Solution of a mixed boundary value problem for the Moisil–Teodoresku system in an infinite layer

Abstract

Aim. The purpose of the paper is to find exact solutions of a mixed boundary value problem for a system of Moisil–Teodorescu equations in an infinite layer.

Methodology. The paper considers mixed boundary value problems for the Moisil–Teodorescu system of equations in a layer and for the Cauchy–Riemann system in a strip. These problems are reduced to mixed Dirichlet–Neumann boundary value problems for the Laplace equation in a layer and in a strip, respectively, whose explicit solutions were previously obtained by the authors using the Fourier transform of generalized functions of slow growth.

Results. Exact solutions of mixed boundary value problems for the Moisil–Teodorescu system and for the Cauchy–Riemann system are obtained, which are written as convolutions of rapidly decreasing, infinitely differentiable functions (kernels) with boundary functions that are considered to be generalized functions of slow growth. If the boundary functions are ordinary functions of slow growth, then the solutions are written using integral formulas, which can be considered analogous to the Keldysh–Sedov formulas. In particular, if the boundary functions are polynomials, then the solutions are also polynomials.

Research implications. Exact solutions of mixed boundary value problems for the Moisil–Teodorescu system and for the Cauchy–Riemann system are obtained.