ON AN INVERSE PROBLEM OF THE BITSADZE-SAMARSKY TYPE FOR PARABOLIC EQUATION OF FRACTIONAL ORDER
DOI:
https://doi.org/10.52754/16948645_2023_1_90Keywords:
inverse problem, Gerasimov-Caputo operator, equation of fractional order, Riesz basis, adjoint problem.Abstract
In this paper, we study a nonlocal inverse problem of the Bitsadze-Samarskii type for a degenerate parabolic equation of fractional order with the Gerasimov-Caputo operator. 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 a spatial variable. The spectral questions of the obtained problem, as well as of the adjoint problem, are investigated, and the differential operator corresponding to the adjoint problem is being discontinuous. Eigenvalues and eigenfunctions of the problems are found, completeness and the Riesz basis property of the obtaining systems are proved. Further, under certain conditions on the given functions, uniqueness and existence theorems for a solution to the posed problem are proved. When proving the uniqueness of the solution to the problem, we use the completeness of the system of eigenfunctions of the corresponding spectral problem, and construct the solution of the problem in the form of an absolutely and uniformly descending series.
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