Dirichlet Problem Polynomial Solutions for the Tricomi Equation in a Strip

Abstract

In the paper we consider a 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 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. There are many papers on finding conditions to have a well-posed Dirichlet problem for a mixed-type equation in a mixed domain.The paper objective is to find the 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. The paper presents an algorithm for constructing this polynomial solution and discusses examples. 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.

Keywords

• tricomi equation
• dirichlet problem
• fourier transform
• generalized functions of slow growth
• polyno-mial solutions