On approximate open boundary conditions for the wave equation and the Klein-Gordon equation

When implementing numerical simulations of non-stationary processes in spatially unbounded domains, in a number of cases the original problem can be reduced to a problem in a fixed subdomain by imposing certain conditions on the so-called open boundary that separates the subdomain from the rest of the whole domain. Due to this, the demands for computational system resources the simulation requires decrease considerably. One of the promising methods for generating the conditions of this type is based on approximation of integral operator kernels in the exact equalities that associate the values of the original problem solution and its partial derivatives on the open boundary. Some problems related to the substantiation of such approximate conditions as well as to the optimal parameter choice for their realization need to be studied via analytical methods. The above questions are discussed in this paper for a model problem.