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Find the steady-state temperature $u(r,\theta)$ in a quarter semi-circular plate of radius $r = 4$ subject to the heat equation in polar coordinates

$\displaystyle \frac{\partial^2 u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r} + \frac{1}{r^2}\frac{\partial^2 u}{\partial\theta^2} = 0$

$\displaystyle u(r,0) = 0, 0 < r < 4,$

$\displaystyle u\left(r,\frac{\pi}{6}\right) = 0, 0 < r < 4,$

$\displaystyle u(4,\theta) = -6\sin\left(6\theta\right), 0 < \theta < \frac{\pi}{6}$

As in the circular plate model we can separate the PDE into two ODE's with

$\displaystyle r^2R^{\prime\prime}+rR^\prime-\lambda R = 0$

$\displaystyle \Theta^{\prime\prime}+\lambda\Theta = 0$

with

$\displaystyle \Theta(0) =$

$\displaystyle \Theta\left(\frac{\pi}{6}\right) =$

We can solve the BVP in $\Theta$ : $\Theta(\theta)$ = (without a coefficient). Therefore $\sqrt{\lambda}$ is a multiple of an integer n, and $\lambda =$

Now that we know $\lambda$ , we can rewrite the differential equation in R as (using the prime notation for derivatives): $= 0$ .
This is a Cauchy-Euler equation. Substituting the guessed solution $r^m$ into this equation we get the auxiliary equation = 0. The only bounded solution at $r=0$ is $R(r) =$

Therefore $u(r,\theta) = \sum_{n=1}^\infty B_n$
Note We don't need $n = 0$ since that term is zero.

The coefficients $B_n$ are Fourier sine coefficients, therefore

$\displaystyle B_n=$

$\hskip 25pt 2$
------------------

$\Bigg($

$\displaystyle\int$
$0$

$d\theta\Bigg)$

$\displaystyle\hskip 20pt =$ if n =

$\displaystyle\hskip 20pt = 0$ if n $\neq$

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