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8.1: Example

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Consider a circular plate of radius c m, insulated from above and below. The temperature on the circumference is 100 C on half the circle, and 0 C on the other half.

circ_T.png
Figure 8.1.1: The boundary conditions for the temperature on a circular plate.

The differential equation to solve is ρ22uρ2+ρuρ+2uϕ2u=0, with boundary conditions u(c,ϕ)={100if 0<ϕ<π0if π<ϕ<2π.

8.1.1: Periodic BC

There is no real boundary in the ϕ direction, but we introduce one, since we choose to let ϕ run from 0 to 2π only. So what kind of boundary conditions do we apply? We would like to see “seamless behaviour”, which specifies the periodicity of the solution in ϕ, u(ρ,ϕ+2π)=u(ρ,ϕ),uϕ(ρ,ϕ+2π)=uϕ(ρ,ϕ). If we choose to put the seem at ϕ=π we have the periodic boundary conditions u(ρ,2π)=u(ρ,0),uϕ(ρ,2π)=uϕ(ρ,0).

We separate variables, and take, as usual u(ρ,ϕ)=R(ρ)Φ(ϕ). This gives the usual differential equations ΦλΦ=0,ρ2R+ρR+λR=0. Our periodic boundary conditions gives a condition on Φ, Φ(0)=Φ(2π),Φ(0)=Φ(2π). The other boundary condition involves both R and Φ.


This page titled 8.1: Example is shared under a CC BY-NC-SA 2.0 license and was authored, remixed, and/or curated by Niels Walet via source content that was edited to the style and standards of the LibreTexts platform.

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