The dissolution of glassy polymers is described by a phenomenological model of the motion of two boundaries: the liquid-gel boundary and the gel-glass boundary. The motion of these boundaries, as well as the concentration profile in the layers of a dissolving polymer, was obtained by numerical solution of the Stefan boundary value problem. Confirmation of the experimental program written to simulate the problem is established by its good agreement with direct observation of the dissolution dynamics of polystyrene in methyl ethyl ketone. A potential application of this model to the study of the dissolution dynamics of other polymer-solvent systems is done by simulating the dissolution of three types of polymer-solvent pairs: 1) swelling of rubber, 2) high glass transition concentration, and 3) low glass transition concentration. Contrasting dissolution characteristics are shown for the effect of different types of polymer-solvent pairs as well as for the effect of different molecular weights for the same type of polymer-solvent pair.