  A class of inverse problems for restoring the right-hand side of a parabolic
equation for a large class of positive operators with discrete spectrum is
considered. The results on existence and uniqueness of solutions of these
problems as well as on the fractional time diffusion (subdiffusion) equations
are presented. Consequently, the obtained results are applied for the similar
inverse problems for a large class of subelliptic diffusion and subdiffusion
equations (with continuous spectrum). Such problems are modelled by using
general homogeneous left-invariant hypoelliptic operators on general graded Lie
groups. A list of examples is discussed, including Sturm-Liouville problems,
differential models with involution, fractional Sturm-Liouville operators,
harmonic and anharmonic oscillators, Landau Hamiltonians, fractional
Laplacians, and harmonic and anharmonic operators on the Heisenberg group. The
rod cooling problem for the diffusion with involution is modelled numerically,
showing how to find a "cooling function", and how the involution normally slows
down the cooling speed of the rod.
