Abstract <p>A multidimensional boundary value problem for a high order equation with fractional derivative with mixed boundary conditions is studied. The existence and uniqueness theorem of the problem in Sobolev classes is proved. The solution of the problem under consideration is constructed in the form of a series sum over the system of eigenfunctions of the multidimensional spectral problem, for which its eigenvalues are found as roots of the transcendental equation and the corresponding system of eigenfunctions is constructed. It is shown that this system of eigenfunctions is complete and forms the Ries basis in Sobolev spaces. On the basis of the completeness of the system of eigenfunctions the uniqueness theorem of the solution of the initial boundary value problem is obtained.</p>