Derivation of heat equation What are the main physical laws to derive the following heat equation:
$$u_t -\Delta u=f(t,x)?$$
I'm wondering about the interpretation of the Laplacian $\Delta$ and its role in heat process.
 A: I think a very good intuition about what is the Laplacian "doing" is to look at the way of implementing it in a computational simulation, e.g., its finite-difference method's implementation. Since it is the sum of second, non-crossed derivatives, if the system is approximated with a square grid of stepsize $h$, the Laplacian is approximated as (you can think about this yourself as it is fairly easy or just google it):
$$\Delta u(x,y) \approx \frac{u(x-h, y) + u(x+h, y) + u(x, y-h)+ u(x, y+h) - 4u(x,y)}{h^2}$$
Looking carefully at this expression, it means computing the sum of the difference between all neighbouring sides. Imagine now that there is a gradient in heat $u$ from left to right. This will mean that the difference $u(x-h, y)-u(x, y)$ is positive while $u(x+h, y)-u(x, y)$ is negative. Thus the new value after a time $dt$ on the site $(x,y)$ will be a balance between heat that flows in from the left and heat that flows out to the right. So, basically, what the Laplacian is doing is "homogenizing" the value of $u$ in space. This is why you find the Laplacian operator in any equation that involves diffusion.  
In the case where there are no sources or sinks in the system ($f(x,y) = 0$, $\forall(x,y)$), the equation basically comes to be $\partial_t u = \Delta u$ (diffusion equation), and the system distributes heat as homogenously as possible. With sources or sinks, the system tries to but the sources/sinks $f(x,y)$ keep them from doing it completely.
The subject is usually treated in books on Partial Differential Equations, usually it's one of the first (interesting) cases presented. It allows for a good introduction to Fourier series (historically originating in the problem) and Green's functions. The one by J. David Logan (Springer) has a treatment of the matter, and you can find the finite-difference approximations at the last chapter.
