Inverse Problems of Recovering the Boundary Data with Integral Overdetermination Conditions

S.G. Pyatkov, M.A. Verzhbitskii Yugra State University, Khanty-Mansyisk, Russian Federation E-mail: s_pyakov@ugrasu.ru. С.Г. Пятков, М.А. Вержбицкий Югорский государственный университет, г. Ханты-Мансийск, Российская Федерация E-mail: s_pyakov@ugrasu.ru In the present article we examine an inverse p...

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Bibliographic Details
Published in:Bulletin of the South Ural State University series "Mathematics. Mechanics. Physics"
Main Authors: Pyatkov, S.G., Verzhbitskii, M.A., Пятков, С.Г., Вержбицкий, М.А.
Format: Article in Journal/Newspaper
Language:unknown
Published: Издательский центр ЮУрГУ 2018
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Online Access:http://dspace.susu.ru/xmlui/handle/0001.74/27057
https://doi.org/10.14529/mmph180204
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Summary:S.G. Pyatkov, M.A. Verzhbitskii Yugra State University, Khanty-Mansyisk, Russian Federation E-mail: s_pyakov@ugrasu.ru. С.Г. Пятков, М.А. Вержбицкий Югорский государственный университет, г. Ханты-Мансийск, Российская Федерация E-mail: s_pyakov@ugrasu.ru 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. Рассматривается обратная задача об определении вместе с решением начально-краевой задачи для параболического уравнения второго порядка неизвестных функций, входящих в граничное условие Дирихле. Задачи такого вида об определении граничных данных возникают в самых различных задачах математической физики: управление процессами теплообмена и проектирование тепловой защиты, диагностика и идентификация теплопередачи в сверхзвуковых гетерогенных потоках, идентификация и моделирование теплопереноса в теплозащитных ...