Boundary Value Problem for the Time-Fractional Telegraph Equation with Caputo Derivatives

In this paper the Green formula for the operator of fractional differentiation in Caputo sense is proved. By using this formula the integral representation of all regular in a rectangular domains solutions is obtained in the form of the Green formula for operator generating the time-fractional telegraph equation. The unique solutions of the initial-boundary value problem with boundary conditions of first kind is constructed. The proposed approach can be used to study the more general evolution FPDE as well as ODE with Caputo derivatives.