Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz., 2018, Volume 10, Issue 2, Pages 37–46
Inverse problems of recovering the boundary data with integral overdetermination conditions
S. G. Pyatkov, M. A. Verzhbitskii
Yugra State University, Khanty-Mansyisk, Russian Federation
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.
inverse problem, parabolic equation, boundary and initial condition, Sobolev space, existence and uniqueness theorem, solvability.
|The authors were supported by the grant on development of scientific schools with participation of
young scientists of the Yugra State University.
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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
\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.
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