Methods | Part IB, 2005

Find the Fourier Series of the function

f(θ)={10θ<π1πθ<2πf(\theta)= \begin{cases}1 & 0 \leq \theta<\pi \\ -1 & \pi \leq \theta<2 \pi\end{cases}

Find the solution ϕ(r,θ)\phi(r, \theta) of the Poisson equation in two dimensions inside the unit disk r1r \leq 1

2ϕ=1rr(rϕr)+1r22ϕθ2=f(θ)\nabla^{2} \phi=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial \phi}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} \phi}{\partial \theta^{2}}=f(\theta)

subject to the boundary condition ϕ(1,θ)=0\phi(1, \theta)=0.

[Hint: The general solution of r2R+rRn2R=r2r^{2} R^{\prime \prime}+r R^{\prime}-n^{2} R=r^{2} is R=arn+brnr2/(n24).R=a r^{n}+b r^{-n}-r^{2} /\left(n^{2}-4\right) . ]

From the solution, show that

r1fϕdA=4πn odd 1n2(n+2)2\int_{r \leq 1} f \phi d A=-\frac{4}{\pi} \sum_{n \text { odd }} \frac{1}{n^{2}(n+2)^{2}}

Typos? Please submit corrections to this page on GitHub.