A method of holomorphic generalization of singularly perturbed problems

Kachalov, V. I.

doi:10.3103/s1066369x17060068

Cited by 4 publications

(3 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…. + F k ∂ w k is the linear partial differential operator of the first order in partial derivatives: U = {U [1] , . .…”

Section: Concrete Implementations Of the Theorymentioning

confidence: 99%

“…The solution to the second equation of the series (22) is the vector function [k] } where {V [1] , . .…”

Section: Concrete Implementations Of the Theorymentioning

confidence: 99%

“…As a rule, spaces of holomorphic functions (of one or several variables) are used. In this regard, it was possible to formulate the main principles for the theory of singularly perturbed differential equations and systems-under fairly general assumptions that they possess holomorphics in small parameter first integrals [1,2]. Moreover, a connection between the first integrals and homomorphisms of algebras of holomorphic functions with various numbers of variables was established.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Axiomatic Approach in the Analytic Theory of Singular Perturbations

Besova

Kachalov

2020

Axioms

View full text Add to dashboard Cite

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.

show abstract

“…. + F k ∂ w k is the linear partial differential operator of the first order in partial derivatives: U = {U [1] , . .…”

Section: Concrete Implementations Of the Theorymentioning

confidence: 99%

“…The solution to the second equation of the series (22) is the vector function [k] } where {V [1] , . .…”

Section: Concrete Implementations Of the Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation