I've been thinking about the following problem for a little while and can't seem to find a way around it. Perhaps you'll be able to help me out:
Suppose I want to solve the 1-D heat equation over two domains with distinct diffusivity: $\Omega _1 = (-\pi, 0)$ with diffusivity $\alpha _1$, and $\Omega _2 = [0, \pi)$ with $\alpha _2$, where each domain starts with their respective initial profiles given by $g_1 (x)$, $g_2 (x)$.
However, assume now that both systems are insulated on their opposite ends, but are able to interact over an interface at $x=0$. That is, if $u_1$ and $u_2$ are solutions to the heat equation in domains $\Omega _1$, $\Omega _2$ respectively, then $$ \partial _x u_1 (-\pi, t) = \partial _x u_2 (\pi, t) = 0 $$ and $$ \lim _{\epsilon \rightarrow 0^{-}} u_1(\epsilon, t) = \lim _{\epsilon \rightarrow 0^{+}} u_2 (\epsilon, t), \quad \lim _{\epsilon \rightarrow 0^{-}} \partial _x u_1(\epsilon, t) = \lim _{\epsilon \rightarrow 0^{+}} \partial _x u_2 (\epsilon, t) $$ for $t > 0$. This is where I ran into a problem:
I attempted solving $$ \partial _t u_1 - \alpha _1 \Delta u_1 = 0, \quad \partial _t u_2 - \alpha _2 \Delta u_2 = 0 $$ separately and then finding relations between the fourier series solutions for $u_1$, $u_2$, but it seems impossible because I end up with overdetermined fourier coefficients; that is, I can relate both solutions or make them approximate their initial profiles, but not both.
At this point, I started suspecting that perhaps I was solving the wrong equation, because I was missing a forcing term at the boundary. But, how can this be? Surely $u_1$, for instance, must satisfy $\partial _t u_1 - \alpha _1 \Delta u_1 = 0$ for all $x$ except at the interface, since there are no terms forcing anywhere else. However, it seems silly to me to think about a modified heat equation that's only valid at a single point! How can we make sense of this?
I've tried looking for literature on the matter, but all I'm able to find are papers regarding numerical solutions using finite difference methods and such. This can't be that hard... can it? :)