Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics V / 5 / b / i / Solution

Inward heat flux at the left end points in the positive $x$ direction, so the boundary conditions areFor uniform thermal conductivity, the steady heat equation gives $T_s^{\prime \prime }=0$. The heat flux is constant and equal to $Q$, so $T_s^{\prime }=-Q/k$. Imposing the right-end temperature gives