Exterior Dirichlet problems for two-dimensional (2D) lattice waves on the infinite triangular lattice are considered. Namely, we study Dirichlet problems for the 2D discrete Helmholtz equation in a plane with a hole. New results are obtained for the existence and uniqueness of the solution in the case of the real wave number without passing to a complex wave number. Besides, Green’s representation formula for the solution is derived with the help of difference potentials. To demonstrate the results, we propose a method for numerical calculation.
BrillouinL.Wave propagation in periodic structures. Electric filters and crystal lattices (international series in pure and applied physics). New York: McGraw-Hill, 1946.
2.
CalozCItohT.Electromagnetic metamaterials: transmission line theory and microwave applications: the engineering approach. Hoboken, NJ: John Wiley & Sons Inc., 2006.
3.
DoveMT.Structure and dynamics: an atomic view of materials. Oxford: Oxford University Press, 2002.
4.
SlepyanLI.Models and phenomena in fracture mechanics. New York: Springer, 2002.
5.
ColtonDKressR.Inverse acoustic and electromagnetic scattering theory (applied mathematical sciences). New York: Springer, 2019.
6.
BornMHuangK.Dynamical theory of crystal lattices. Oxford: Clarendon Press, 1954.
7.
BurkeJG.Origins of the science of crystals. Los Angeles, CA: University of California Press, 1966.
8.
KapanadzeD.Exterior diffraction problems for two-dimensional square lattice. Z Angew Math Phys2018; 69: 123.
9.
BurtonAMillerG.Application of integral equation methods to numerical solution of some exterior boundary value problems. Proc R Soc A1971; 323: 201–210.
10.
MartinssonPGRodinGJ.Boundary algebraic equations for lattice problems. Proc R Soc A2009; 465: 2489–2503.
11.
Poblet-PuigJValyaevVYuShaninAV.Boundary element method based on preliminary discretization. Math Models Comput Simul2014; 6: 172–182.
12.
Poblet-PuigJValyaevVYuShaninAV.Suppression of spurious frequencies in scattering problems by means of boundary algebraic and combined field equations. J Integral Equations Appl2015; 27: 233–272.
13.
AndoKIsozakiHMoriokaH.Spectral properties of Schrödinger operators on perturbed lattices. Ann Henri Poincaré2016; 17: 2103–2171.
14.
KapanadzeD.The far-field behaviour of Green’s function for a triangular lattice and radiation conditions. Math Meth Appl Sci2021; 44 : 12746–12759.
15.
SharmaBL.Edge diffraction on triangular and hexagonal lattices: existence, uniqueness, and finite section. Wave Motion2016; 65: 55–78.
16.
BerciuMCookAM.Efficient computation of lattice Green’s functions for models with nearest-neighbor hopping. Europhys Lett2010; 92: 40003.
17.
HoriguchiT. Lattice Green’s functions for the triangular and honeycomb Lattices. J Math Phys1972; 13: 1411.
18.
IsozakiHMoriokaH.A Rellich type theorem for discrete Schrödinger operators. Inverse Probl Imaging2014; 8: 475–489.
