Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / 2 / b / Solution

Project the initial condition onto the orthogonal spatial eigenfunctions. Their squared norms in the ordinary inner product areThe latter follows from the integrals of ${\cos }^2(n\pi x)$ and ${\sin }^2(n\pi x)$ being $1/2$, with the sine-cosine cross term zero. ThereforeFor square-integrable data the expansion converges to the data in the L2 norm at $t=0$; for $t>0$ the decaying high modes are smoothed. Initial data with zero $e^x$ projection have no growing term.