= Solution
For a sufficiently smooth exact solution, the centered spatial residual is
$$
D_{xx}u-\alpha D_xu-(u_{xx}-\alpha u_x)
=h^2\left(\frac{u_{xxxx}}{12}-\frac{\alpha u_{xxx}}6\right)+O(h^4).
$$
Let $e=U-R_hu$ be the nodal error. It satisfies $e'=A_he+\tau_h$, with $A_h=D_{xx}-\alpha D_x$ and $\|\tau_h\|_h\leq Ch^2$ on the fixed time interval. Part (a) gives $\|e^{tA_h}\|\leq1$, so Duhamel's formula yields
$$
\boxed{\|e(t)\|_h\leq\|e(0)\|_h+\int_0^t\|\tau_h(s)\|_hds
\leq\|e(0)\|_h+Cth^2.}
$$
Thus the method is convergent, with second-order spatial error for smooth data and second-order compatible initialization. The homogeneous endpoint values introduce no boundary residual. For nonsmooth L2 data, stable approximating projections and a density argument extend <numerical convergence>, although the same order need not hold without the <derivative> bounds used above.
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