Classical solution of the mixed problem for the wave equation on a graph with two edges and a cycle

In this paper, using the Fourier method, we obtain a classical solution of the mixed problem for the wave equation on the simplest geometric graph consisting of two edges, one of which forms a cycle. We apply an approach based on the method of contour integration of the resolvent of an operator, which allows one to obtain a classical solution to the problem under minimal conditions on the initial data and, at the same time, to avoid a laborious study of the refined asymptotics of the eigenvalues and eigenfunctions of the corresponding operator. The cases of continuous and summable potentials are considered.