One nonlocal problem of determination of the temperature and density of heat sources
This paper considers the nonlocal problem of the determination of the temperature and density of heat sources from given values of the initial and final temperatures. The nonlocal boundary conditions generalize the well-known Samarskii-Ionkin boundary conditions. The mathematical statement of this problem leads to the inverse problem for the heat equation, where it is required to find not only a solution, but also the right-hand side of the equation that depends only on a spatial variable. The use of the Fourier transform for solving the initial problem leads to the spectral problem for the transform operator, the boundary conditions of which are regular, not strongly regular boundary conditions. Thus, the specific feature of the considered problem is that the system of eigenfunctions of the corresponding spectral problem for the multiple differentiation operator subject to boundary conditions of the initial problem does not have the basis property. The main result of the paper is the proof of the theorem on the unique existence of a generalized solution to the considered problem.
not strongly regular boundary conditions
scientific article; zbMATH DE number 6108784