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

2019/01/09 by Dorodnitsyn, Vladimir A., Kozlov, Roman, Meleshko, Sergey V. +1
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1901.06251

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 ≤ n ≤ 6 for nonlinear DODSs and must be n=∞ 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.

Related

• Symmetry Approach to Integration of Ordinary Differential Equations with Retarded Argument
• The exact solutions of differential equation with delay
• Lie group symmetries for first order ODEs
• Some second-order partial differential equations associated with Lie groups
• Group classification of second order neutral differential equations