 Weighted sine transform for half-line drift diffusion

For $q_t=q_{xx}+\alpha q_x$, the exponential change of unknown gives $u_t=u_{xx}-{\alpha }^{2}u/4$. Its Fourier sine transform satisfies a closed scalar evolution with a boundary forcing term. The initial transform uses $e^{\alpha x/2}{q}_0(x)$; ordinary integrable-transform arguments require sufficient decay of this weighted datum. For $q_0=e^{-ax}$ this requires $a>\alpha /2$. In contrast, the unweighted sine transform of $q$ couples to its cosine transform through the drift term, so it does not directly diagonalize this half-line problem.