On the solvability of a boundary value problem for a quasilinear equation of mixed type with two degeneration lines

The article investigates the existence of a generalized solution to one boundary value problem for an equation of mixed type with two lines of degeneration in the weighted space of S.L. Sobolev. In proving the existence of a generalized solution, the spaces of functions U(Ω) and V (Ω) are introduced, the spaces H1(Ω) and H 1 * (Ω) are defined as the completion of these spaces of functions, respectively, with respect to the weighted norms, including the functions K(y) and N(x). Using an auxiliary boundary value problem for a first order partial differential equation, Kondrashov’s theorem on the compactness of the embedding of W 2 1 (Ω) in L2(Ω) and Vishik’s lemma, the existence of a solution to the boundary value problem is proved.

A BOUNDARY VALUE PROBLEM WITH AN OFFSET FOR A MODEL EQUATION OF MIXED TYPE IN AN UNBOUNDED DOMAIN

В данной работе для уравнения смешанного типа в неограниченной области эллиптическая часть которой является горизонтальной полосой исследуется задача со смещением на характеристиках разных семейств. Единственность решения задачи доказывается методом интегралов энергии, а существование решения задачи методом функций Грина и методом интегральных уравнений. In this paper, for a mixed type equation in an unbounded region, the elliptical part of which is a horizontal strip, we study the problem with a shift on the characteristics of different families. The uniqueness of the solution of the problem is proved by the method of energy integrals, and the existence of a solution of the problem by the method of Green functions and the method of integral equations.

Keldysh problem for a three-dimensional equation of mixed type with three singular coefficients in a semi-infinite parallelepiped

This article studies the Keldysh problem for a three-dimensional equation of mixed type with three singular coefficients in a semi-infinite parallelepiped. Based on the completeness property of eigenfunction systems of two one-dimensional spectral problems, the uniqueness theorem is proved. To prove the existence of a solution to the problem, the Fourier spectral method based on the separation of variables is used. The solution to this problem is constructed in the form of a sum of a double Fourier-Bessel series. In substantiating the uniform convergence of the constructed series, we used asymptotic estimates of the Bessel functions of the real and imaginary argument. Based on them, estimates were obtained for each member of the series, which made it possible to prove the convergence of the series and its derivatives to the second order inclusive, as well as the existence theorem in the class of regular solutions.

Polynomial Solutions of the Dirichlet Problem for the Tricomi Equation in a Strip

In the paper we consider the Tricomi equation of mixed type. This equation is elliptic in the upper half-plane, hyperbolic in the lower half-plane and parabolically degenerate on the boundary of half-planes. Equations of a mixed type are used in transonic gas dynamics. The Dirichlet problem for an equation of mixed type in a mixed domain is, in general, ill- posed. Many papers has been devoted to the search for conditions for the well-posednes of the Dirichlet problem for a mixed-type equation in a mixed domain.This paper is devoted to finding exact polynomial solutions of the inhomogeneous Tricomi equation in a strip with a polynomial right-hand side. The Fourier transform method shows that the Dirichlet boundary value problem with polynomial boundary conditions has a polynomial solution. An algorithm for constructing this polynomial solution is given and examples are considered. If the strip lies in the ellipticity region of the equation, then this solution is unique in the class of functions of polynomial growth. If the strip lies in a mixed domain, then the solution of the Dirichlet problem is not unique in the class of functions of polynomial growth, but it is unique in the class of polynomials.