= Solution
Use the <spectral parameter for a linear boundary value problem> $k\in\mathbb C$ and define the <dispersion relation> $\omega(k)=k^2-i\alpha k$. Direct differentiation gives the <divergence form>
$$
\boxed{\partial_t(e^{-ikx+\omega t}u)-\partial_x\left(e^{-ikx+\omega t}[u_x+(ik+\alpha)u]\right)=0.}
$$
Indeed the coefficient of $u$ left over after expansion is $\omega-k^2+i\alpha k=0$, and the remaining factor is $u_t-u_{xx}-\alpha u_x$. Thus this is equivalent to the <advection-diffusion equation> for every $k$.
Introduce the <Half-range Fourier transforms> and <finite-time spectral boundary transforms>
$$
\widehat u(k,t)=\int_0^\infty e^{-ikx}u(x,t)\,dx,\qquad
\widehat u_0(k)=\int_0^\infty e^{-ikx}u_0(x)\,dx,\qquad
G_j(k,t)=\int_0^t e^{\omega(k)s}g_j(s)\,ds,
$$
where $g_1(t)=u_x(0,t)$ is the unknown <normal derivative> with the positive-$x$ convention. The outward normal at zero instead gives $-g_1$. The spatial transforms are analytic for $\operatorname{Im}k<0$ and continuous on the real axis under the stated decay assumptions. Integrating the <divergence form> on $(0,\infty)\times(0,t)$ gives the <global relation>
$$
\boxed{\widehat u_0(k)-e^{\omega(k)t}\widehat u(k,t)
=G_1(k,t)+(ik+\alpha)G_0(k,t),\qquad \operatorname{Im}k\leq0.}
$$
The sign follows from the lower spatial endpoint: the integrated spatial derivative is minus its value at zero. This sign will determine the boundary-forcing term in the solution.
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