 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ib/paper-4/14a/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

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) .

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

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