On holomorphic regularization of strongly nonlinear singularly perturbed problems

Abstract

The method of holomorphic regularization, being a logical continuation of the method of S.A. Lomova, allows one to construct solutions to nonlinear singularly perturbed initial problems as series in powers of a small parameter converging in the usual sense. The method is based on a generalization of the Poincare decomposition theorem: in the regular case, solutions depend holomorphically on a small parameter, in the singular case the first integrals inherit this dependence. Having arised in the framework of the regularization method, S.A. Lomov's concept of a pseudo-analytic (pseudo-holomorphic) solution of singularly perturbed problems initiated the formation of the analytic theory of singular perturbations. This theory is designed to equalize the rights of regular and singular theories. In the first case, under sufficiently general assumptions, the series obtained in the solution of problems in powers of the small parameter converge in the usual sense, and in the second case they are basically asymptotic. A vivid example of the holomorphic dependence on a parameter of the solution to a differential equation is given by Poincare's decomposition theorem. In the present paper, the holomorphic regularization method is applied for constructing pseudo-holomorphic solutions to a singularly perturbed first order equation and to a second order Tikhonov system.

Asymptotic and Pseudoholomorphic Solutions of Singularly Perturbed Differential and Integral Equations in the Lomov’s Regularization Method

“…and we seek its solution in the form of a series in powers of ε , assuming the operator ∂ z to be a subordinate operator f ∂ w . We have [8], for an arbitrary function ϕ(z) ∈ A z 0 , that…”

“…The second part of our paper is devoted to the construction of approximate solutions of singularly perturbed problems using the method of holomorphic regularization [8,9]. The analysis of asymptotic methods for solving singularly perturbed problems shows that the solutions of such problems depend in two ways on a small parameter: regularly and singularly.…”

Axiomatic Approach in the Analytic Theory of Singular Perturbations

“…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.…”