Linear differential-algebraic equations perturbed by Volterra integral operators
type: Item
instance of: scholarly article
title: Linear differential-algebraic equations perturbed by Volterra integral operators
zbMATH Open document ID: 1386.45009
DOI: 10.1134/S0012266117100044
published in: Differential Equations
publication date: 2018-02-27T00:00:00Z
review text: In the present work, the authors study the initial problem (IP) for linear inhomogeneous vector ordinary differential equations of arbitrary order in the form \[ {{\Lambda }_{k}}x:={{A}_{k}}(t){{x}^{(k)}}(t)+{{A}_{k-1}}(t){{x}^{(k-1)}}(t)+\dots+{{A}_{0}}(t)x(t)=f(t),\tag{1} \] \[ {{x}^{(j)}}(\alpha )={{\alpha }_{j}},\;j=1,\dots,k-1,\tag{2} \] where \(t\in T=[\alpha ,\beta ]\), \(k,n\in \mathbb{N}\), \({{\alpha }_{j}}\in {{\mathbb{R}}^{n}}\) are given vectors, \({{A}_{j}}:T\to {{\mathbb{R}}^{n\times n}}\) and \(\det {{A}_{k}}(t)=0,\;t\in T\). Along with IP (1), (2) is studied the an auxiliary IP (3), (4) obtained by perturbations with a Volterra operator (not necessarily small) of equation (1) in the form \[ ({{\Lambda }_{k}}+\lambda V)z:={{A}_{k}}(t){{z}^{(k)}}(t)+\dots+{{A}_{0}}(t)x(t)+\lambda \int\limits_{\alpha }^{t}{K(t,s)z(s)ds}=f(t),\tag{3} \] \[ {{z}^{(j)}}(\alpha )={{\alpha }_{j}},\;j=1,\dots,k-1.\tag{4} \] In the work under natural conditions are obtained sufficient conditions for the existence and uniqueness of the solutions of the IP (1), (2) and IP (3), (4) respectively, as well as and representations of their general solutions. The influence of small perturbations of the free term and the initial data on the solutions of the IP (1), (2) and IP (3),(4) are considered. A numerical method is suggested and single out a class of differential-algebraic equations for which one, when computing the index does not need to find derivatives of the input data and hence can be applyed a broad class of finite-difference methods. The results are illustrated with numerical experiments.
reviewed by: Andrey Zahariev
Mathematics Subject Classification ID: 45J05
zbMATH DE Number: 6844046
zbMATH Keywords: Linear differential-algebraic equations
zbMATH Keywords: Volterra integral operators
author: V. F. Chistyakov
author: E. V. Chistyakova
MaRDI profile type: Publication
cites work: Q3900330
cites work: Q4868578
cites work: Algebro-differential operators with a finite-dimensional kernel.
cites work: Q4344158
cites work: Q4267915
cites work: Q5708359
cites work: Collocation Methods for Volterra Integral and Related Functional Differential Equations
cites work: Differential-algebraic equations. A projector based analysis
cites work: Q5839071
cites work: Q4302108
cites work: Q4003936
cites work: Application of matrix polynomials to the analysis of linear differential-algebraic equations of higher order
cites work: On the structure of the general solution and conditions of solvability of the Cauchy problem for degenerate linear systems of higher-order differential equations
cites work: Transformation of high order linear differential-algebraic systems to first order
cites work: Regularizing properties of difference schemes for singular integral-differential equations
cites work: Q3317364
cites work: Q3741301
cites work: Q4528895
full work available at URL: s0012266117100044
OpenAlex ID: W2770420930
description: scientific article; zbMATH DE number 6844046