Abstract
We study exterior Dirichlet problems for the two-dimensional discrete Helmholtz equation with the real wave number $$k\in (0,2\sqrt{2})\backslash \{2\}$$ . The investigation is carried out without passing to the complex wave number. Similarly to the continuum theory, we use the notion of radiating solution. Then, the unique solvability result and Green’s representation formula are obtained with the help of difference potentials. Finally, we proposed a method for numerical calculation. Efficiency of our approach is demonstrated on an example related to the propagation problems in the left-handed 2D inductor–capacitor metamaterial with a hole.
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