Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics V resit / Question 6 / c / ii

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- Solution ii

Solution

The boundary data consist of the third angular Fourier mode. Its regular radial factor in the harmonic expansion on a disk is

r^{3}

, so

T (r, θ) = r^{3} sin (3 θ) .

For a direct check of Laplace's equation,

T_{rr} + \frac{1}{r} T_{r} + \frac{1}{r ^{2}} T_{θθ} = 6 r sin (3 θ) + 3 r sin (3 θ) - 9 r sin (3 θ) = 0.

Also

T (1, θ) = sin (3 θ)

. At the origin the apparent coordinate singularity is harmless: this is the harmonic polynomial

3 x^{2} y - y^{3}

. The allowed uniqueness for the Dirichlet problem completes the identification.

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