The regularization method is applied for the construction of algorithm for an asymptotical solution for linear singular perturbed systems with the irreversible limit operator. The main idea of this method is based on the analysis of dual singular points of investigated equations and passage in the space of the larger dimension, what reduces to study of systems of first-order partial differential equations with incomplete initial data.

/pdf/asymptotic-solutions-of-singular-perturbed-problems-with-an-4hdt5wgyui.pdf

Asymptotic Solutions of Singular Perturbed Problems with an Instable Spectrum of the Limiting Operator

CONTENTS Introduction ??1. The Cauchy problem in a Banach space for a diagonalizable limit operator ??2. The Cauchy problem in a Banach space for a non-diagonalizable limit operator Conclusion References

Asymptotic integration of singularly perturbed problems

In this paper we consider the Cauchy problem for systems of integral - dierential equations with zero points of the spectrum of the limit operator. The presence of singularities in the integral term with a rapidly decreasing kernel generates in a solution of the original problem of essentially special singularities by a small parameter, which describes the internal boundary layer. To establish mathematical theory, criterion formulation of correctness of mathematical description of the boundary layer and to develop a regular theory for singularly perturbed problems we use the regularization method of S.A.Lomov. Normal and unique solvability of iterative tasks is proved, the asymptotic convergence of formal solutions is proved.

/pdf/internal-boundary-layer-for-integral-differential-equations-3sx9ttq2me.pdf

Internal boundary layer for integral-differential equations with zero spectrum of the limit operator and rapidly changing kernel

An asymptotic solution of the linear Cauchy problem in the presence of a “weak” turning point for the limit operator is constructed by the method of S. A. Lomov regularization. The main singularities of this problem are written out explicitly. Estimates are given for e that characterize the behavior of singularities for ϵ→0. The asymptotic convergence of a regularized series is proven. The results are illustrated by an example. Bibliography: six titles.

https://www.mdpi.com/2075-1680/9/3/86/pdf

On the Regularized Asymptotics of a Solution to the Cauchy Problem in the Presence of a Weak Turning Point of the Limit Operator

The aim of the research is to develop the regularization method. By Lomov’s regularization method, we constructed a uniform asymptotic solution of the singularly perturbed Cauchy problem for a parabolic equation in the case of violation of stability conditions of the limit-operator spectrum. The problem with a “simple” turning point is considered in the case, when the eigenvalue vanishes at t=0 and has the form tm/na(t). The asymptotic convergence of the regularized series is proved.

/pdf/singularly-perturbed-cauchy-problem-for-a-parabolic-equation-13p7cqkoev.pdf

Singularly Perturbed Cauchy Problem for a Parabolic Equation with a Rational “Simple” Turning Point

A new method of asymptotic integration is developed - the method of regularization - in the case when the spectrum of the variable limit operator is zero at isolated points. To describe the singular dependence of a solution on the perturbation, additional independent variables are introduced; the space of resonance-free solutions is introduced, in which the coefficients of regularized kind (the solution of the extended problem) are defined. Asymptotic convergence of the series thus obtained to the exact solution of the original singularly perturbed problem is proved.Bibliography: 14 titles.

Theory of Singular Perturbations in the Case of Spectral Singularities of the Limit Operator

Development of the Lomov Regularization Method for a Singularly Perturbed Cauchy Problem and a Boundary Value Problem on the Half-Line for Parabolic Equations with a “Simple” Rational Turning Point

Regularized asymptotics of the solution of a singularly perturbedmixed problem on the semiaxis for an equation of Schrodinger type in the presence of a strong turning point for the limit operator

Singularly Perturbed Cauchy Problem in Which the Limit Operator has Multiple Spectrum and a Weak First-Order Turning Point

The article is devoted to the development of the regularization method by S.A. Lomov on singularly perturbed problems in the presence of spectral singularities of the limit operator. In particular, a regularized asymptotic solution is constructed for the singularly perturbed inhomogeneous Cauchy problem that arises in the quasiclassical approximation in the Schroedinger equation in the coordinate representation. The potential energy profile chosen in the paper leads to a singularity in the spectrum of the limit operator in the form of a <> turning point. Based on the ideas of asymptotic integration of problems with unstable spectrum, S.A. Lomov and A.G. Eliseev, it is indicated how and from what considerations regularizing functions and additional regularizing operators should be introduced, the formalism of the regularization method for the indicated type of singularity is described in detail, this algorithm is substantiated, and an asymptotic solution of any order with respect to a small parameter is constructed.

/pdf/a-regularized-asymptotic-solution-of-the-cauchy-problem-for-vk58dt1v.pdf

A Regularized Asymptotic Solution of the Cauchy Problem for the Nonhomogeneous Schroedinger Equation in the Quasiclassical Approximation in the Presence of a "strong" Turning Point of the Limit Operator

A G Eliseev

Papers

Theory of Singular Perturbations in the Case of Spectral Singularities of the Limit Operator

Development of the Lomov Regularization Method for a Singularly Perturbed Cauchy Problem and a Boundary Value Problem on the Half-Line for Parabolic Equations with a “Simple” Rational Turning Point

Regularized asymptotics of the solution of a singularly perturbedmixed problem on the semiaxis for an equation of Schrodinger type in the presence of a strong turning point for the limit operator

Singularly Perturbed Cauchy Problem in Which the Limit Operator has Multiple Spectrum and a Weak First-Order Turning Point

A Regularized Asymptotic Solution of the Cauchy Problem for the Nonhomogeneous Schroedinger Equation in the Quasiclassical Approximation in the Presence of a "strong" Turning Point of the Limit Operator