(a) Verify that the PDE
$$
u_t=u_{x x x}
$$
is well posed as an initial value problem.
(b) Consider solving it numerically using the scheme
$$
\frac{u(t+k, x)-u(t-k, x)}{2 k}=\frac{-\frac{1}{2} u(x-2 h, t)+u(x-h, t)-u(x+h, t)+\frac{1}{2} u(x+2 h, t)}{h} .
$$

Determine this scheme's stability condition.