On the theory of the known inverse problems for the heat transfer equation

The inverse problems for finding the initial condition and the right-hand side were studied for the heat transfer equation. A solution of the initial boundary value problem for the inhomogeneous heat transfer equation with sufficient conditions for the solvability of the problem was constructed in the first place. On the basis of the solution of the initial boundary value problem, a criterion for the uniqueness of the solution of the inverse problem to determine the initial condition was established. The study of the inverse problem of finding the right-hand side of the component, which depends on time, is equivalent to reducing to the unique solvability of the Volterra integral equation of the second kind. In view of the unique solvability of the given integral equation in the class of continuous functions, we obtained theorems for the unique solvability of the inverse problem. The solution of the inverse problem to determine the factor of the right-hand side, depending on the spatial coordinate, was constructed as a sum of the series in the system of eigenfunctions of the corresponding one-dimensional spectral problem; the criterion of uniqueness was established, and the existence and stability theorems of the solution of the problem were proved.