Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 2 / 20 / b / Solution

For the Gaussian ansatz, partial differentiation givesCancel the positive exponential and compare the constant and quadratic powers of $x$ in the diffusion equation. The two coupled ordinary differential equations areThe initial profile requires $F(0)=T_0$ and $H(0)=L^{-2}$. The second equation is a separable differential equation:Substitute this into the first equation and integrate its logarithmic derivative:Thusand the solution isThis is Gaussian diffusion of a smooth initial profile: its width grows while its peak decreases. The Gaussian integral gives ${\int }_{-\infty }^{\infty }T(x,t)dx=\sqrt{\pi }{T}_{0}L$, independent of time, which checks conservation of the total amount under the diffusion equation.