On a Boundary Value Problem for a Third-Order Equation of Parabolic-Hyperbolic Type with a Fractional Order Operator

In the paper, we consider a boundary value problem for a third-order mixed differential equation of parabolic-hyperbolic type with a fractional Gerasimov–Caputo operator. Under certain conditions on the data of a problem, applying the methods of the theory of integral equations and the Green function, we prove the unique solvability of the problem. The uniqueness of the solution is proved by the method of the extremum principle, and the existence of this problem is proved by reducing it to a boundary value problem for a fractional order differential equation, as well as to the Volterra integral equation of the second kind.