An Eigenfunction Solution of a Problem in Heat Conduction

The time-dependent one-dimensional equation of heat conduction is solved for a slab of two layers possessing different thermal characteristics and with the following boundary conditions. One face is kept at a constant temperature, and at the other face there is a heat flux which depends linearly on the temperature there. The initial temperature distribution is left arbitrary. The solution is obtained by the classical method of separation of variables and several significant concepts inherent in this method are put in evidence. In particular, the nature and properties of the eigenvalues and eigenfunctions are investigated in some detail.