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Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia


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Volume 22, Issue 2

Issues

Inverse problems for a class of degenerate evolution equations with Riemann – Liouville derivative

Vladimir E. Fedorov
  • Mathematical Analysis Dept., Chelyabinsk State University Kashirin Brothers St., 129, Chelyabinsk, – 454001, Russia
  • Laboratory of Functional Materials, South Ural State University Lenin Av., 76, Chelyabinsk, – 454080, Russia
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Roman R. Nazhimov
  • Mathematical Analysis Dept., Chelyabinsk State University Kashirin Brothers St., 129, Chelyabinsk, – 454001, Russia
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  • Other articles by this author:
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Published Online: 2019-05-11 | DOI: https://doi.org/10.1515/fca-2019-0018

Abstract

Unique solvability and well-posedness issues are studied for linear inverse problems with a constant unknown parameter to fractional order differential equations with Riemann – Liouvlle derivative in Banach spaces. Firstly, well-posedness criteria for the inverse problem with the Cauchy type initial conditions to the differential equation in a Banach space that solved with respect to the fractional derivative is obtained. This result is applied to search of sufficient conditions for the unique solution existence of the inverse problem for equation with linear degenerate operator at the Riemann – Liouville fractional derivative. It is shown that the presence of the matching conditions for the data of the problem excludes the possibility of the well-posedness consideration for the degenerate inverse problem with the Cauchy type condition. But for the inverse problem with the Showalter – Sidorov type conditions it is found the criteria of the well-posedness. Abstract results are used to the search of conditions of the unique solvability for an inverse problem to a class of partial differential equations of time-fractional order with polynomials of elliptic differential operators with respect to the spatial variables.

MSC 2010: Primary 35R30; Secondary 34G10; 35R11; 34A08

Key Words and Phrases: fractional Riemann – Liouville derivative; inverse problem; degenerate evolution equation; (L, p)-bounded operator

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About the article

Received: 2018-03-26

Published Online: 2019-05-11

Published in Print: 2019-04-24


Citation Information: Fractional Calculus and Applied Analysis, Volume 22, Issue 2, Pages 271–286, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.1515/fca-2019-0018.

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