ISSN 1995-0802, Lobachevskii Journal of Mathematics, 2023, Vol. 44, No. 10, pp. 4304–4314. c  Pleiades Publishing, Ltd., 2023. Boundary Value Problems for a Parabolic-Hyperbolic Equation with a Superposition of Operators of the First and Second Orders B. I. Islomov 1* , T. K. Yuldashev 2** , and G. K. Kylyshbayeva 1*** (Submitted by A. M. Elizarov) 1 National University of Uzbekistan, Tashkent, 100174, Uzbekistan 2 Tashkent State University of Economics, Tashkent, 100066, Uzbekistan Received May 16, 2023; revised July 24, 2023; accepted July 28, 2023 Abstract—The paper proposes a method for solving the problem for a parabolic-hyperbolic equation of the third order in a rectangular domain, when the main part of equation contains a first-order operator. A criterion for the uniqueness of the solution is established. When justifying the uniform convergence of the Fourier series, the problem of small denominators arises. In this regard, estimates of small denominators about the distance from zero with the corresponding asymptotics are established. These estimates made it possible to prove the convergence of the series in the class of regular solutions of this equation. Estimates on the stability of the solution from given boundary functions are proved. DOI: 10.1134/S1995080223100189 Keywords and phrases: Third-order equation, boundary value problem, small denominators, uniqueness and existence of solution, stability of solution. 1. INTRODUCTION Boundary-value problems for equations of parabolic-hyperbolic and elliptic-hyperbolic types of the third order, when the main part of the equation contains an operator with respect to x or y, were first studied by analytical methods in the works of A.V. Bitsadze and M.S. Salakhitdinov [1], M.S. Salakhitdinov [2], T.D. Djuraev [3], T.D. Djuraev, A. Sopuev and M. Mamazhonov [4], where the domain was a mixed region, consisting of characteristic triangles and a rectangle (or a semicircle). In these papers, the solution was found in the class of functions represented as u(x, y)= υ(x, y)+ ω(x) or u(x, y)= υ(x, y)+ ω(y), where υ(x, y) is an arbitrary regular solution of a second-order equation Lυ =0,ω(x),ω(y) are arbitrary functions. Such kind of representation is important for equations composed of a product of commutative differential operators. But, for a mixed-type equation with a generalized operator containing minor terms, this method is not always correct [5]. Further, this direction for various third-order partial differential equations was developed in [2, 3, 5–11] and others. Various inverse problems for certain types of partial differential equations have been studied in many works. We note here, first of all, the works of A.N. Tikhonov [12], M.M. Lavrent’ev [13], S.I. Kabanikhin [14], V.K. Ivanov [15], A.M. Denisov [16] and others. It is interesting to study by the method of spectral analysis [17] the unique solvability and stability of the solution of direct and inverse problems for partial differential equations, in particular, for a model equation of the second and third orders of mixed parabolic-hyperbolic and elliptic-hyperbolic types with integer and fractional orders in a rectangular domain. Here we can note works [18–30]. In this paper, we propose a method for solving the problem for a third-order parabolic-hyperbolic equation in a rectangular domain, when the main part contains a general first-order operator. * E-mail: islomovbozor@yandex.com ** E-mail: tursun.k.yuldashev@gmail.com *** E-mail: kalbaevna85@mail.ru 4304