 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-v-resit/question-6/c/iii/solution

sin^{3} θ = \frac{3}{4} sin θ - \frac{1}{4} sin (3 θ)

shows that only angular Fourier modes

1

and

3

are required. Differentiating the harmonic expansion on a disk at

r = 1

multiplies the coefficient of each nonconstant Fourier mode by

n

. Thus

B_{1} = 3/4

and

3 B_{3} = - 1/4

T (r, θ) = C + \frac{3}{4} r sin θ - \frac{1}{12} r^{3} sin (3 θ), C \in R .

Each term is a regular harmonic function, and the radial derivative is

T_{r} (1, θ) = \frac{3}{4} sin θ - \frac{1}{4} sin (3 θ) = sin^{3} θ .

\int_{0}^{2 π} sin^{3} θ d θ = 0,

The additive constant is undetermined. This Neumann boundary condition is unchanged by

T \mapsto T + C

, so the paper's blanket uniqueness assertion requires a normalization in this case. The uniqueness of the Neumann problem gives uniqueness up to a constant; prescribing

T (0) = 0

, or zero mean temperature, selects the displayed answer with

C = 0

 New to topics? Read the docs here!