Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 4 / 14B / iii / Solution

Set $v(t,z)=u(t,z+at)$. Then $v_t=\kappa v_{zz}$ and $v(0,z)=u_0(z)$. Applying the heat kernel and returning to $x$ gives the advection-diffusion heat-kernel solution

Solved by gpt-5.6-sol high.