On the convergence of iterative methods for general differential-algebraic systems
type: Item
instance of: scholarly article
title: On the convergence of iterative methods for general differential-algebraic systems
zbMATH Open document ID: 1054.65130
published in: Journal of Computational and Applied Mathematics
publication date: 2004-08-16T00:00:00Z
review text: The main object of the paper is the system of functional-integral-algebraic equations \[ \dot {x}(t) = x^0 + \int_{0}^{t} f(t,s,x(\cdot),\lambda(\cdot))\,ds, \quad t\in J=[0,T], \tag{1} \] \[ \lambda(t)=g(t,x(\cdot),\lambda(\cdot)), \quad t\in J, \tag{2} \] where all functions and functionals are continuous (in spaces with the maximum norms). The purpose of the paper is the investigation of conditions that provide existence and uniqueness of a solution to (1), (2) as well as convergence of different iterative procedures to it. The most attention is devoted to the waveform relaxation (WR) algorithm for resolving nonlinear equations, that is based on the transition from the initial nonlinear functions of the system to be solved to equivalent splitting ones via their variables set expansion. This paper is developing results obtained earlier by two of the authors for differential-algebraic equations (DAEs), which are defined by operators of Volterra type [cf. \textit{Z. Jackiewicz} and \textit{M. Kwapisz}, SIAM J. Numer. Anal. 33, No. 6, 2303--2317 (1996; Zbl 0889.34064)]. Here this property is omitted. Besides the WR the classical Picard and Gauss-Seidel algorithms are considered. Special cases of problems (1), (2), particularly a quasi-linear DAEs are also investigated and numerical results are presented.
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L80
Mathematics Subject Classification ID: 45J05
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 34A09
zbMATH DE Number: 2091417
zbMATH Keywords: functional-integral-algebraic equations
zbMATH Keywords: iterative processes
zbMATH Keywords: waveform relaxation
zbMATH Keywords: convergence
zbMATH Keywords: Picard algorithm
zbMATH Keywords: numerical examples
zbMATH Keywords: differential-algebraic equations
zbMATH Keywords: Gauss-Seidel algorithms
author: Zbigniew Bartoszewski
author: Tadeusz Jankowski
author: Marian Kwapisz
reviewed by: Vladimir Gorbunov
MaRDI profile type: Publication
full work available at URL: j.cam.2003.12.032
OpenAlex ID: W2023128239
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DOI: 10.1016/J.CAM.2003.12.032
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description: scientific article; zbMATH DE number 2091417