One Class of Inverse Problems for Reconstructing the Process of Heat Conduction from Nonlocal Data

We consider one class of inverse problems for a one-dimensional heat equation with involution and with periodic type boundary conditions with respect to a space variable. This problem simulates the process of heat propagation in a thin closed wire wrapped around a weakly permeable insulation. The inverse problem consists in the restoration (simultaneously with the solution) of the unknown right-hand side of the equation, which depends only on the spatial variable. The conditions for overdetermination are initial and final states. Problems for a classical heat equation, for an equation with fractional derivatives with respect to a time variable, and for a degenerate equation are considered. Existence and uniqueness results for the given problem are obtained via the method of separation of variables.