Axiomatic Approach in the Analytic Theory of Singular Perturbations

Abstract

Introduced by S.A. Lomov, the concept of a pseudoanalytic (pseudoholomorphic) solution laid the foundation for the development of the singular perturbation analytical theory. In order for this concept to work in case of linear problems, an apparatus for the theory of exponential type vector spaces was developed. When considering nonlinear singularly perturbed problems, an algebraic approach is currently used. This approval is based on the properties of algebra homomorphisms for holomorphic functions with various numbers of variables, as a result of which it is possible to obtain pseudoholomorphic solutions. In this paper, formally singularly perturbed equations are considered in topological algebras, which allows the authors to formulate the main concepts of the singular perturbation analytical theory from the standpoint of maximal generality.

Analytical Aspects of the Theory of Tikhonov Systems

“…The results presented in this article allow us to introduce the possibility of constructing an analytical theory of singular perturbations as a section of differential equations [13,14]. It should also be noted that the algebraic approach is actively used in the holomorphic regularization method [9].…”

“…In this paper, the number of fast and slow variables can be arbitrary. In addition, boundary value problems are considered, which means that the use of the pseudoholomorphic continuation algorithm [9] is required.…”