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An iterative method for a Cauchy problem for the heat equation
Linköping University, The Institute of Technology. Linköping University, Department of Science and Technology.ORCID iD: 0000-0001-9066-7922
2006 (English)In: IMA Journal of Applied Mathematics, ISSN 0272-4960, E-ISSN 1464-3634, Vol. 71, no 2, 262-286 p.Article 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.

Place, publisher, year, edition, pages
2006. Vol. 71, no 2, 262-286 p.
Keyword [en]
Cauchy problem, Heat equation, Iterative regularization method, Mixed problem, Weighted Sobolev space
National Category
Engineering and Technology
Identifiers
URN: urn:nbn:se:liu:diva-50241DOI: 10.1093/imamat/hxh093OAI: oai:DiVA.org:liu-50241DiVA: diva2:271137
Available from: 2009-10-11 Created: 2009-10-11 Last updated: 2017-12-12
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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