Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2024 / Paper 1 / 2 / b / Solution

By orthogonality and the sampling property of the Dirac delta distribution,Consequently the reaction-diffusion equation solution isFor every $t>0$ this eigenfunction expansion is smoothed by the Gaussian decay in $n$. As $t\downarrow 0$, it recovers the specified initial condition in the sense of distributions.