On one problem for restoring the density of sources of the fractional heat conductivity process with respect to initial and final temperatures In this paper we consider inverse problems for a fractional heat equation, where the fractional time derivative is taken into account in Riemann-Liouville sense. For the solution of this equation, we have to find an unknown right-hand side depending only on a spatial variable. The problem modeling the process of determining the temperature and density of sources in the process of fractional heat conductivity with respect to given initial and final temperatures is considered. Problems with general boundary conditions with respect to the spatial variable that are not strongly regular are investigated. The existence and uniqueness of classical solution to the problem are proved. The problem is considered independent from a corresponding spectral problem for an operator of multiple differentiation with not strongly regular boundary conditions has the basis property of root functions.