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arXiv · 1901.06251

Second-order delay ordinary differential equations, their symmetries and application to a traffic problem

Vladimir A. Dorodnitsyn·Roman Kozlov·Sergey V. Meleshko·Pavel Winternitz
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Abstract

This article is the third in a series the aim of which is to use Lie group theory to obtain exact analytic solutions of Delay Ordinary Differential Systems (DODSs). Such a system consists of two equations involving one independent variable $x$ and one dependent variable $y$. As opposed to ODEs the variable $x$ figures in more than one point (we consider the case of two points, $x$ and $x_-$). The dependent variable $y$ and its derivatives figure in both $x$ and $x_-$. Two previous articles were devoted to {\it first}-order DODSs, here we concentrate on a large class of {\it second}-order ones. We show that within this class the symmetry algebra can be of dimension $n$ with $0 \leq n \leq 6$ for nonlinear DODSs and must be $n=\infty$ for linear or linearizable ones. The symmetry algebras can be used to obtain exact particular group invariant solutions. As a specific application we present some exact solutions of a DODS model of traffic flow.

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Vladimir A. Dorodnitsyn, Roman Kozlov, Sergey V. Meleshko, Pavel Winternitz. 2020-07-08. Second-order delay ordinary differential equations, their symmetries and application to a traffic problem. https://arxiv.org/abs/1901.06251

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