12.4E: Laplace's Equation in Polar Coordinates (Exercises)
( \newcommand{\kernel}{\mathrm{null}\,}\)
Q12.4.1
1. Define the formal solution of
urr+1rur+1r2uθθ=0,ρ0<r<ρ,−π≤θ<π,u(ρ0,θ)=f(θ),u(ρ,θ)=0,−π≤θ<π,
where 0<ρ0<ρ.
2. Define the formal solution of
urr+1rur+1r2uθθ=0,ρ0<r<ρ,0<θ<γ,u(ρ0,θ)=0,u(ρ,θ)=f(θ),0≤θ≤γ,u(r,0)=0,u(r,γ)=0,ρ0<r<ρ,
where 0<γ<2π and 0<ρ0<ρ.
3. Define the formal solution of
urr+1rur+1r2uθθ=0,ρ0<r<ρ,0<θ<γ,u(ρ0,θ)=0,ur(ρ,θ)=g(θ),0≤θ≤γ,uθ(r,0)=0,uθ(r,γ)=0,ρ0<r<ρ,
where 0<γ<2π and 0<ρ0<ρ.
4. Define the bounded formal solution of
urr+1rur+1r2uθθ=0,0<r<ρ,0<θ<γ,u(ρ,θ)=f(θ),0≤θ≤γ,uθ(r,0)=0,u(r,γ)=0,0<r<ρ,
where 0<γ<2π.
5. Define the formal solution of
urr+1rur+1r2uθθ=0,ρ0<r<ρ,0<θ<γ,ur(ρ0,θ)=g(θ),ur(ρ,θ)=0,0≤θ≤γ,u(r,0)=0,uθ(r,γ)=0,ρ0<r<ρ,
where 0<γ<2π and 0<ρ0<ρ.
6. Define the bounded formal solution of
urr+1rur+1r2uθθ=0,0<r<ρ,0<θ<γ,u(ρ,θ)=f(θ),0≤θ≤γ,uθ(r,0)=0,uθ(r,γ)=0,0<r<ρ,
where 0<γ<2π.
7. Show that the Neumann problem
urr+1rur+1r2uθθ=0,0<r<ρ,−π≤θ<π,ur(ρ,θ)=f(θ),−π≤θ<π
has no bounded formal solution unless ∫π−πf(θ)dθ=0. In this case it has infinitely many solutions. Find those solutions.