Solutions of helmholtz equation in complex domain

Abstract

A harmonic equation and the Helmholtz equation are elliptic type equations and describe important physical processes (the first – stationary, the second - stationary and dynamic). Effective solutions of boundary value problems for harmonic equation (in different regions in the plane) are constructed by the methods of the theory of analytic functions of a complex variable. These methods can not be applied directly to solving problems for the Helmholtz equation. In the scientific literature, solutions of boundary value problems for this equation are known only in certain areas that are represented by cumbersome formulas. In the paper, using the solution of the Helmholtz equation in a circle through the functions (not analytical) of complex variable and conformal mapping of a given area on the circle, a general approach to building a solution of the corresponding boundary value problem is formulated. An important prerequisite for presenting this solution as functional series is finding the solution of harmonic equation in a given region that satisfies the given boundary conditions and an analytic function in this region respectively. The solutions of the Helmholtz equation in the plane with an elliptic hole and half-plane are constructed. For effective formulation of boundary value prob­lems and finding analytic functions in these areas, systems of basic functions in the corresponding spaces of analytic functions are found.