Differential Equations and Mathematical Physics

Inverse problem for an equation of mixed parabolic-hyperbolic type with a characteristic line of change

D. K. Durdiev<sup>ab

a</sup> Bukhara Branch of the Institute of Mathematics named after V.I. Romanovskiy at the Academy of Sciences of the Republic of Uzbekistan, Bukhara, 705018, Uzbekistan
^b Bukhara State University, Bukhara, 705018, Uzbekistan

Abstract: This study investigates direct and inverse problems for a model equation of mixed parabolic-hyperbolic type. In the direct problem, an analogue of the Tricomi problem is considered for this equation with a characteristic line of type change. The unknown in the inverse problem is a variable coefficient of the lower-order term in the parabolic equation. To determine it relative to the solution defined in the parabolic part of the domain, an integral overdetermination condition is specified. Local theorems of unique solvability of the posed problems in terms of classical solutions are proven.

Keywords: parabolic-hyperbolic equation, characteristic, Green's function, inverse problem, principle of compressed mappings

UDC: 517.956.6

MSC: 35R11

Received: May 30, 2023
Revised: November 10, 2023
Accepted: December 13, 2023
First online: December 25, 2023

DOI: 10.14498/vsgtu2027

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