NDLI: Zu einer speziellen Klasse nichtlinearer Fredholm-Integralgleichungen 1. Art

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Zu einer speziellen Klasse nichtlinearer Fredholm-Integralgleichungen 1. Art

Content Provider	Springer Nature Link
Author	Schröter, Torsten
Copyright Year	1997
Abstract	In this paper a special class of nonlinear Fredholm integral equations of the first kind, the so-called Urysohn equation, is considered, where the kernel depends ont only via the unknown functionx(t). To overcome the ambiguity, a decreasing rearrangement approach is used. Moreover, a constrained least squares method helps regularizing the problem. As a specific property, the equation can be decomposed into a well-posed nonlinear part, the inversion of a function, and an ill-posed linear part, a linear Fredholm integral equation of the first kind. The linear part of this two-stage procedure was already discussed in [8]. In the present paper the two-stage procedure is compared with a one-stage nonlinear least squares approximation which is directly applied to the nonlinear original integral equation. The comparison is explained by means of a computational case study for a specific example arising in optics.In dieser Arbeit wird eine spezielle Klasse von nichtlinearen Fredholm-Integralgleichungen 1. Art untersucht, die sogenannten Urysohn-Gleichungen. Dabei soll die Kernfunktion nur über die Funktionx(t) von der Integrationsgrößet abhängen. Um die Eindeutigkeit der Lösung zu gewährleisten, wird das Prinzip der fallenden Umordnung benutzt. Die Regularisierung des Problems erfolgt unter Benutzung der Methode der Kleinsten Quadrate mit Nebenbedingungen. Als eine spezielle Eigenschaft kann die Gleichung in einen korrekt gestellten nichtlinearen Teil, die Umkehrung einer Funktion, und einen inkorrekt gestellten linearen Teil, eine lineare Fredholm-Integralgeichung 1. Art, zerlegt werden. Der lineare Teil dieser 2-stufigen Methode wurde bereits in [8] näher untersucht. In der vorliegenden Arbeit wird die 2-stufige Methode mit einer 1-stufigen nichtlinearen Kleinste Quadrate Näherung unter Nebenbedingungen verglichen. Der Vergleich erfolgt anhand einer numerischen Fallstudie für ein spezielles Beispiel aus der Optik.
Ending Page	279
Page Count	21
Starting Page	259
File Format	PDF
ISSN	0010485X
e-ISSN	14365057
Journal	Computing
Issue Number	3
Volume Number	58
Language	German
Publisher	Springer-Verlag
Publisher Date	1997-01-01
Publisher Place	Vienna
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Improperly posed problems Computational Mathematics and Numerical Analysis regularization Ill-posed-integral equation Particular nonlinear operators (superposition, Hammerstein, NemytskiÄ, Uryson, etc.)
Content Type	Text
Resource Type	Article
Subject	Theoretical Computer Science Computational Theory and Mathematics Computational Mathematics Numerical Analysis Computer Science Applications Software

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Zu einer speziellen Klasse nichtlinearer Fredholm-Integralgleichungen 1. Art

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