A general theory is devised for determining the temperature and heat flux at the surface of a solid when the temperature at an interior location is a prescribed function of lime. The theory is able to accommodate an initial temperature distribution which varies arbitrarily with position throughout the solid. Detailed analytical treatment is extended to the sphere, the plane slab, and the long cylinder; and it is additionally shown that the semi-infinite solid is a particular case of the general formulation. The accuracy of the method is demonstrated by a numerical example. In addition, a numerical calculation procedure is devised which appears to provide smooth, nonoscillatory results.