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An iterative procedure for solving a Cauchy problem for second order elliptic equations
Linköping University, The Institute of Technology. Linköping University, Department of Science and Technology.ORCID iD: 0000-0001-9066-7922
2004 (English)In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 272, 46-54 p.Article in journal (Refereed) Published
Abstract [en]

An iterative method for reconstruction of solutions to second order elliptic equations by Cauchy data given on a part of the boundary, is presented. At each iteration step, a series of mixed well-posed boundary value problems are solved for the elliptic operator and its adjoint. The convergence proof of this method in a weighted L2 space is included. © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Place, publisher, year, edition, pages
2004. Vol. 272, 46-54 p.
Keyword [en]
Cauchy problem, Ill-posed, Iterative method
National Category
Engineering and Technology
Identifiers
URN: urn:nbn:se:liu:diva-45843DOI: 10.1002/mana.200310188OAI: oai:DiVA.org:liu-45843DiVA: diva2:266739
Available from: 2009-10-11 Created: 2009-10-11 Last updated: 2017-12-13
In thesis
1. Reconstruction of flow and temperature from boundary data
Open this publication in new window or tab >>Reconstruction of flow and temperature from boundary data
2003 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis, we study Cauchy problems for elliptic and parabolic equations. These include the stationary Stokes system and the heat equation. Data are given on a part of the boundary of a bounded domain. The aim is to reconstruct the solution from these data. These problems are ill-posed in the sense of J. Hadamard.

We propose iterative regularization methods, which require solving of a sequence of well-posed boundary value problems for the same operator. Methods based on this idea were _rst proposed by V. A. Kozlov and V. G. Maz'ya for a certain class of equations which do not include the above problems. Regularizing character is proved and stopping rules are proposed.

The regularizing character for the heat equation is proved in a certain weighted L2 space. In each iteration the Zaremba problem for the heat equation is solved. We also prove well-posedness of this problem in a weighted Sobolev space. This result is of independent interest and is presented as a separate paper.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2003. 13 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 832
Keyword
Partiella differentialekvationer, Operatorteori
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-140145 (URN)91-7373-682-1 (ISBN)
Public defence
2003-10-24, TP2, Täppan, Campus Norrköping, Norrköping, 10:15 (English)
Opponent
Supervisors
Available from: 2017-08-31 Created: 2017-08-31 Last updated: 2017-09-08Bibliographically approved

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Johansson, Tomas

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