2024/05/29 by Kapanadze, David, Vashakidze, Zurab
#35J05 #39A14 #39A60 #74S20 #78A45 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2405.18806
This paper examines the propagation of time-harmonic waves in a two-dimensional triangular lattice with a lattice constant a = 1. The sources are positioned along line segments within the lattice. Specifically, we investigate the discrete Helmholtz equation with a wavenumber k ∈ ( 0,2√(2) ), where input data is prescribed on finite rows or columns of lattice sites. We focus on two main questions: the efficacy of the numerical methods employed in evaluating the Green's function, and the necessity of the cone condition. Consistent with a continuum theory, we employ the notion of radiating solution and establish a unique solvability result and Green's representation formula using difference potentials. Finally, we propose a numerical computation method and demonstrate its efficiency through examples related to the propagation problems in the left-handed two-dimensional inductor-capacitor metamaterial.