Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / 2

Remove the first spatial derivative by the Dirichlet gauge transform for constant drift:The nonzero exponential factor preserves the zero side and top values; it also preserves $v_y=0$ on the bottom. A separated product $v=X(x)Y(y)T(t)$ has the spatial eigenvalue problemsThe nontrivial solutions are $X_m=\sin (m\pi x)$ for $m\ge 1$ and $Y_n=\cos ((n-1/2)\pi y)$ for $n\ge 1$. There is no zero spatial mode in either direction. The temporal equation gives $T=e^{-{\lambda }_{mn}t}$, whereTherefore the mixed-boundary drift-diffusion eigenmodes on a rectangle giveAt $t=0$ the temporal factors are one, giving the required expansion. The sine and mixed-end cosine bases are complete and orthogonal for $v$. In particular,The factor four comes from the product of the two squared basis norms, each $1/2$.

After multiplying the initial field by $e^{\beta x/2}$, the $y$ dependence is exactly the $n=2$ cosine mode. Its orthogonality therefore gives $A_{mn}=0$ when $n\ne 2$. For $n=2$, the remaining Fourier coefficient in the Fourier sine series isThusThe coefficients are independent of $\beta$, because the initial exponential exactly cancels the drift-removing weight. Substitution into the preceding expansion gives the full subsequent solution, with decay rates $m^2{\pi }^2+9{\pi }^2/4+{\beta }^2/4$.