Interior nonlocal problem for a third-order mixed parabolic-hyperbolic type equation

Yu. P. Apakov^ab, A. A. Sopuev<sup>c

a</sup> Romanovsky Institute of Mathematics AS RUz, ul. Universitetskaya, 46, Tashkent, 100174, Uzbekistan
^b Namangan State Technical University, ul. Karimova, 12, Namangan 160103, Uzbekistan
^c Osh State University, ul. Lenina, 331, Osh 723500, Kyrgyzstan

Abstract: The existence of a unique solution to a nonlocal conjugation problem for a third-order partial differential equation of mixed parabolic-hyperbolic type is established. In the upper half-plane, the characteristic equation has a triple root, while in the lower half-plane, it has one simple root and two multiple roots. By applying the method of order reduction, Green’s and Riemann’s functions, and the method of integral equations, the problem is equivalently reduced to a nonlocal problem with an integral condition imposed on the trace of the unknown function along the type-changing line of the equation. This, in turn, is reduced to solving a Fredholm integral equation of the second kind, the solvability of which is proven using the method of successive approximations. The solution in the parabolic part of the domain is constructed using Green’s function, whereas in the hyperbolic part, the Riemann function method is employed, reducing the problem to a two-dimensional Volterra integral equation of the second kind. Examples are provided.

Keywords: differential and integral equations, third order, multiple characteristics, conjugation problem, nonlocal problem with an integral condition, uniqueness, existence, Green's function, Riemann function.

UDC: 517.956.6

Received: 30.10.2023
Revised: 23.06.2025
Accepted: 23.06.2025

DOI: 10.33048/SIBJIM.2025.28.201