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Time-dependent identification problem for a fractional Telegraph equation with the Caputo derivative

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Abstract

This study investigates the inverse problem of determining the right-hand side of a telegraph equation given in a Hilbert space. The main equation under consideration has the form \((D_{t}^{\rho })^{2}u(t)+2\alpha D_{t}^{\rho }u(t)+Au(t)=p( t)q+f(t)\), where \(0<t\le T\), \(0<\rho <1\) and \(D_{t}^{\rho }\) is the Caputo derivative. The equation contains a self-adjoint positive operator A and a time-varying multiplier p(t) in the source function, which, like the solution of the equation, is unknown. To solve the inverse problem, an additional condition \(B[u(t)] = \psi (t)\) is imposed, where B is an arbitrary bounded linear functional. The existence and uniqueness of a solution to the problem are established and stability inequalities are derived. It should be noted that, as far as we know, such an inverse problem for the telegraph equation is considered for the first time. Examples of the operator A and the functional B are discussed.

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Acknowledgements

The authors are grateful to A. O. Ashyralyev for posing the problem and they convey thanks to Sh. A. Alimov for discussions of these results. We acknowledge financial support from the Ministry of Innovative Development of the Republic of Uzbekistan, Grant No F-FA-2021-424.

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Authors and Affiliations

  1. V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Science, University str., 9, Olmazor district, 100174, Tashkent, Uzbekistan

    Ravshan Ashurov & Rajapboy Saparbayev

  2. Department of Mathematics, New Uzbekistan University, Movarounnahr Street 1, 100000, Tashkent, Uzbekistan

    Ravshan Ashurov

  3. Tashkent State University of Economics (TSUE), 49, Karimov street, 100066, Tashkent, Uzbekistan

    Rajapboy Saparbayev

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  1. Ravshan Ashurov
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Correspondence to Ravshan Ashurov.

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Ashurov, R., Saparbayev, R. Time-dependent identification problem for a fractional Telegraph equation with the Caputo derivative. Fract Calc Appl Anal 27, 652–676 (2024). https://doi.org/10.1007/s13540-024-00240-0

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