The Helmholtz equation in the exterior of slits in a plane with different impedance boundary conditions on opposite sides of the slits

Abstract

The boundary value problem for the Helmholtz equation is studied outside slits in a plane. The impedance boundary conditions are specified on the slits. In general, the impedance conditions may be different at different sides of each slit. In a particular case, the impedance conditions may be the same on the sides of each slit. It is proved that the classical solution to the problem exists, and it is unique. The integral representation for a solution to the problem is obtained in the form of potentials, the densities in which are uniquely determined from the uniquely solvable system of the Fredholm integral equations of the second kind and index zero.

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“…In [15], the solution to (2.2) in K is constructed in the form of sum of an angular potential and single layer potential:…”

“…For these density functions, (2.7) gives a function in K solving (2.2), except the boundary condition on Γ ± . These conditions are used in [15] to determine µ 1 , µ 2 . To this end, let us introduce some known kernel functions from [15].…”

“…If we specify two sides of Γ by Γ + and Γ − , then the boundary impedance coefficient as well as the boundary data in Γ + and Γ − may be different. The numerical method developed in the present paper is based on the boundary integral equation approach proposed in [15], where the integral representation for a solution of the problem is obtained in the form of a sum of a single layer potential and an angular potential. In the case of an arc with the same impedance condition on both sides, the computational scheme has been given in [11,13], where the analytic property of the arc and the hyper-singular integral are required.…”

“…Although the integral equation for the density functions has been derived in [15] and its solvability is proven there, the efficient numerical realization of this scheme is still open. More precisely, the integral system consists of a Cauchy singular integral equation of the 1st kind with additional integral condition and an integral equation of the second kind with smooth kernels.…”

“…Since the Cauchy singular operator has additional weak singularities at the ends, then we need to be more careful about the method of computation. On the other hand, the arc Γ is represented in [15][16][17] using the arc length for the parametrization, which is the restriction on this scheme. The novelty of our paper is the new numerical method for computing the integrals with high accuracy and therefore solving this very complicated system of integral equations efficiently.…”