Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2010 / Paper 2 / 5 / ii

Solving the transformed ordinary differential equation with its initial condition gives $\tilde{\theta }(k,t)=e^{-k^{2}t}\tilde{\theta }(k,0)$. The Fourier inversion theorem therefore yieldsFor $t>0$ the Gaussian multiplier suppresses large spatial frequencies, which explains the smoothing produced by heat conduction.