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 Zh. Vychisl. Mat. Mat. Fiz., 2015, Volume 55, Number 1, Pages 89–104 (Mi zvmmf10137)

Recovery of the coefficient of $u_t$ in the heat equation from a condition of nonlocal observation in time

A. B. Kostin

National Research Nuclear University “MEPhI”, Kashirskoe sh. 31, Moscow, 115409, Russia

Abstract: The inverse problem of finding the coefficient $\rho(x)=\rho_0+r(x)$ multiplying $u_t$ in the heat equation is studied. The unknown function $r(x)\geqslant0$ is sought in the class of bounded functions, and $\rho_0$ is a given positive constant. In addition to the initial and boundary conditions (data of the direct problem), a nonlocal observation condition is specified in the form $\int\limits_0^T u(x,t)d\mu(t)=\chi(x)$ with a given measure $d\mu(t)$ and a function $\chi(x)$. The case of integral observation (i.e., $d\mu(t)=\omega(t)dt$) is considered separately. Sufficient conditions for the existence and uniqueness of a solution to the inverse problem are obtained in the form of easy-to-check inequalities. Examples of inverse problems are given for which the assumptions of the theorems proved in this work are satisfied.

Key words: coefficient inverse problems, inverse problem for the heat equation, nonlocal observation (or overdetermination) condition, sufficient conditions for the existence and uniqueness of a solution.

DOI: https://doi.org/10.7868/S0044466915010123

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English version:
Computational Mathematics and Mathematical Physics, 2015, 55:1, 85–100

Bibliographic databases:

UDC: 519.633.9
Revised: 14.07.2014

Citation: A. B. Kostin, “Recovery of the coefficient of $u_t$ in the heat equation from a condition of nonlocal observation in time”, Zh. Vychisl. Mat. Mat. Fiz., 55:1 (2015), 89–104; Comput. Math. Math. Phys., 55:1 (2015), 85–100

Citation in format AMSBIB
\Bibitem{Kos15} \by A.~B.~Kostin \paper Recovery of the coefficient of $u_t$ in the heat equation from a condition of~nonlocal observation in time \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2015 \vol 55 \issue 1 \pages 89--104 \mathnet{http://mi.mathnet.ru/zvmmf10137} \crossref{https://doi.org/10.7868/S0044466915010123} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3304926} \elib{http://elibrary.ru/item.asp?id=22908449} \transl \jour Comput. Math. Math. Phys. \yr 2015 \vol 55 \issue 1 \pages 85--100 \crossref{https://doi.org/10.1134/S0965542515010121} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000348997900008} \elib{http://elibrary.ru/item.asp?id=23970392} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84922021762} 

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This publication is cited in the following articles:
1. Kh. M. Gamzaev, “Chislennyi metod resheniya koeffitsientnoi obratnoi zadachi dlya uravneniya diffuzii–konvektsii–reaktsii”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2017, no. 50, 67–78
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