Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 67 / 1 / b / Solution

Use negative-exponential spatial Fourier transforms and finite-time spectral boundary transforms:Write $\mathcal{F}=k^2{F}_0-ik{F}_1-F_2$ and $\mathcal{G}=k^2{G}_0-ik{G}_1-G_2$. Integrate the local relation first over $0<x<L$ and then over time. The left and right endpoint fluxes have opposite signs, giving the finite-interval Airy global relationAll these spatial and temporal integral transforms are entire functions of $k$ for sufficiently smooth data on the finite intervals. Put $\alpha =e^{2\pi i/3}$ and $k_j={\alpha }^{j}k$, $j=0,1,2$. The dispersion relation obeys $\omega (k_j)=\omega (k)$; hence the same global relation holds at all three $k_j$. In particular, the missing traces satisfy the concrete linear systemwhere, with a common upper time limit $t$,The finite-time spectral boundary transforms $F_j,G_j$ are invariant under these rotations because their time kernel depends only on $k^3$. The coefficients multiplying them are not invariant; these differences make the three equations useful for eliminating the missing boundary traces.