Second-Order Delay Ordinary Differential Equations, Their Symmetries And Application To A Traffic Problem

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 ordinary differential equations (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 first-order DODSs, here we concentrate on a large class of 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 infinite-dimensional 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.