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Inverse Problem for Determining the Order of the Fractional Derivative in Mixed-Type Equations

Authors
  • Ashurov, R. R.1
  • Zunnunov, R. T.1
  • 1 V. I. Romanovskii Institute of Mathematics of the Academy of Sciences of Uzbekistan, Tashkent, Uzbekistan , Tashkent (Uzbekistan)
Type
Published Article
Journal
Lobachevskii Journal of Mathematics
Publisher
Pleiades Publishing
Publication Date
Dec 13, 2021
Volume
42
Issue
12
Pages
2714–2729
Identifiers
DOI: 10.1134/S1995080221120052
Source
Springer Nature
Keywords
Disciplines
  • Article
License
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Abstract

AbstractIn this paper the inverse problem of determining the fractional orders in mixed-type equations is considered. In one part of the domain, the considered equation is the subdiffusion equation with a fractional derivative in the sense of Gerasimov–Caputo of the order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha\in(0,1)$$\end{document} and in the other part—a wave equation with a fractional derivative of the order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta\in(1,2)$$\end{document}. The elliptic part of the equation is a second-order operator, considered in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N$$\end{document}-dimensional domain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega$$\end{document}. Assuming the parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document} to be unknown, additional conditions are found that provide an unambiguous determination of the required parameters.

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