An inverse problem in unsteady heat conduction is one for which boundary conditions are prescribed internally, the surface conditions being unknown. By specifying the boundary conditions at a single location, an exact solution is obtained as a rapidly convergent series with the well-known, lumped capacitance approximation as the leading term. Specific forms of the series are determined for sample inverse problems: solid slab, cylinder, sphere, and transpiration-cooled slab. The solution also is applied to direct problems, involving two-point boundary conditions. By truncating the series, approximate solutions of simple form result. The one-term and two-term approximations compare well with exact solutions.

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