Generalized boundary value problem for a linear ordinary differential equation with a discretely distributed fractional differentiation operator

Abstract

This paper formulates and solves a generalized boundary value problem for a linear ordinary differential equation with a discretely distributed fractional differentiation operator. The fractional derivative is understood as the Gerasimov–Caputo derivative. The boundary conditions are given in the form of linear functionals, which makes it possible to cover a wide class of linear local and non-local conditions. A representation of the solution is found in terms of special functions. A necessary and sufficient condition for the solvability of the problem under study is obtained, as well as conditions under which the solvability condition is certainly satisfied. The theorem of existence and uniqueness of the solution is proved.