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## Archive of Issues

Uzbekistan Ferghana
Year
2020
Volume
30
Issue
2
Pages
270-289
 Section Mathematics Title Unique solvability of a nonlocal problem with shift for a parabolic-hyperbolic equation Author(-s) Urinov A.K.a, Mamanazarov A.O.a Affiliations Ferghana State Universitya Abstract In the paper, a parabolic-hyperbolic equation with a singular coefficient and a spectral parameter in the domain which consists of a characteristic triangle and a half strip has been considered. A nonlocal problem connecting the values of the desired function at the two points of boundary characteristics and the line of equation type changing by means of two operators, the first of which depends on the coefficient of the singularity and the second one - on the spectral parameters, is formulated. The considered problem is investigated by reducing it to the system of equations in the trace of the desired function and its derivative with respect to $x$ on the line of equation type changing. The uniqueness of the solution is proved by the method of energy integrals, for this we use integral representations of Euler gamma-function and Bessel function of the first kind. The existence of the solution is proved by the method of integral equations, for this we equivalently reduce the considered problem to the Fredholm integral equation of the second kind which solvability follows from the uniqueness of the problem solution. Sufficient conditions for unique solvability of the considered problem are found. Keywords parabolic-hyperbolic equation, singular coefficient, spectral parameter, noncharacteristic line of type changing, nonlocal problem, unique solvability UDC 517.956.6 MSC 35M10, 35M12 DOI 10.35634/vm200210 Received 4 February 2020 Language Russian Citation Urinov A.K., Mamanazarov A.O. Unique solvability of a nonlocal problem with shift for a parabolic-hyperbolic equation, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. 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