Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 336 / 3 / b / Solution

Away from the thin layer, set $ϵ=0$. The outer equation isand the boundary condition selected by this first-order problem is the right-end condition. Thuswhich satisfies $y_{\mathrm{out}}(1)=1$ but has $y_{\mathrm{out}}(0)=e^{3/8}$.

The missing left condition is supplied by an outflow boundary layer. Put $X=x/ϵ$ and $y=Y(X)$. After multiplication by $ϵ$, the equation isAt leading order,The conditions $Y_0(0)=1$ and $Y_0\to y_{\mathrm{out}}(0)=e^{3/8}$ as $X\to \infty$ give

The common overlap is $e^{3/8}$. The additive composite expansion is thereforeIt satisfies both endpoint values up to exponentially small or higher-order errors and is uniformly valid to $O(1)$ on $0\le x\le 1$.