(i) The diffusion equation

$$\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}, \quad 0 \leqslant x \leqslant 1, \quad t \geqslant 0$$

with the initial condition $u(x, 0)=\phi(x), 0 \leqslant x \leqslant 1$, and with zero boundary conditions at $x=0$ and $x=1$, can be solved by the method

$$u_{m}^{n+1}=u_{m}^{n}+\mu\left(u_{m-1}^{n}-2 u_{m}^{n}+u_{m+1}^{n}\right), \quad m=1,2, \ldots, M, \quad n \geqslant 0,$$

where $\Delta x=1 /(M+1), \mu=\Delta t /(\Delta x)^{2}$, and $u_{m}^{n} \approx u(m \Delta x, n \Delta t)$. Prove that $\mu \leqslant \frac{1}{2}$ implies convergence.

(ii) By discretizing the same equation and employing the same notation as in Part (i), determine conditions on $\mu>0$ such that the method

$$\begin{aligned} &\left(\frac{1}{12}-\frac{1}{2} \mu\right) u_{m-1}^{n+1}+\left(\frac{5}{6}+\mu\right) u_{m}^{n+1}+\left(\frac{1}{12}-\frac{1}{2} \mu\right) u_{m+1}^{n+1}= \\ &\left(\frac{1}{12}+\frac{1}{2} \mu\right) u_{m-1}^{n}+\left(\frac{5}{6}-\mu\right) u_{m}^{n}+\left(\frac{1}{12}+\frac{1}{2} \mu\right) u_{m+1}^{n} \end{aligned}$$

is stable.