Lie group classification of first-order delay ordinary differential equations

TL;DR

This paper classifies first-order delay differential equations using Lie group analysis, revealing symmetry structures and providing methods to find exact solutions through symmetry reduction.

Contribution

It offers a comprehensive Lie group classification for first-order delay ODEs, including linear, nonlinear, and linearizable cases, and demonstrates how to obtain solutions via symmetry methods.

Findings

Linear DODSs have infinite-dimensional symmetry algebras.

Nonlinear DODSs have symmetry algebras of dimension 0 to 3.

Exact solutions can be derived using symmetry reduction techniques.

Abstract

A group classification of first-order delay ordinary differential equation (DODE) accompanied by an equation for delay parameter (delay relation) is presented. A subset of such systems (delay ordinary differential systems or DODSs) which consists of linear DODEs and solution independent delay relations have infinite-dimensional symmetry algebras, as do nonlinear ones that are linearizable by an invertible transformation of variables. Genuinely nonlinear DODSs have symmetry algebras of dimension $n$, $0 \leq n \leq 3$. It is shown how exact analytical solutions of invariant DODSs can be obtained using symmetry reduction.