Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 3 / 14A / b / iii

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- Solution iii

Solution

Green's identity with

\partial G / \partial n = 0

leaves only the prescribed normal derivative. Since

\partial u / \partial n = - p

and, on the plane,

G = - \frac{1}{2 π ( x - x _{0} ) ^{2} + ( y - y _{0} ) ^{2} + z _{0}^{2}},

the decaying solution is

u (x_{0}, y_{0}, z_{0}) = - \frac{1}{2 π} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \frac{p ( x , y )}{( x - x _{0} ) ^{2} + ( y - y _{0} ) ^{2} + z _{0}^{2}}, d x, d y .

Thus

f (X, Y, z_{0}) = - \frac{1}{2 π X ^{2} + Y ^{2} + z _{0}^{2}} .

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