I'm stuck on an old exam question:
Let $\Omega = \{(x,y) \in R^2 : 1 < x^2 + y^2 < a \}$. Determine the unique solution for the following boundary condition problem:
$\Delta u = 1$ for $(x,y) \in \Omega$
$\frac{\delta u}{\delta n}(x,y) = 0$ for $x^2+y^2=a$
$u(x,y) = 0$ for $x^2 + y^2 = 1$
What I figured out:
You need to express the problem in polar coordinates:
$\frac{\delta^2 u}{\delta r^2} + \frac{1}{r}\frac{\delta{u}}{\delta r} + \frac{1}{r^2}\frac{\delta^2 u}{\delta \phi^2} = 1$
$\frac{\delta u}{\delta r}(\phi, a) = 0$ for $\phi \in [0,2\pi)$
$u(\phi, 1) = 0$ for $\phi \in [0,2\pi)$
I searched on the internet for solutions, but only found solution methods for the second boundary condition, the first boundary condition gives me a hard time trying to adapt these. Also my book does only briefly mention poisson equations, though la place equations are explained thoroughly.
Can someone point me in the right direction?