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Inverse problem with final overdetermination for time-fractional differential equation in a Banach space

  • Dmitry Orlovsky EMAIL logo and Sergey Piskarev

Abstract

We consider in a Banach space E the inverse problem

( 𝐃 t Ξ± ⁒ u ) ⁒ ( t ) = A ⁒ u ⁒ ( t ) + β„± ⁒ ( t ) ⁒ f , t ∈ [ 0 , T ] , u ⁒ ( 0 ) = u 0 , u ⁒ ( T ) = u T ,  0 < Ξ± < 1

with operator A, which generates the analytic and compact Ξ±-times resolvent family { S Ξ± ⁒ ( t , A ) } t β‰₯ 0 , the function β„± ⁒ ( β‹… ) ∈ C 1 ⁒ [ 0 , T ] and u 0 , u T ∈ D ⁒ ( A ) are given and f ∈ E is an unknown element. Under natural conditions we have proved the Fredholm solvability of this problem. In the special case for a self-adjoint operator A, the existence and uniqueness theorems for the solution of the inverse problem are proved. The semidiscrete approximation theorem for this inverse problem is obtained.

MSC 2010: 34M50; 35R30; 65M32

Award Identifier / Grant number: N20-11-20085

Award Identifier / Grant number: 02.a03.21.0005

Funding statement: The support of Russian Science Foundation (RSF) N20-11-20085 (for substantiating Section 4) and the program of competitiveness increase of the National Research Nuclear University MEPhI contract 02.a03.21.0005 are gratefully acknowledged.

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Received: 2020-08-04
Revised: 2020-10-13
Accepted: 2020-10-14
Published Online: 2020-11-13
Published in Print: 2022-04-01

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