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Reconstruction of flow and temperature from boundary data
Linköping University, Department of Science and Technology, Communications and Transport Systems. Linköping University, The Institute of Technology.ORCID iD: 0000-0001-9066-7922
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. , p. 13
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 832
Keywords [en]
Partiella differentialekvationer, Operatorteori
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-140145ISBN: 91-7373-682-1 (print)OAI: oai:DiVA.org:liu-140145DiVA, id: diva2:1137626
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
List of papers
1. An alternating method for the stationary Stokes system
Open this publication in new window or tab >>An alternating method for the stationary Stokes system
2006 (English)In: Zeitschrift für angewandte Mathematik und Mechanik, ISSN 0044-2267, E-ISSN 1521-4001, Vol. 86, no 4, p. 268-280Article in journal (Refereed) Published
Abstract [en]

An alternating procedure for solving a Cauchy problem for the stationary Stokes system is presented. A convergence proof of this procedure and numerical results are included. © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-41016 (URN)10.1002/zamm.200410238 (DOI)54933 (Local ID)54933 (Archive number)54933 (OAI)
Available from: 2009-10-10 Created: 2009-10-10 Last updated: 2017-12-13
2. An iterative procedure for solving a Cauchy problem for second order elliptic equations
Open this publication in new window or tab >>An iterative procedure for solving a Cauchy problem for second order elliptic equations
2004 (English)In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 272, p. 46-54Article 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.

Keywords
Cauchy problem, Ill-posed, Iterative method
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-45843 (URN)10.1002/mana.200310188 (DOI)
Available from: 2009-10-11 Created: 2009-10-11 Last updated: 2017-12-13
3. An iterative method for a Cauchy problem for the heat equation
Open this publication in new window or tab >>An iterative method for a Cauchy problem for the heat equation
2006 (English)In: IMA Journal of Applied Mathematics, ISSN 0272-4960, E-ISSN 1464-3634, Vol. 71, no 2, p. 262-286Article in journal (Refereed) Published
Abstract [en]

An iterative method for reconstruction of the solution to a parabolic initial boundary value problem of second order from Cauchy data is presented. The data are given on a part of the boundary. At each iteration step, a series of well-posed mixed boundary value problems are solved for the parabolic operator and its adjoint. The convergence proof of this method in a weighted L2-space is included. © 2006. Oxford University Press.

Keywords
Cauchy problem, Heat equation, Iterative regularization method, Mixed problem, Weighted Sobolev space
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-50241 (URN)10.1093/imamat/hxh093 (DOI)
Available from: 2009-10-11 Created: 2009-10-11 Last updated: 2017-12-12

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Johansson, Tomas

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Citation style
  • apa
  • harvard1
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