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A number of exact solutions for the surface temperature of a semi-infinite solid which is heated over a region of its surface is given. These have applications to edge effects in surface induction heating, to the heating of the anode of an X-ray tube, and to heating by friction. The regions discussed are a half-plane, a strip, a square, a rectangle, and a circle. Temperatures within the solid and the way in which they approximate to those of a continuous point source are discussed. Holm's approximation (heating of the region bounded internally by a sphere) and heat generation over a spherical region are also studied.
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JC Jaeger (1952) studied this question.