 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 formIndeed 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 transformswhere $g_1(t)=u_x(0,t)$ 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 $\mathrm{Im}k<0$ and continuous on the real axis under the stated decay assumptions. Integrating the divergence form on $(0,\infty )\times (0,t)$ gives the global relationThe 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.