 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-1/10/a/solution

Substitute

T = e^{- ν t} R (r)

into the axisymmetric heat equation and cancel the exponential:

- ν R = \frac{1}{r} (r κ R^{'})^{'} .

(r κ R^{'})^{'} + ν r R = 0, - (p R^{'})^{'} = ν wR, p (r) = r κ (r), w (r) = r .

w = r

R (1) = 0

, and the physical solution is regular at the centre. The factor

r

in the weighted inner product is also the radial part of the polar area element.

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