
Nonlocal problem for a mixed type fourthorder differential equation with Hilfer fractional operator
Tursun K. Yuldashev^{a}, Bakhtiyor J. Kadirkulov^{b} ^{a} National University of Uzbekistan named after Mirzo Ulugbek
^{b} Tashkent State Institute of Oriental Studies
Abstract:
In this paper, we consider a nonselfadjoint boundary value problem for a fourthorder differential equation of mixed type with Hilfer operator of fractional integrodifferentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain. The mixed type differential equation under consideration is a fourth order differential equation with respect to the second variable. Regarding the first variable, this equation is a fractional differential equation in the positive part of the segment, and is a secondorder differential equation with spectral parameter in the negative part of this segment. A rational method of solving a nonlocal problem with respect to the Hilfer operator is proposed. Using the spectral method of separation of variables, the solution of the problem is constructed in the form of Fourier series. Theorems on the existence and uniqueness of the problem are proved for regular values of the spectral parameter. For sufficiently large positive integers in unique determination of the integration constants in solving countable systems of differential equations, the problem of small denominators arises. Therefore, to justify the unique solvability of this problem, it is necessary to show the existence of values of the spectral parameter such that the quantity we need is separated from zero for sufficiently large n. For irregular values of the spectral parameter, an infinite number of solutions in the form of Fourier series are constructed. Illustrative examples are provided.
Keywords:
Mixed type equation, Nonselfadjoint boundary value problem, Hilfer operator, MittagLeffler function, Spectral parameter, Solvability.
DOI:
https://doi.org/10.15826/umj.2020.1.013
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Tursun K. Yuldashev, Bakhtiyor J. Kadirkulov, “Nonlocal problem for a mixed type fourthorder differential equation with Hilfer fractional operator”, Ural Math. J., 6:1 (2020), 153–167
Citation in format AMSBIB
\Bibitem{YulKad20}
\by Tursun~K.~Yuldashev, Bakhtiyor~J.~Kadirkulov
\paper Nonlocal problem for a mixed type fourthorder differential equation with Hilfer fractional operator
\jour Ural Math. J.
\yr 2020
\vol 6
\issue 1
\pages 153167
\mathnet{http://mi.mathnet.ru/umj119}
\crossref{https://doi.org/10.15826/umj.2020.1.013}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=MR4128768}
\elib{http://elibrary.ru/item.asp?id=43793632}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2s2.085088935310}
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