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

Use the outer expansionsThe reduced leading equation isFor the symmetric fundamental mode, $y_0=a\cos ({\omega }_{0}x)$. The reduced second-order problem can impose the displacement conditions $y_0(\pm 1)=0$ but not both clamped-slope conditions. The smallest positive root is

Near the right endpoint introduce the stretched coordinateand near the left endpoint use $X_{-}=(1+x)/ϵ$. The leading outer solution is $O(ϵ)$ in either inner region, so put $y=ϵY_0+\cdots$. The dominant inner equation isAt either endpoint, clamping gives $Y_0(0)=Y_0^{\prime }(0)=0$, bounded matching excludes the growing exponential, and the outer behavior requiresThus the two leading inner solutions have the same form:They show explicitly that each clamped endpoint has a boundary layer of width $O(ϵ)$.

At order $ϵ$, the outer equation isThe symmetric solution iswhich is the stated resonant particular solution plus a homogeneous normalization term.

As $x=1-ϵX_{+}$ approaches the right endpoint from the outer region,The large-$X_{+}$ inner expansion isMatching the constant terms givesThe left endpoint gives the same condition by symmetry. Therefore

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 $0\le x\le 1$.