Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
134 tokens/sec
GPT-4o
9 tokens/sec
Gemini 2.5 Pro Pro
47 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

On Exact and Approximate Symmetries of Algebraic and Ordinary Differential Equations with a Small Parameter (2104.00049v1)

Published 31 Mar 2021 in math-ph, math.DS, and math.MP

Abstract: The framework of Baikov-Gazizov-Ibragimov approximate symmetries has proven useful for many examples where a small perturbation of an ordinary differential equation (ODE) destroys its local symmetry group. For the perturbed model, some of the local symmetries of the unperturbed equation may (or may not) re-appear as approximate symmetries, and new approximate symmetries can appear. Approximate symmetries are useful as a tool for the construction of approximate solutions. We show that for algebraic and first-order differential equations, to every point symmetry of the unperturbed equation, there corresponds an approximate point symmetry of the perturbed equation. For second and higher-order ODEs, this is not the case: some point symmetries of the original ODE may be unstable, that is, they do not arise in the approximate point symmetry classification of the perturbed ODE. We show that such unstable point symmetries correspond to higher-order approximate symmetries of the perturbed ODE, and can be systematically computed. Two detailed examples, including a fourth-order nonlinear Boussinesq equation, are presented. Examples of the use of higher-order approximate symmetries and approximate integrating factors to obtain approximate solutions of higher-order ODEs are provided.

Summary

We haven't generated a summary for this paper yet.