= Solution
{c}
Conservation of <thermal energy> and <Fourier's law> give the <heat equations>
$$
\boxed{\rho_1c_1\,T_{1t}=k_1T_{1xx}\quad(-L<x<0),\qquad
\rho_2c_2\,T_{2t}=k_2T_{2xx}\quad(0<x<L).}
$$
The <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,
$$
-k_1T_{1x}(0_-,t)=-k_2T_{2x}(0_+,t)
$$
expresses continuity of the signed <heat flux> in the positive $x$ direction: the interface neither stores nor produces <heat>.
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