Half-line advection-diffusion global relation ($\hat{u}=e^{-\omega t}{\hat{u}}_0-(ik+\beta )f_0-f_1$)

For $u_t=u_{xx}+\beta u_x$ on $x>0$, integration by parts in its half-line Fourier transform gives ${\hat{u}}_t+(k^2-i\beta k)\hat{u}=-(ik+\beta )u(0,t)-u_x(0,t)$. Its time-integrated identity relates initial data and boundary values. The spectral involution $k\mapsto -k+i\beta$ preserves $\omega =k^2-i\beta k$. Evaluating the relation at both arguments eliminates the unknown Neumann transform from a Dirichlet problem; the residual unknown solution transform vanishes upon closing the upper contour for $x>0$.