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.