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

Exact Solutions to the Boundary-Value Problems for the Helmholtz Equation in a Layer With Polynomials in the Right-Hand Sides of the Equation and of the Boundary Conditions

“…If the parameter of the Helmholtz equation tends to zero, then the Helmholtz equations turn into the Poisson equation, and the quasipolynomial solutions of the boundary value problems turn into polynomial solutions of the boundary value problems for the Poisson equation [2]. In the same way, exact polynomial solutions of boundary value problems for the Tricomi equation in a strip are obtained [3], [4]. The search for solutions of partial differential equations in the form of polynomials or quasipolynomials has been the subject of work by many authors [5] - [9].…”

Boundary Value Problems for the Poisson Equation in a Half-Space With Polynomial Data and the Cauchy Problem

“…Поиску решений уравнений с частными производными в виде полиномов или квазиполиномов посвящены работы многих авторов [8][9][10][11][12], см . также [13][14][15] .…”

Modal approach in structural-dynamic equations

“…The method of expansion in eigenfunctions of a linear homogeneous operator with constant coefficients has no such restrictions, which form the system of orthogonal functions [4], satisfying the conditions of theorem Steklov [4,5,6], i.e. it is possible to build an absolutely convergent generalized Fourier series [7].…”