Approximate Symmetries and Approximate Solutions of Some Perturbed ODE Models

TL;DR

This paper develops methods to find approximate symmetries and solutions for perturbed ordinary differential equations, specifically applying to the Boussinesq and Benjamin-Bona-Mahony models, enhancing analytical tools for nonlinear ODEs.

Contribution

It introduces a systematic approach to identify approximate symmetries and solutions for perturbed second-order ODEs, including higher-order symmetries linked to unstable point symmetries.

Findings

Derived approximate point symmetries for the Boussinesq ODE.

Constructed general approximate solutions using these symmetries.

Applied approximate integrating factors to solve the Benjamin-Bona-Mahony ODE.

Abstract

We find Baikov-Gazizov-Ibragimov approximate point symmetries of the second-order Boussinesq ODE, and we find the higher-order approximate symmetries corresponding to the unstable point symmetries (the point symmetries that disappear fron the classification of the BGI approximate point symmetries) of the unperturbed equation. Approximate local symmetries are used to construct a general approximate solution of the Boussinesq ODE. We use approximate integrating factors to find a general approximate solution of the Benjamin-Bona-Mahony ODE reduction.