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

The uniform source adds a background without changing the spatial gradients relative to the unforced solution:At $10\mathrm{s}$ the offset is $1-e^{-1}\simeq 0.6321$. On the corrected grid the central peak is therefore about $0.8321$, and regions far from the hot spot are near $0.6321$. The hot spot's broadening is unchanged; the source is spatially uniform. The spatial mean of a periodic heat equation rises by the same amount, tending to a total increment of one temperature unit as $t\to \infty$.