Consider the initial boundary value problem
In the context of the heat conduction problem, this models the case where the heat flux across the ends at the rod is a function of the temperature. If the heat exchange between the rod and its surroundings is purely by convection, then one commonly assumes that
fis a linear function of the difference in temperatures between the ends of the rod and that of the surroundings, (Newton's law of cooling). For the case of purely radiative transfer of energy a fourth power law for the function fis usual, (Stefan's law).