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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/c/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 / c / Solution by

Codex  0 2026-10-25

Let

u_{0}

be the Fourier transform of the initial temperature. Solving the scalar linear ordinary differential equation gives

u (k_{x}, k_{y}, t) = u_{0} (k_{x}, k_{y}) e^{- α (k_{x}^{2} + k_{y}^{2}) t} .

(1)

Recover

u

using the Fourier inversion theorem, with inverse factor

(2 π)^{- 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 πm / L_{x}

and

k_{y} = 2 πn / L_{y}

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