Singular Localised Boundary-Domain Integral Equations of Acoustic Scattering by Inhomogeneous Anisotropic Obstacle

TL;DR

This paper develops a new boundary-domain integral equation approach for modeling acoustic wave scattering by inhomogeneous, anisotropic obstacles, handling discontinuities and establishing mathematical properties for solution existence and uniqueness.

Contribution

It introduces a localized quasi-parametrix and derives singular boundary-domain integral equations for anisotropic scattering, proving their Fredholm properties and invertibility.

Findings

Established Fredholm properties of the integral operator.

Proved invertibility and unique solvability of the integral equations.

Extended the approach to arbitrary frequency parameters.

Abstract

We consider the time-harmonic acoustic wave scattering by a bounded {\it anisotropic inhomogeneity} embedded in an unbounded {\it anisotropic} homogeneous medium. The material parameters may have discontinuities across the interface between the inhomogeneous interior and homogeneous exterior regions. The corresponding mathematical problem is formulated as a transmission problems for a second order elliptic partial differential equation of Helmholtz type with discontinuous variable coefficients. Using a localised quasi-parametrix based on the harmonic fundamental solution, the transmission problem for arbitrary values of the frequency parameter is reduced equivalently to a system of {\it singular localised boundary-domain integral equations}. Fredholm properties of the corresponding {\it localised boundary-domain integral operator} are studied and its invertibility is established in appropriate Sobolev-Slobodetskii and Bessel potential spaces, which implies existence and uniqueness results for the localised boundary-domain integral equations system and the corresponding acoustic scattering transmission problem.