On a Nonlocal Problem for a Mixed-Type Equation with a Fractional Order Operator

In this paper, a nonlocal problem of the Bitsadze-Samarskii type for a parabolic-hyperbolic equation with the Gerasimov-Caputo operator is studied. The spectral method is used to solve the problem. Using this method, the problem under consideration is reduced to the study of a spectral boundary value problem for a second-order ordinary differential equation with respect to the spatial variable. The spectral properties of the obtained, as well as the adjoint problem, are investigated. Eigenvalues are found, as are the corresponding root functions, and their basis property is proved. Further, under certain conditions on given functions, uniqueness and existence theorems for the solution of the problem are proved. At the proof of uniqueness of the solution, the completeness of the system of eigenfunctions is used, and the solution of the problem is constructed in the form of an absolutely and uniformly convergent biorthogonal series.