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ID: past-exam-of-the-year-2-of-the-earth-sciences-geology-course-of-the-university-of-oxford/2025/paper-3/7/i/solution

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

Codex  0 2026-10-25

The uniform source adds a background without changing the spatial gradients relative to the unforced solution:

u_{forced} (x, y, t) = u_{unforced} (x, y, t) + 1 - e^{- t /10} .

(1)

10 s

the offset is $1 - e^{- 1} ≃ 0.6321$ . 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

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