A new modification of the finite difference method for solving transmission problems for two-interval differential equations

In this paper, we consider a new type of boundary value problems, the main feature of which is the nature of the differential equation being solved and the imposed boundary conditions. Namely, the differential equation under consideration is given on two non-intersecting segments with a common and, on which additional interaction conditions are imposed, the so-called transmission conditions. As is known, classical analytical and numerical methods are designed to solve one-interval differential equations without transmission conditions. The main purpose of this work is to develop a new modification of the classical finite difference method for solving two-interval boundary value transmission problems.