Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 4 / 14D (Complex Methods)

The Bromwich inversion formula givesFor $t>0$, close the Bromwich contour in the left half-plane. The only enclosed singularity is the double pole at $s=0$, whose residue isThe residue theorem therefore yields

Taking the Laplace transform in $t$ and using the initial condition $u(r,0)=u_0$, the radial heat equation in three dimensions becomesPut $W=rU$ and $q=\sqrt{s}$. ThensoFiniteness at the centre forces $B(s)=0$.

The transformed boundary condition isSincesubstitution and cancellation of the two $u_0/s$ terms givesHenceand the required explicit transform is

Taking $r\to 0$ and using $\sinh (r\sqrt{s})/r\to \sqrt{s}$ givesSet $x=a\sqrt{s}$. The Taylor series of the denominator isConsequentlyThe singular part at $s=0$ is thereforeUsing long-time asymptotics from a Laplace transform and part (a), the centre temperature has late-time behaviour