Half-line drift global relation (${\hat{u}}_0-e^{\omega t}\hat{u}=G_1+(ik+\alpha )G_0$)

For the advection-diffusion equation $u_t=u_{xx}+\alpha u_x$ on $x>0$, the spectral divergence form uses $\omega =k^2-i\alpha k$ and factor $e^{-ikx+\omega t}$. With negative-exponential Half-range Fourier transforms, integration gives the displayed global relation for $\mathrm{Im}k\le 0$. The boundary transform $G_1$ refers to $u_x(0,t)$, the negative of the outward normal derivative.

For $q_t=q_{xx}+\alpha q_x$ with $\alpha >0$, the dispersion relation is $\omega =k^2-i\alpha k$ and its symmetry is $\nu =i\alpha -k$. On the upper domain $\mathrm{Re}\omega <0$, $\mathrm{Im}k>\alpha$ and therefore $\mathrm{Im}\nu <0$. After removing the unknown trace through the global relation, deform the boundary contour to the displayed two rays, with $0<\theta <\pi /4$, oriented from left infinity to $i\alpha$ to right infinity. The factor $e^{ikx}$ decays like $e^{-rx\sin \theta }$, while $e^{-\omega t}$ has Gaussian decay for large $r$. Apparent poles introduced by splitting a finite-time forcing transform must be treated as removable singularities of the unsplit expression.