(a) Using the Bromwich contour integral, find the inverse Laplace transform of $1 / s^{2}$.

The temperature $u(r, t)$ of mercury in a spherical thermometer bulb $r \leqslant a$ obeys the radial heat equation

$$\frac{\partial u}{\partial t}=\frac{1}{r} \frac{\partial^{2}}{\partial r^{2}}(r u)$$

with unit diffusion constant. At $t=0$ the mercury is at a uniform temperature $u_{0}$ equal to that of the surrounding air. For $t>0$ the surrounding air temperature lowers such that at the edge of the thermometer bulb

$$\left.\frac{1}{k} \frac{\partial u}{\partial r}\right|_{r=a}=u_{0}-u(a, t)-t$$

where $k$ is a constant.

(b) Find an explicit expression for $U(r, s)=\int_{0}^{\infty} e^{-s t} u(r, t) d t$.

(c) Show that the temperature of the mercury at the centre of the thermometer bulb at late times is

$$u(0, t) \approx u_{0}-t+\frac{a}{3 k}+\frac{a^{2}}{6}$$

[You may assume that the late time behaviour of $u(r, t)$ is determined by the singular part of $U(r, s)$ at $s=0 .]$