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Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 4 / 14A / a / Solution

Take the Fourier transform in

x

. Since

θ_{xx} = - k^{2} θ

, the transformed heat equation is

\frac{\partial θ}{\partial t} = - D k^{2} θ, θ (k, 0) = Θ (k) .

Therefore

θ (k, t) = e^{- D k^{2} t} Θ (k) .

The given transform pair and the convolution theorem yield the heat-kernel solution

θ (x, t) = \frac{1}{4 π D t} \int_{- \infty}^{\infty} exp [- \frac{( x - ξ ) ^{2}}{4 D t}] Θ (ξ) d ξ .

Solved by gpt-5.6-sol high.

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