BOUNDARY VALUE PROBLEMS FOR THE POISSON EQUATION IN A HALF-SPACE WITH POLYNOMIAL DATA AND THE CAUCHY PROBLEM

Abstract

Aim. The purpose is to find exact solutions of boundary value problems and the Cauchy problem for the Poisson equation in a half-space with polynomial data. Methodology. The paper considers the Dirichlet and Neumann boundary value problems in a half-space and the Cauchy problem with polynomial data for the Poisson equation. These problems are solved using the Fourier transform of generalized functions of slow growth. Results. It is shown that the Cauchy problem with polynomial data for the Poisson equation has a solution that is a polynomial. This solution is the only one in the class of functions of slow growth in hyperplanes parallel to the hyperplane on which the initial conditions are specified. The polynomial solution is obtained explicitly. Each solution from an infinite set of solutions to the Dirichlet or Neumann problem is a solution to some Cauchy problem. Research implications. We have obtained exact solutions to boundary value problems and the Cauchy problem with polynomial data for the Poisson equation.