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12.4E: Laplace's Equation in Polar Coordinates (Exercises)

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Q12.4.1

1. Define the formal solution of

 urr+1rur+1r2uθθ=0,ρ0<r<ρ,πθ<π,u(ρ0,θ)=f(θ),u(ρ,θ)=0,πθ<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),0θγ,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),0θγ,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),0θγ,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)=0,0θγ,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),0θγ,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,πθ<π,ur(ρ,θ)=f(θ),πθ<π

has no bounded formal solution unless ππf(θ)dθ=0. In this case it has infinitely many solutions. Find those solutions.


This page titled 12.4E: Laplace's Equation in Polar Coordinates (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench via source content that was edited to the style and standards of the LibreTexts platform.

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