Boundary Value Problems for the Three-Dimensional Helmholtz Equation in the Unbounded Octant, Square and Half Space

Abstract

At present, the results of the study of boundary value problems for the two-dimensional Helmholtz equation with one and two singular coefficients are known. In the presence of two positive singular coefficients in the two-dimensional Helmholtz equation, explicit solutions of the Dirichlet, Neumann and Dirichlet-Neumann problems in a quarter plane are expressed through a confluent hypergeometric function of two variables. The established properties of the confluent hypergeometric function of two variables allow us to prove the theorem of uniqueness and existence of a solution to the problems posed.In this paper, we study the Dirichlet, Neumann, and Dirichlet-Neumann problems for the three-dimensional Helmholtz equation at zero values of singular coefficients in an octant, a quarter of space, and a half-space. Uniqueness and existence theorems are proved under certain restrictions on the data. The uniqueness of solutions of which is proved using the extremum principle for elliptic equations. Using the known fundamental (singular) solution of the Helmholtz equation, solutions to the problems under study are written out in explicit forms.

Keywords

• confluent hypergeometric function of three variables
• system of partial differential equations
• asymptotic formula
• three-dimensional helmholtz equation with three singular coefficients
• dirichlet problem in the first infinite octant
• вырожденная гипергеометрическая функция трех переменных
• система уравнений в частных производных
• асимптотическая формула
• трехмерное уравнение гельмгольца с тремя сингулярными коэффициентами
• задача дирихле в первом бесконечном октанте