Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2002 / Paper 1 / 12 / iii

Expand the constant right-edge value in a Fourier sine series. Its coefficients are

b_{n} = \frac{2}{π} \int_{0}^{π} sin (n y) d y = \frac{2 [ 1 - ( - 1 ) ^{n} ]}{nπ} .

(1)

The superposition principle combines the normalized modes, giving

f (x, y) = \frac{4}{π} m = 0 \sum \infty \frac{sinh [( 2 m + 1 ) x ]}{( 2 m + 1 ) sinh [( 2 m + 1 ) π ]} sin [(2 m + 1) y] .

(2)

This is a harmonic function in a rectangle with one constant boundary value, with the coordinate roles interchanged. The series and its derivatives converge uniformly on compact subsets of

0 < x < π

, so it satisfies Laplace's equation there. It vanishes on the other three edges and approaches one at every interior point of the right edge. At the two right corners the one-sided prescribed values disagree; no continuous extension can satisfy both. The stated open-edge condition avoids that incompatibility, and isolated corner assignments do not change the bounded interior Dirichlet problem.

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2002 / Paper 1 / 12 / iii

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