On a regularization of index 3 differential-algebraic equations
type: Item
instance of: scholarly article
title: On a regularization of index 3 differential-algebraic equations
zbMATH Open document ID: 1113.65083
author: Hong Liu
author: Yongzhong Song
published in: Applied Mathematics and Computation
publication date: 2006-12-07T00:00:00Z
review text: \textit{M. Hanke} [J. Math. Anal. Appl. 151, 236--253 (1990; Zbl 0722.34013)] introduced a regularisation for linear systems of differential algebraic equations (DAEs) with time-dependent coefficients \(A(t) x'(t) + B(t) x(t) = q(t)\) in case of tractability index 2. In the paper at hand, a corresponding regularisation is performed for such systems in case of tractability index 3, where linear two-point boundary conditions are considered. Based on an according matrix chain with smooth projectors, the authors arrange a modified system of DAEs depending on a parameter \(\varepsilon\). This system is proved to be transferable for sufficiently small parameters. Furthermore, assuming some properties, a solution of the modified system yields a solution of another larger system of DAEs, which represents a singular perturbation problem with respect to the parameter \(\varepsilon\). Vice versa, a solution of the singular perturbation problem forms a solution of the modified system of DAEs under analogue conditions. The authors prove that the norm of a projected difference between the solution of the original index-3 DAEs and the solution of the modified parameter-dependent system exhibits the magnitude \(\mathcal{O}(\sqrt{\varepsilon})\). Finally, two examples of initial value problems involving index-3 DAEs are discussed, where the exact solution is known explicitly. The corresponding modified systems are solved numerically via the implicit Euler scheme using several parameters \(\varepsilon\). The resulting differences verify that the predicted order of convergence is achieved or represents at least a lower bound.
reviewed by: Roland Pulch
Mathematics Subject Classification ID: 65L80
Mathematics Subject Classification ID: 34A09
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 34B05
zbMATH DE Number: 5078424
zbMATH Keywords: differential algebraic equations
zbMATH Keywords: tractability index
zbMATH Keywords: regularisation
zbMATH Keywords: singular perturbation problem
zbMATH Keywords: numerical examples
zbMATH Keywords: linear two-point boundary conditions
zbMATH Keywords: initial value problems
zbMATH Keywords: convergence
MaRDI profile type: Publication
full work available at URL: j.amc.2006.03.006
OpenAlex ID: W2029106091
cites work: Q3993493
cites work: Q3852479
cites work: Q3940107
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DOI: 10.1016/J.AMC.2006.03.006
description: scientific article; zbMATH DE number 5078424