  In this paper we consider a Robin problem for the Klein-Gordon equation in a
doubly connected domain. The solution domain considered is a bounded smooth
doubly connected planar domain bounded by two simple disjoint closed curves.
The analysis of the problem is based on the indirect integral equations method.
The solution is represented as a sum of two single-layer potentials defined on
each of the two boundary curves with unknown densities. To find out the
densities the representation is matched with the given Robin data to generate a
system of linear integral equations of the second kind with continuous and
weakly-singular kernels. It is shown that the operator corresponding to this
system is injective and due to its compactness according to Riesz theory there
exists a unique solution. To discretize the system we apply Nystrom method with
a specifically chosen quadrature rules to obtain an exponential order of
convergence of the approximate solution. Numerical experiments are conducted
for three testing examples that back up the theoretical reasoning.
