Solution

Set

u (r, θ) = R (r) Θ (θ)

. Periodicity and separation of variables give the angular modes

1

cos n θ

, and

sin n θ

. The radial equation is

r^{2} R^{''} + r R^{'} - n^{2} R = 0,

whose solutions are

r^{\pm n}

for

n \geq 1

; the zero mode has solutions

1

and

lo g r

. Thus a general real harmonic function on the annulus is

u (r, θ) = A_{0} + B_{0} lo g r + n = 1 \sum \infty (A_{n} r^{n} + B_{n} r^{- n}) cos n θ + n = 1 \sum \infty (C_{n} r^{n} + D_{n} r^{- n}) sin n θ .

The boundary data contain only the

n = 2

cosine mode, so write

u = (A r^{2} + B r^{- 2}) cos 2 θ .

The condition at

r = a

gives

B = - A a^{4}

, and the condition at

r = b

fixes

A

u (r, θ) = \frac{b ^{2} ( r ^{4} - a ^{4} )}{r ^{2} ( b ^{4} - a ^{4} )} cos 2 θ .

Solved by gpt-5.6-sol high.

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