A methodology is set forth for the numerical solution of transient two-dimensional diffusion-type problems (e.g. Heat conduction) in which one of the boundaries of the solution domain moves with time. The moving boundary is immobilized by a coordinate transformation, but the transformed coordinates are, in general, not orthogonal. Furthermore, with respect to a given control volume in the new coordinate system, mass appears to pass through the control surface which bounds the volume, and this mass movement brings about a convection-like transport of energy. The energy equation for a moving, nonorthogonal control volume is derived in general and then specialized to the transformed coordinate system associated with the immobilization of the moving boundary. A fully implicit scheme is used to discretize the control volume energy equation. The spatial derivatives are discretized by either of two schemes depending on the size of the pseudo-convection relative to the diffusion. The energy balance at the moving boundary of the solution domain is also transformed and discretized. A numerical procedure is then developed for solving the discretized energy equations. The use of the control volume formulation and the solution methodology will be illustrated for a specific physical situation in a companion paper that follows this paper in the journal.