The 2D Dirichlet Problem for the Propagative Helmholtz Equation in an Exterior Domain with Cracks and Singularities at the Edges

P A Krutitskii

doi:10.1155/2012/340310

Abstract

The Dirichlet problem for the 2D Helmholtz equation in an exterior domain with cracks is studied. The compatibility conditions at the tips of the cracks are assumed. The existence of a unique classical solution is proved by potential theory. The integral representation for a solution in the form of potentials is obtained. The problem is reduced to the Fredholm equation of the second kind and of index zero, which is uniquely solvable. The asymptotic formulae describing singularities of a solution gradient at the edges (endpoints) of the cracks are presented. The weak solution to the problem may not exist, since the problem is studied under such conditions that do not ensure existence of a weak solution.

Highlights

The 2D Dirichlet boundary value problem for the Helmholtz equation in an exterior multiply connected domain bounded by closed curves is considered in monographs on mathematical physics, for instance, in 1–3
We study Dirichlet problem in an exterior domain bounded by closed curves and cracks
The integral representation for a solution presented in the present paper enables us to derive asymptotic formulae for singularities of a gradient of the solution at the tips of the cracks

Summary

Introduction

The 2D Dirichlet boundary value problem for the Helmholtz equation in an exterior multiply connected domain bounded by closed curves is considered in monographs on mathematical physics, for instance, in 1–3. We study Dirichlet problem in an exterior domain bounded by closed curves and cracks. Since the derived integral equation of the 2nd kind is uniquely solvable, we may obtain its numerical solution in a very simple way, just by discretization and inversion of the matrix. Substituting numerical solution of the integral equation into potentials, we obtain numerical solution to the exterior Dirichlet problem in a very simple way as well. It is important to stress that the boundary data on the closed curves in the present paper is assumed to be only continuous This means that weak solution may not exist in our problem, though classical solution exists. The Dirichlet problem for elasticity equations in an exterior of several arbitrary curvilinear cracks in a plane has been reduced to the uniquely solvable integral equations in 14

Formulation of the Problem

Integral Equations at the Boundary

Γ1 i 4 μ σ H01

The Fredholm Integral Equation and the Solution to the Problem

Singularities of the Gradient of the Solution at the Endpoints of the Cracks