(a) Given the finite-difference recurrence

$$\sum_{k=r}^{s} a_{k} u_{m+k}^{n+1}=\sum_{k=r}^{s} b_{k} u_{m+k}^{n}, \quad m \in \mathbb{Z}, n \in \mathbb{Z}^{+}$$

that discretises a Cauchy problem, the amplification factor is defined by

$$H(\theta)=\left(\sum_{k=r}^{s} b_{k} e^{i k \theta}\right) /\left(\sum_{k=r}^{s} a_{k} e^{i k \theta}\right)$$

Show how $H(\theta)$ acts on the Fourier transform $\hat{u}^{n}$ of $u^{n}$. Hence prove that the method is stable if and only if $|H(\theta)| \leqslant 1$ for all $\theta \in[-\pi, \pi]$.

(b) The two-dimensional diffusion equation

$$u_{t}=u_{x x}+c u_{y y}$$

for some scalar constant $c>0$ is discretised with the forward Euler scheme

$$u_{i, j}^{n+1}=u_{i, j}^{n}+\mu\left(u_{i+1, j}^{n}-2 u_{i, j}^{n}+u_{i-1, j}^{n}+c u_{i, j+1}^{n}-2 c u_{i, j}^{n}+c u_{i, j-1}^{n}\right)$$

Using Fourier stability analysis, find the range of values $\mu>0$ for which the scheme is stable.