2023/02/08 by Charles L. Epstein, Epstein, Charles L. · 3 citations
Mathematics · Engineering · #Differential Equations and Numerical Methods #Physics and Engineering Research Articles #Numerical methods for differential equations
paper · pdf · doi:10.2140/paa.2026.8.535
We introduce a layer potential representation for the solution of the transmission problem defined by two dielectric channels, or open wave-guides, meeting along the straight-line interface, \x1=0\. The main observation is that the outgoing fundamental solution for the operator Δ+k12+q(x2), acting on functions defined in \mathbb R2, is easily constructed using the Fourier transform in the x1-variable and the elementary theory of ordinary differential equations. These fundamental solutions can then be used to represent the solution to the transmission problem in half planes. The transmission boundary conditions lead to integral equations along the intersection of the half planes, which, in our normalization, is the x2-axis. We show that, in appropriate Banach spaces, these integral equations are Fredholm equations of second kind, which are therefore generically solvable. We analyze the representation of the guided modes in our formulation.