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Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz., 2018, Volume 10, Issue 2, Pages 37–46 (Mi vyurm373)  

Mathematics

Inverse problems of recovering the boundary data with integral overdetermination conditions

S. G. Pyatkov, M. A. Verzhbitskii

Yugra State University, Khanty-Mansyisk, Russian Federation

Abstract: In the present article we examine an inverse problem of recovering unknown functions being part of the Dirichlet boundary condition together solving an initial boundary problem for a parabolic second order equation. Such problems on recovering the boundary data arise in various tasks of mathematical physics: control of heat exchange prosesses and design of thermal protection systems, diagnostics and identification of heat transfer in supersonic heterogeneous flows, identification and modeling of heat transfer in heat-shielding materials and coatings, modeling of properties and heat regimes of reusable heat protection of spacecrafts, study of composite materials, etc. As the overdetrermination conditions we take the integrals of a solution over the spatial domain with weights. The problem is reduced to an operator equation of the Volterra-type. The existence and uniqueness theorem for solutions to this inverse problem is established in Sobolev spaces. A solution is regular, i. e., all generalized derivatives occuring into the equation exists and are summable to some power. The proof relies on the fixed point theorem and bootstrap arguments. Stability estimates for solutions are also given. The solvability conditions are close to necessary conditions.

Keywords: inverse problem, parabolic equation, boundary and initial condition, Sobolev space, existence and uniqueness theorem, solvability.

Funding Agency
The authors were supported by the grant on development of scientific schools with participation of young scientists of the Yugra State University.


DOI: https://doi.org/10.14529/mmph180204

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Document Type: Article
UDC: 517.956
Received: 06.02.2018
Language: English

Citation: S. G. Pyatkov, M. A. Verzhbitskii, “Inverse problems of recovering the boundary data with integral overdetermination conditions”, Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz., 10:2 (2018), 37–46

Citation in format AMSBIB
\Bibitem{PyaVer18}
\by S.~G.~Pyatkov, M.~A.~Verzhbitskii
\paper Inverse problems of recovering the boundary data with integral overdetermination conditions
\jour Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz.
\yr 2018
\vol 10
\issue 2
\pages 37--46
\mathnet{http://mi.mathnet.ru/vyurm373}
\crossref{https://doi.org/10.14529/mmph180204}
\elib{http://elibrary.ru/item.asp?id=32855768}


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