= Mixed-boundary drift-diffusion eigenmodes on a rectangle
{title2=$u_{mn}=e^{-\beta x/2}\sin(m\pi x/L_x)\cos((n-1/2)\pi y/L_y)e^{-\lambda_{mn}t}$}
For $u_t=u_{xx}+u_{yy}+\beta u_x$ on $(0,L_x)\times(0,L_y)$, impose zero <Dirichlet boundary conditions> on the two vertical sides and the top, and a zero <Neumann boundary condition> at the bottom. The <Dirichlet gauge transform for constant drift> $u=e^{-\beta x/2}v$ gives $v_t=\Delta v-\beta^2v/4$. <Separation of variables> produces the displayed modes, with $m,n\geq1$ and $\lambda_{mn}=(m\pi/L_x)^2+((n-1/2)\pi/L_y)^2+\beta^2/4$. Their spatial parts are orthogonal with weight $e^{\beta x}$; equivalently, expand $e^{\beta x/2}u_0$ in the ordinary sine-cosine basis.
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