 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2004/paper-1/2/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

With constant positive thermal diffusivity

ν

, no internal heat source and one spatial coordinate, the heat equation is

\partial_{t} θ = ν \partial_{x}^{2} θ, 0 < x < L .

Separation of variables makes each spatial mode decay by

e^{- ν k^{2} t}

. The endpoint conditions determine whether the initial triangular profile is expanded in a Fourier sine series or a Fourier cosine series.

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