Past exam of the year 2 of the earth sciences (geology) course of the University of Oxford / 2023 / Paper 1 / 10 / a

For $t>0$, fixed zero endpoint temperatures are homogeneous Dirichlet boundary conditions:An initial condition is also needed to determine a particular solution of the heat equation.

By Fourier's law, $q_x=-k{T}_x$. With nonzero thermal conductivity $k$, zero outward heat flux at both endpoints gives homogeneous Neumann boundary conditions:The outward normal has opposite signs at the two ends, but a zero-flux condition gives zero derivative at each.