
# 8.1: Example

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Consider a circular plate of radius $$c \text{ m}$$, insulated from above and below. The temperature on the circumference is $$100^\circ~\text{C}$$ on half the circle, and $$0^\circ~\text{C}$$ on the other half.

The differential equation to solve is $\rho^2 \frac{\partial^2 u}{\partial \rho^2} + \rho \frac{\partial u}{\partial \rho} + \frac{\partial^2 u}{\partial \phi^2}{u} = 0,$ with boundary conditions $u(c,\phi) = \begin{cases} 100 & \text{if 0 < \phi < \pi} \\ 0 & \text{if \pi < \phi < 2\pi } \end{cases}\quad .$

## 8.1.1: Periodic BC

There is no real boundary in the $$\phi$$ direction, but we introduce one, since we choose to let $$\phi$$ run from $$0$$ to $$2\pi$$ 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 $$\phi$$, \begin{aligned} u(\rho,\phi+2\pi)&=u(\rho,\phi),\\ \frac{\partial u}{\partial \phi}(\rho,\phi+2\pi)&=\frac{\partial u}{\partial \phi}(\rho,\phi).\end{aligned} If we choose to put the seem at $$\phi=-\pi$$ we have the periodic boundary conditions \begin{aligned} u(\rho,2\pi)&=u(\rho,0),\\ \frac{\partial u}{\partial \phi}(\rho,2\pi)&=\frac{\partial u}{\partial \phi}(\rho,0).\end{aligned}

We separate variables, and take, as usual $u(\rho,\phi) = R(\rho) \Phi(\phi).$ This gives the usual differential equations \begin{aligned} \Phi''-\lambda\Phi &=0,\\ \rho^2 R'' + \rho R' + \lambda R &= 0.\end{aligned} Our periodic boundary conditions gives a condition on $$\Phi$$, $\Phi(0)=\Phi(2\pi),\;\;\Phi'(0)=\Phi'(2\pi). \label{eq:phiBC}$ The other boundary condition involves both $$R$$ and $$\Phi$$.