= Half-line advection-diffusion global relation
{title2=$\widehat u=e^{-\omega t}\widehat u_0-(ik+\beta)f_0-f_1$}
For $u_t=u_{xx}+\beta u_x$ on $x>0$, integration by parts in its half-line <Fourier transform> gives $\widehat u_t+(k^2-i\beta k)\widehat u=-(ik+\beta)u(0,t)-u_x(0,t)$. 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 $x>0$.
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