Advanced Mathematical Models & Applications

In this paper is considered the Cauchy problem for a discrete analogue of a second- order hyperbolic equation with a periodic coefficient. The value of the solution at each point is considered as the value of some linear functionality on the initial data. The concept of Riemann function is introduced for discrete hyperbolic type equations. Using the eigenfunctions of the discrete Hill equation, the Riemann function of a discrete analogue of a second-order hyperbolic equation is constructed. For this purpose, some facts related to the spectral theory of the discrete Hill equation are proved. The properties of the Riemann function are studied. A representation of the solution to the Cauchy problem through Riemann functions is found.