Continuous Models of Diffusion and Vibration

We shall describe the diffusion of heat energy through a long thin rod G with uniform cross section S. As before, we identify G with the open interval (a, b), and we assume the rod is perfectly insulated along its length. Let u(x, t) denote the temperature within the rod at a point x ∈ G and at time t > 0. Our previous experience shows that the heat flux q(x, t) at the point x, i.e., the flow rate to the right per unit area, is proportional to the temperature gradient ∂u ∂x . The precise statement of this experimental fact is Fourier’s law of heat conduction,