Heat equation -- Modelling a real-life situation I have read through a lot of books and lecture notes that cover the heat equation and I am still not sure how I would model the easiest real world situations.
For example, take a rod at constant temperature $0$. Suppose someone turns on heaters on both ends of the rod at time $0$. What would the equation be?
Is it
$$
u_t =\Delta u+\delta_{0}+\delta_{1}\\
u(0,x)=0\\
u_x(t,0)=u_x(t,1)=0\\
$$? (I have read that generally heat sources can be modeled by the right-hand side, the delta peaks are just suggestions and may be replaced by something similar that is mathematically more convenient) Or is it
$$
u_t =\Delta u\\
u(0,x)=0\\
u_x(t,0)=u(t,0)-1\\
u_x(t,1)=1-u(t,1)
$$? (I have seen this a lot. But I wonder: If we put an additional heater in the middle, then in analogy this would mean that we expect two different derivative conditions at $x=1/2$ (different signs) which is only possible if the derivative is zero, i.e. if the temperature at the middle is always exactly say $1$. Or which is also possible if we have a hat like thing in the middle which seems unintuitive to me) Or is it
$$
u_t =\Delta u\\
u(0,x)=0\\
u_x(t,0)=u(t,0)^4-1\\
u_x(t,1)=1-u(t,1)^4
$$? (I have seen this, with a connection to the Stefan-Boltzmann law mentioned)
I have little physical knowledge, and I do not expect you to explain the world to me. But perhaps you can point out some obvious flaws in some of the suggestions and suggest texts that say more about the physics "Neumann means isolation, Dirichlet means fixed temperature" and yet can be read by someone with no physical background.
 A: There are some physical limits which are often hidden in the usual presentations. The "physically reasonable" boundary condition is the heat flux boundary condition: there is a thermal conductivity coefficient $k$ for the boundary, and the system is in thermal contact with a temperature $u_{ext}$ at each point of the boundary. Then the boundary condition says that the outward heat flux at a point $x$ on the boundary is $k(u(t,x)-u_{ext}(t,x))$. This outward heat flux is $\nabla_x u \cdot n$. So you wind up with
$$\begin{cases} u_t(t,x) = \Delta u(t,x), & x \in \Omega,t>0 \\
(\nabla_x u \cdot n)(t,x) = k(t,x) (u(t,x)-u_{ext}(t,x)), & x \in \partial \Omega \\
u(0,x)=f(x). & \end{cases}.$$
The Dirichlet boundary condition occurs when you send $k \to \infty$; in this case the boundary is in perfect thermal contact with the heat reservoir, and so $u$ is forced to be equal to $u_{ext}$. The homogeneous Neumann boundary condition occurs when you send $k \to 0$; in this case the boundary is ideally isolated from the heat reservoir, and so the total heat in the system is held fixed. Note also that when we take material properties into account, $k$ could be a source of nonlinearity, in that the heat flux could be a more complicated function of $u$ and $u_{ext}$.
Responding to some of your middle remarks: 


*

*A heater in the middle is just an external forcing, it is not a boundary condition. 

*With delta functions, you can to some degree interchange between boundary conditions and forcing. For instance, in a first order ODE, it is the same to have $y'(t)=F(t,y),y(0)=y_0$ as it is to have $y'(t)=F(t,y)+y_0 \delta_0(t),y(0)=0$.

