ISSN 2079-6900 (Print)
ISSN 2587-7496 (Online)

#### Middle Volga Mathematical Society Journal

MSC2020 35K20, 35K51, 35K58

### On the solvability of a mixed problem for a fractional partial differential equation with delayed time argument and Laplace operators with nonlocal boundary conditions in Sobolev classes

#### M. M. Babayev1

Annotation In this paper, we study a problem with initial functions and boundary conditions for partial differential equations of fractional order with Laplace operators. The boundary conditions of the problem are nonlocal, and the solution is supposed to belong to one of Sobolev classes. The solution of the initial boundary value problem is constructed as the sum of a series of multidimensional spectral problem’s eigenfunctions. The eigenvalues of the spectral problem are found and the corresponding system of eigenfunctions is constructed. It is shown that this system is complete and forms a Riesz basis in the subspaces of Sobolev spaces. Basing on the completeness of the eigenfunctions’ system, the uniqueness theorem for the solution of the problem is proved. The existence of a regular solution of the initial boundary value problem is proved in Sobolev subspaces. partial differential equation with delayed argument, fractional time derivative, initial boundary value problem, spectral method, eigenvalues, eigenfunctions, completeness, Riess basis, uniqueness, existence, series, nonlocal boundary conditions, Sobolev class, fractional derivative, mixed problem

1Mahkambek M. Babayev, Doctoral Student, Department of Differential Equations and Mathematical Physics, National University of Uzbekistan named after Mirzo Ulugbek, (4 Universitetskaya St., Tashkent 100174, Uzbekistan), ORCID: https://orcid.org/0000-0002-1799-0413, babayevm@mail.ru

Citation: M. M. Babayev, "[On the solvability of a mixed problem for a fractional partial differential equation with delayed time argument and Laplace operators with nonlocal boundary conditions in Sobolev classes]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,22:1 (2020) 13–23 (In Russian)

DOI 10.15507/2079-6900.22.202001.13-23