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 The Bulletin of Irkutsk State University. Series Mathematics: Year: Volume: Issue: Page: Find

 The Bulletin of Irkutsk State University. Series Mathematics, 2019, Volume 28, Pages 123–137 (Mi iigum377)

Inverse problem for evolutionary equation with the Gerasimov–Caputo fractional derivative in the sectorial case

V. E. Fedorov, A. V. Nagumanova

Chelyabinsk State University, Chelyabinsk, Russian Federation

Abstract: We investigates the unique solvability of a class of linear inverse problems with a time-independent unknown coefficient in an evolution equation in Banach space, which is resolved with respect to the fractional Gerasimov–Caputo derivative. We assume that the operator in the right-hand side of the equation generates a family of resolving operators of the corresponding homogeneous equation, which is exponentially bounded and analytic in a sector containing the positive semiaxis. It is shown that for the well-posedness of the inverse problem it is necessary to choose as the space of initial data the definition domain of the generating operator endowed with its graph norm. Sufficient conditions for the unique solvability of the inverse problem are found. The obtained abstract results are applied to the unique solvability study of an inverse problem for a class of time-fractional order partial differential equations. The considered example, in particular, shows that when choosing as the source data space not the domain of definition of the generating operator, but the whole space, the inverse problem is ill-posed.

Keywords: inverse problem, fractional Gerasimov–Caputo derivative, evolution equation, resolving family of operators.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 1.6462.2017/Á× Russian Foundation for Basic Research 19-41-450001_ð_à

DOI: https://doi.org/10.26516/1997-7670.2019.28.123

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Bibliographic databases:

UDC: 517.95, 517.98
MSC: 35R30, 35R11, 34G10

Citation: V. E. Fedorov, A. V. Nagumanova, “Inverse problem for evolutionary equation with the Gerasimov–Caputo fractional derivative in the sectorial case”, The Bulletin of Irkutsk State University. Series Mathematics, 28 (2019), 123–137

Citation in format AMSBIB
\Bibitem{FedNag19} \by V.~E.~Fedorov, A.~V.~Nagumanova \paper Inverse problem for evolutionary equation with the Gerasimov--Caputo fractional derivative in the sectorial case \jour The Bulletin of Irkutsk State University. Series Mathematics \yr 2019 \vol 28 \pages 123--137 \mathnet{http://mi.mathnet.ru/iigum377} \crossref{https://doi.org/10.26516/1997-7670.2019.28.123}