DIRICHLET–NEUMANN BOUNDARY VALUE PROBLEM FOR A FRACTIONAL ORDER DIFFERENTIAL EQUATION WITH DELAY

In this paper, we develop Green’s function method for solving a boundary value problem with generalized conditions of the Dirichlet–Neumann type for a linear ordinary differential equation of arbitrary fractional order. A condition that guarantees the unique solvability of the problem is obtained. Green’s function is constructed, and its properties are proved. The representation of Green’s function is obtained in terms of a special function, whose properties are also studied in the article. The solution to the problem is written out explicitly in terms of Green’s function. The existence and uniqueness theorem to the problem is proved.