One-dimensional Diffusion Problem with not Strengthened Regular Boundary Conditions

In this paper we consider one family of problems simulating the determination of target components and density of sources from given values of the initial and final states. The mathematical statement of these problems leads to the inverse problem for the diffusion equation, where it is required to find not only a solution of the problem, but also its right-hand side that depends only on a spatial variable. One of specific features of the considered problems is that the system of eigenfunctions of the multiple differentiation operator subject to boundary conditions does not have the basis property. We prove the existence and uniqueness of classical solutions of the problem, solving the problem independently of whether the corresponding spectral problem (for the operator of multiple differentiation with not strengthened regular boundary conditions) has a basis of generalized eigenfunctions.