WeBWorK Standalone Renderer

Find the steady-state temperature $u(r,\theta)$ in a circular plate of radius $r = 2$ subject to the heat equation in polar coordinates

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

$\displaystyle 0 < r < 2, 0 < \theta < 2\pi$

$\displaystyle u(2,\theta) = \begin{cases} \theta, &0 < \theta < \pi\\ \pi -\theta, &\pi<\theta<2\pi\\ \end{cases}$

Now we can solve the PDE using the series solution $u(r,\theta) = \displaystyle A_0 + \sum\limits_{n=1}^\infty r^n \left( A_n \cos\left(n\theta\right) + B_n \sin\left(n\theta\right) \right)$ . The coefficients $A_n$ and $B_n$ are Fourier coefficients:

$\displaystyle A_0=$

----------

$\Bigg($

$\displaystyle\int$

$d\theta+$

$\displaystyle\int$

$d\theta\Bigg)$

$\hskip 15pt =$

----------

$\Bigg($

$+$

$\Bigg)$

$\hskip 17pt =$

$\displaystyle A_n=$

----------

$\Bigg($

$\displaystyle\int$

$d\theta+$

$\displaystyle\int$

$d\theta\Bigg)$

$\hskip 15pt =$

----------

$\Bigg($

$+$

$\Bigg)$

$\hskip 17pt =$

$\displaystyle B_n=$

----------

$\Bigg($

$\displaystyle\int$

$d\theta+$

$\displaystyle\int$

$d\theta\Bigg)$

$\hskip 15pt =$

----------

$\Bigg($

$+$

$\Bigg)$

$\hskip 17pt =$

You can earn partial credit on this problem.