On geometry of solutions to approximate equations and their symmetries

Abstract

The paper is devoted to developing a geometric approach to the theory of approximate equations (including ODEs and PDEs) and their symmetries. We introduce dual Lie algebras, manifolds over dual numbers and dual Lie group. We describe some constructions applied for these objects. On the basis of these constructions, we show how one can formulate basic concepts and methods in the theory of approximate equations and their symmetries. The proofs of many general results here can be obtained almost immediately from classical ones, unlike the methods used for studying the approximate equations.