Consider the problem of transient diffusion in a composite slab with two different thermal conductivities. Thus region 1 extending from 0 to \(\kappa\) has a thermal conductivity \(k_{1}\), while the region 2 from \(\kappa\) to 1 has a conductivity of \(\kappa_{2}\). The slab is initially at a dimensionless temperature of 1 , and both ends are exposed to a zero temperature.

Set up the problem and solve by separation of variables.