several examples are worked out, and open problems are described.
Mathematics Subject Classification ID: 12H20
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 47J25
zbMATH DE Number: 3924934
zbMATH Keywords: algebraic differential equations
zbMATH Keywords: generalized solutions
zbMATH Keywords: ADE
zbMATH Keywords: N-solution
zbMATH Keywords: IN-solution
zbMATH Keywords: Ritt's differential Nullstellensatz
zbMATH Keywords: Schwartz distribution
zbMATH Keywords: non-linear equation
reviewed by: Hans Ade
MaRDI profile type: Publication
full work available at URL: 0022-0396(86)90100-2
OpenAlex ID: W2027928494
Wikidata QID: Q56812657
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description: scientific article; zbMATH DE number 3924934