= Solution
Set $\omega(k)=k^2-i\alpha k$ and $\nu(k)=i\alpha-k$, so $\omega(\nu)=\omega(k)$. For boundary traces $h_j(s)=\partial_x^jq(0,s)$, define
$$
Q(k)=\int_0^\infty e^{-ikx}q_0(x)dx,\qquad G_j(k,t)=\int_0^t e^{\omega(k)s}h_j(s)ds.
$$
The <Half-range Fourier transform> of the equation is obtained by two integrations by parts:
$$
\partial_t\widehat q=-\omega(k)\widehat q-h_1(t)-(ik+\alpha)h_0(t).
$$
Solving this scalar equation gives the <drift-diffusion half-line global relation>
$$
e^{\omega(k)t}\widehat q(k,t)=Q(k)-G_1(k,t)-(ik+\alpha)G_0(k,t),\qquad\operatorname{Im}k\leq0.
$$
In particular the reflected relation is valid on $L$, since $\operatorname{Im}\nu(k)\leq0$ there.
In the region between the real axis and $L$, $\operatorname{Re}\omega\geq0$. Thus a boundary contribution $e^{-\omega t}G_j=\int_0^t e^{-\omega(t-s)}h_j(s)ds$ admits an upward <contour deformation> from the real axis to $L$. Large-contour estimates follow from the spatial exponential and the time integral; at the upper time endpoint one can integrate in time once or use a small contour displacement. The initial-data transform is retained on the real axis. Hence, if $I_C[F]=(2\pi)^{-1}\int_C e^{ikx-\omega(k)t}F(k)dk$, ordinary inversion first gives
$$
q=I_{\mathbb R}[Q]-I_L[G_1+(ik+\alpha)G_0].
$$
For the <Dirichlet boundary condition>, take $h_0=g_0$. Subtract the null integral of the reflected time-dependent transform, using the inversion from part (a) with $c=-1$. The unknown derivative transform cancels because $i\nu+\alpha=-ik$. This derives the <Dirichlet spectral formula for half-line constant drift>:
$$
\boxed{q(x,t)=\frac1{2\pi}\int_{\mathbb R}e^{ikx-\omega(k)t}Q(k)dk-\frac1{2\pi}\int_L e^{ikx-\omega(k)t}\left[Q(\nu(k))+(2ik+\alpha)G_0(k,t)\right]dk.}
$$
Only initial and prescribed boundary data occur in this expression.
For the <Neumann boundary condition>, take $h_1=g_1$. The reflected <global relation> reads $e^{\omega t}\widehat q(\nu,t)=Q(\nu)-G_1+ikG_0$. Solve for $G_0$ and substitute it in the preliminary representation. The leftover term is a null integral with multiplier $(ik+\alpha)/(ik)$, analytic above $L$; its pole at zero is below the contour. Thus the <Neumann spectral formula for half-line constant drift> is
$$
\boxed{q(x,t)=\frac1{2\pi}\int_{\mathbb R}e^{ikx-\omega(k)t}Q(k)dk+\frac1{2\pi}\int_L e^{ikx-\omega(k)t}\left[\frac{ik+\alpha}{ik}Q(\nu(k))-\frac{2ik+\alpha}{ik}G_1(k,t)\right]dk.}
$$
There is no residue from the origin, since this elimination never moves the contour across it. The given derivative is the inward-coordinate derivative $q_x$, not the outward derivative $-q_x$. The initial compatibility for this case is $q_0'(0)=g_1(0)$.
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