Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics V / 6 / a / Solution

Let $A$ be the constant cross-sectional area, and suppose that the temperature is uniform on each cross-section. Consider a slice $[a,b]$. Its stored thermal energy is $\rho cA{\int }_a^{b}T(x,t)dx$. Since the lateral surface is insulated and there is no internal heat generation, conservation of energy giveswhere $q$ is the heat flux in the positive $x$ direction. By Fourier's law, $q=-k{T}_x$, soThe slice is arbitrary; therefore the local heat equation isThe constant $\kappa$ is the thermal diffusivity. The cancellation of $A$ uses a rod of uniform cross-section; the one-dimensional model assumes no transverse temperature variation.