Past exam of the year 2 of the earth sciences (geology) course of the University of Oxford / 2025 / Paper 3 / 7 / c / Solution

Let ${\hat{u}}_0$ be the Fourier transform of the initial temperature. Solving the scalar linear ordinary differential equation givesRecover $u$ using the Fourier inversion theorem, with inverse factor $(2\pi {)}^{-2}$ for this two-dimensional convention. Equivalently, take the convolution of the initial data with the two-dimensional heat kernel. On a periodic square, the same exponential applies to each discrete Fourier mode, with $k_x=2\pi m/L_x$ and $k_y=2\pi n/L_y$.