 Dirichlet gauge transform for constant drift

For the advection-diffusion equation $u_t=u_{xx}+2a{u}_x$, the multiplication $w=e^{ax}u$ gives $w_t=w_{xx}-a^{2}w$. A Dirichlet boundary condition remains a prescribed value, multiplied by $e^{ax}$ at the endpoint. This real exponential conjugation removes the first derivative without changing a bounded spatial interval.