(a) The diffusion equation

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

is approximated by the Crank-Nicolson scheme

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

with $m=1, \ldots, M$. Here $\mu=k / h^{2}, k=\Delta t, h=\Delta x=\frac{1}{M+1}$, and $u_{m}^{n}$ is an approximation to $u(m h, n k)$. Assuming that $u(0, t)=u(1, t) \stackrel{M+1}{=} 0$, show that the above scheme can be written in the form

$$B \mathbf{u}^{n+1}=C \mathbf{u}^{n}, \quad 0 \leqslant n \leqslant T / k-1$$

where $\mathbf{u}^{n}=\left[u_{1}^{n}, \ldots, u_{M}^{n}\right]^{T}$ and the real matrices $B$ and $C$ should be found. Using matrix analysis, find the range of $\mu>0$ for which the scheme is stable.

[Hint: All Toeplitz symmetric tridiagonal (TST) matrices have the same set of orthogonal eigenvectors, and a TST matrix with the elements $a_{i, i}=a$ and $a_{i, i \pm 1}=b$ has the eigenvalues $\lambda_{k}=a+2 b \cos \frac{\pi k}{M+1}$. ]

(b) The wave equation

$$\frac{\partial^{2} u}{\partial t^{2}}=\frac{\partial^{2} u}{\partial x^{2}}, \quad x \in \mathbb{R}, \quad t \geqslant 0$$

with given initial conditions for $u$ and $\partial u / \partial t$, is approximated by the scheme

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

with the Courant number now $\mu=k^{2} / h^{2}$. Applying the Fourier technique, find the range of $\mu>0$ for which the method is stable.