Injectivity of boundary integral operator in direct-indirect mixed Burton-Miller equation for wave scattering problems with transmissive circular inclusion

TL;DR

This paper establishes that the injectivity condition for the boundary integral operator in the mixed Burton-Miller equation with a transmissive circular inclusion is equivalent to that of the standard Burton-Miller equation, clarifying its well-posedness.

Contribution

It proves the injectivity condition for the mixed Burton-Miller boundary integral operator in the presence of a transmissive circular inclusion, linking it to the standard case.

Findings

Injectivity condition matches that of the standard Burton-Miller equation.

Clarifies the well-posedness of the mixed Burton-Miller boundary integral equation.

Supports faster numerical methods for wave scattering with transmissive inclusions.

Abstract

This study proves that the injectivity condition for the integral operator of the direct-indirect mixed Burton-Miller (BM) boundary integral equation (BIE) for Helmholtz transmission problems is identical to that for the ordinary BM BIE for Helmholtz transmission problems with a transmissive circular inclusion. Although some numerical methods based on the direct-indirect mixed BM BIE can be computed faster than the ordinary BM BIE, its well-posedness has been unclear. This study resolves a part of the well-posedness, namely the injectivity of the integral operator with a transmissive circular inclusion.