 Solution

Conservation of thermal energy and Fourier's law give the heat equationsThe density times the specific heat capacity, ${\rho }_i{c}_i$, is the volumetric heat capacity, and $k_i$ is the thermal conductivity.

The condition $T_1(-L,t)=0$ holds the left exterior face at a fixed zero reference temperature; $T_2(L,t)=0$ does the same for the right exterior face. The condition $T_1(0_{-},t)=T_2(0_{+},t)$ expresses perfect thermal contact, with no temperature jump at the interface. Finally,expresses continuity of the signed heat flux in the positive $x$ direction: the interface neither stores nor produces heat.