Polynomial solutions of the Dirichlet problem for the Tricomi equation in a strip

TL;DR

This paper proves that the Dirichlet problem for the inhomogeneous Tricomi equation in a strip with polynomial boundary data admits polynomial solutions, providing an algorithm for their construction and analyzing uniqueness depending on the region.

Contribution

It introduces an algorithm to explicitly construct polynomial solutions for the Dirichlet problem of the Tricomi equation in a strip and analyzes their uniqueness in different regions.

Findings

Polynomial solutions exist for polynomial boundary data.

An explicit algorithm for constructing solutions is provided.

Uniqueness depends on the ellipticity or mixed region of the strip.

Abstract

An inhomogeneous Tricomi equation is considered in a strip with a polynomial right-hand side. It is shown 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 region, 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.