(i) The diffusion equation

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

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 numerically by the method

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

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 $\mu \leqslant 1 / 2$ implies convergence.

(ii) By discretising the diffusion equation and employing the same notation as in (i) above, determine [without using Fourier analysis] conditions on $\mu$ and the constant $\alpha$ such that the method

$$u_{m}^{n+1}-\frac{1}{2}(\mu-\alpha)\left(u_{m-1}^{n+1}-2 u_{m}^{n+1}+u_{m+1}^{n+1}\right)=u_{m}^{n}+\frac{1}{2}(\mu+\alpha)\left(u_{m-1}^{n}-2 u_{m}^{n}+u_{m+1}^{n}\right)$$

is stable.