Projector approach to the Butuzov-Nefedov algorithm for asymptotic solution of a class of singularly perturbed problems in a critical case
type: Item
instance of: scholarly article
title: Projector approach to the Butuzov-Nefedov algorithm for asymptotic solution of a class of singularly perturbed problems in a critical case
zbMATH Open document ID: 1465.34073
DOI: 10.1134/S0965542520120076
published in: Computational Mathematics and Mathematical Physics
publication date: 2021-01-08T00:00:00Z
review text: In this paper, an asymptotic solution for a \(m\)-dimensional weakly nonlinear singularly perturbed initial value problem \[ \varepsilon^2\frac{dx}{dt}=A(t)x+\varepsilon f(x,t,\varepsilon),\ x=x(t,\varepsilon)\in X,\ t\in[0,T], \] \[ x(0,\varepsilon)=x^0 \] is constructed using the orthogonal projectors of the space \(X\) onto \(\mathrm{ker}\,A(t)\) and \(\mathrm{ker}\,A^T(t)\). It allows to represent expressions of the boundary function method for finding asymptotic terms of any order in an explicit form. Here, \(\varepsilon\) denotes a small positive parameter, \(A(t)\) is an \(m\times m\) singular matrix with \(k\) zero eigenvalues (\(k < m\)) with equal algebraic and geometric multiplicity for each \(t\in[0,T];\) \(f(x,t,\varepsilon)\) is an \(m\)-dimensional vector function, and \(A(t)\) and \(f(x,t,\varepsilon)\) are assumed to be sufficiently smooth with respect to their arguments. Theory is illustrated by an example of the construction of a first-order asymptotic approximation with numerical simulation.
reviewed by: Robert Vrabel
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 34E05
Mathematics Subject Classification ID: 34E10
Mathematics Subject Classification ID: 34A12
zbMATH DE Number: 7291191
zbMATH Keywords: singular perturbation
zbMATH Keywords: asymptotic solution
zbMATH Keywords: multiple-scale
author: G. A. Kurina
author: Nguyen Thi Xuan Hoai
MaRDI profile type: Publication
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full work available at URL: s0965542520120076
OpenAlex ID: W3121079861
description: scientific article; zbMATH DE number 7291191