Im currently trying to understand the whole derivation of the Heat equation from Fourier's Law, and there is one part that doesn't make sense conceptually, and I would like an explanation.
Heat equation: $u_t=k \Delta u +f$
So it's when we are looking at the rate of change in energy w.r.t time.
$$\frac{\partial}{\partial t} \int \int_{\Omega} u \ dxdy = \int_{\partial \Omega}J \cdot (-\nu) \ dxdy + \int \int_{\Omega} f \ dxdy$$
where $\Omega \subset C$ where $C$ is a conducting plate.
The integration with $J \cdot (- \nu)$ considers the energy flow through $\partial \Omega$.
The last term is said to consider an external source according to the course literature.
However, it's the last term that does not make sense to me. If said to exist a external source that changes the total rate of change in energy, doesn't that go through the edge $\partial \Omega$ and is then in the integration with $J \cdot (- \nu)$?
How can there be an external source that affects the rate of change in energy, but doesn't go through $\partial \Omega$?
Here's an example where heat is generated internally. We derive the equation modelling heat flow in one dimension. We model a metal rod which is insulated except at the ends. The rod is heated uniformly by means of a current and also due to a raised temperature at one of the ends. Now, let $T(x,t)$ be the temperature of the rod at $x$ and time $t$. Let $q$ be the heat generated per unit volume in the rod by the current and be constant throughout the rod.
Now, using Fourier's law of heat conduction, where heat flows proportionally with the temperature gradient we get, from a differential energy balance:
$$ -k A \frac{\partial T}{\partial x} \vert_x + k A \frac{\partial T}{\partial x} \vert_{x + dx} + q A dx = \rho C_p A dx \frac{\partial T}{\partial t} $$
with $A$ the area of the cross-section of the rod and $k$ the conductivity. Dividing both sides by $Adx$ gives us:
$$ k\frac{\partial^2 T}{\partial x^2} + q = \rho C_p \frac{\partial T}{\partial t}$$
So $q$ is a term representing internal heat generated by the flow of current.