The diffusion equation

$$u_{t}=u_{x x}, \quad 0 \leqslant x \leqslant 1, \quad t \geqslant 0,$$

with the initial condition $u(x, 0)=\phi(x), 0 \leqslant x \leqslant 1$, and boundary conditions $u(0, t)=$ $u(1, t)=0$, is discretised by $u_{m}^{n} \approx u(m h, n k)$ with $k=\Delta t, h=\Delta x=1 /(1+M)$. The Courant number is given by $\mu=k / h^{2}$.

(a) The system is solved numerically by the method

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

Prove directly that $\mu \leqslant 1 / 2$ implies convergence.

(b) Now consider the method

$$a u_{m}^{n+1}-\frac{1}{4}(\mu-c)\left(u_{m-1}^{n+1}-2 u_{m}^{n+1}+u_{m+1}^{n+1}\right)=a u_{m}^{n}+\frac{1}{4}(\mu+c)\left(u_{m-1}^{n}-2 u_{m}^{n}+u_{m+1}^{n}\right)$$

where $a$ and $c$ are real constants. Using an eigenvalue analysis and carefully justifying each step, determine conditions on $\mu, a$ and $c$ for this method to be stable.

[You may use the notation $[\beta, \alpha, \beta]$ for the tridiagonal matrix with $\alpha$ along the diagonal, and $\beta$ along the sub-and super-diagonals and use without proof any relevant theorems about such matrices.]