The Neumann problem for the 2‐D Helmholtz equation in a domain, bounded by closed and open curves

P. A. Krutitskii

doi:10.1155/s0161171298000301

Abstract

The Neumann problem for the dissipative Helmholtz equation in a connected plane region bounded by closed and open curves is studied. The existence of classical solution is proved by potential theory. The problem is reduced to the Fredholm equation of the second kind, which is uniquely solvable. Our approach holds for both internal and external domains.

Highlights

The boundary value problems in domains bounded by closed and open curves were not treated in the theory of 2-D PDEs before
Even in the case of Laplace and Helmholtz equations the problems in domains bounded by closed curves [1-2], [5-8] and problems in the exterior of open arcs [5], [9-11] were treated separately, because different methods were used in their analysis
The Neumann problem in the exterior of an open arc was reduced to the hypersingular integral equation [9-10] or to the infinite algebraic system of equations [11], while the Neumann problem in domains bounded by closed curves was reduced to the Fredholm equation of the second kind [1], [6-8]

Summary

INTRODUCTION

The boundary value problems in domains bounded by closed and open curves were not treated in the theory of 2-D PDEs before. Even in the case of Laplace and Helmholtz equations the problems in domains bounded by closed curves [1-2], [5-8] and problems in the exterior of open arcs [5], [9-11] were treated separately, because different methods were used in their analysis. The Neumann problem in the exterior of an open arc was reduced to the hypersingular integral equation [9-10] or to the infinite algebraic system of equations [11], while the Neumann problem in domains bounded by closed curves was reduced to the Fredholm equation of the second kind [1], [6-8]. The approach suggested in the present paper enables to reduce the Neumann problem in domains bounded by closed and open curves to the Fredholm integral equation on the whole boundary with the help of the nonclassical angular potential. Since the boundary integral equation is Fredholm, the solvability theorem follows from the uniqueness theorem, which is ensured for the Neumann problem in the case of the dissipative Helmholtz equation This approach is based on [3-4], where the problems in the exterior of open curves were reduced to the Fredholm integral equations using the angular potential

FORMULATION OF THE PROBLEM

INTEGRAL EQUATIONS AT THE BOUNDARY

THE FREDHOLM INTEGRAL EQUATION AND THE SOLUTION OF THE PROBLEM