Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 2 / 14A / d

Table of contents
- Solution d

Solution

By the superposition principle, interchange

x

and

y

in the solution from part (c) and add the two solutions:

w (x, y) = u (x, y) + u (y, x) .

The first term supplies the required data on the positive

x

-axis and vanishes on the positive

y

-axis; the second does the reverse. Explicitly,

w (x, y) = \frac{4 x y}{π} (\int_{0}^{1} \frac{v d v}{[( x - v ) ^{2} + y ^{2} ] [( x + v ) ^{2} + y ^{2} ]} + \int_{0}^{1} \frac{v d v}{[( y - v ) ^{2} + x ^{2} ] [( y + v ) ^{2} + x ^{2} ]}) .

It is harmonic in the quarter-plane, has the stated boundary values, and decays in both unbounded directions.

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