Conditional symmetries of nonlinear third-order ordinary differential equations
type: Item
instance of: scholarly article
title: Conditional symmetries of nonlinear third-order ordinary differential equations
zbMATH Open document ID: 1385.34028
author: Peng Zhang
published in: Discrete and Continuous Dynamical Systems. Series S
publication date: 2018-01-19T00:00:00Z
review text: The starting point of this study is the following Definition 1: A nonlinear third-order ODE is conditionally classifiable by a symmetry algebra \(\mathcal{A}\) with respect to a nonlinear second-order ODE called the root ODE if the third-order ODE jointly with the second-order ODE forms an overdetermined compatible system (so the solutions of the third-order ODE reduces to the solutions of the second-order ODE) and the second-order ODE has the symmetry algebra \(\mathcal{A}\) which is called the conditional symmetry algebra of the third-order ODE. An algorithm is presented for computing the conditional symmetries of a scalar third-order ODE of the form \(y^{\prime \prime \prime }=f(x, y, y^{\prime }, y^{\prime \prime })\) subject to the nonlinear second-order ODE \(y^{\prime \prime}=g(x, y, y^{\prime })\). Several examples are classified. A main result is that the derived nonlinear ODEs without substitution have in general less inherited symmetries as compared to that of the root second-order ODEs.
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 34A05
zbMATH DE Number: 6827646
zbMATH Keywords: Lie point symmetries
zbMATH Keywords: conditional symmetries
zbMATH Keywords: first integrals
zbMATH Keywords: three symmetry equations
reviewed by: Mircea Crasmareanu
MaRDI profile type: Publication
full work available at URL: dcdss.2018040
OpenAlex ID: W2765177927
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DOI: 10.3934/DCDSS.2018040
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description: scientific article; zbMATH DE number 6827646