Boundary value problem for a differential-difference equation with a fractional derivative

The work is devoted to the study of a differential-difference equation with a fractional derivative of order not exceeding one. For the equation under consideration, a boundary value problem is posed and solved on a manifold that is a countable union of intervals. To solve the problem, we used an analogue of the Green function method, adapted for differential-difference equations. A general representation of the solution to the equation under study has been found, a fundamental solution has been constructed in terms of the Prabhakar function, its properties have been studied, and a theorem on the existence and uniqueness of a solution to the problem under study has been proven.