 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2010/paper-2/5/i/solution

Differentiate the spatial Fourier transform under the integral, and integrate the second spatial derivative by parts twice. For sufficiently decaying temperature and first derivative, the boundary terms vanish:

\partial_{t} θ (k, t) = \int_{R} e^{- ik x} \partial_{x}^{2} θ d x = (ik)^{2} θ (k, t) .

\partial_{t} θ = - k^{2} θ .

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