(a)

(i) For the diffusion equation

$$\frac{\partial \phi}{\partial t}-K \frac{\partial^{2} \phi}{\partial x^{2}}=0 \quad \text { on }-\infty<x<\infty \text { and } t \geqslant 0$$

with diffusion constant $K$, state the properties (in terms of the Dirac delta function) that define the fundamental solution $F(x, t)$ and the Green's function $G(x, t ; y, \tau)$.

You are not required to give expressions for these functions.

(ii) Consider the initial value problem for the homogeneous equation:

$$\frac{\partial \phi}{\partial t}-K \frac{\partial^{2} \phi}{\partial x^{2}}=0, \quad \phi\left(x, t_{0}\right)=\alpha(x) \quad \text { on }-\infty<x<\infty \text { and } t \geqslant t_{0}$$

and the forced equation with homogeneous initial condition (and given forcing term $h(x, t))$ :

$$\frac{\partial \phi}{\partial t}-K \frac{\partial^{2} \phi}{\partial x^{2}}=h(x, t), \quad \phi(x, 0)=0 \quad \text { on }-\infty<x<\infty \text { and } t \geqslant 0$$

Given that $F$ and $G$ in part (i) are related by

$$G(x, t ; y, \tau)=\Theta(t-\tau) F(x-y, t-\tau)$$

(where $\Theta(t)$ is the Heaviside step function having value 0 for $t<0$ and 1 for $t>0$, show how the solution of (B) can be expressed in terms of solutions of (A) with suitable initial conditions. Briefly interpret your expression.

(b) A semi-infinite conducting plate lies in the $\left(x_{1}, x_{2}\right)$ plane in the region $x_{1} \geqslant 0$. The boundary along the $x_{2}$ axis is perfectly insulated. Let $(r, \theta)$ denote standard polar coordinates on the plane. At time $t=0$ the entire plate is at temperature zero except for the region defined by $-\pi / 4<\theta<\pi / 4$ and $1<r<2$ which has constant initial temperature $T_{0}>0$. Subsequently the temperature of the plate obeys the two-dimensional heat equation with diffusion constant $K$. Given that the fundamental solution of the twodimensional heat equation on $\mathbb{R}^{2}$ is

$$F\left(x_{1}, x_{2}, t\right)=\frac{1}{4 \pi K t} e^{-\left(x_{1}^{2}+x_{2}^{2}\right) /(4 K t)}$$

show that the origin $(0,0)$ on the plate reaches its maximum temperature at time $t=3 /(8 K \log 2)$.