### Author Topic: Quiz 6 T5102  (Read 2049 times)

#### Victor Ivrii

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##### Quiz 6 T5102
« on: March 16, 2018, 06:35:33 AM »
Solve
\begin{align*}
& \Delta u:=u_{xx}+u_{yy}=0&& \text{in }  r> a\\[3pt]
& u|_{r=a}=f(\theta),\\[3pt]
& \max |u| <\infty.
\end{align*}
where we use polar coordinates $(r,\theta)$ and f(\theta)=\left\{\begin{aligned} &1 &&0<\theta<\pi\\ -&1 &&\pi<\theta<2\pi. \end{aligned}\right.

The expected answer: solution as a series.

#### Jingxuan Zhang

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##### Re: Quiz 6 T5102
« Reply #1 on: March 16, 2018, 11:43:33 AM »
The general bounded solution of the DE is
$$u=\frac{a_0}{2}+\sum_n r^{-n}(a_n \cos n\theta + b_0\sin n\theta)\label{a}$$
Now $f$ is odd so $a_n\equiv0$ and
$$b_n=\frac{2a^n}{\pi}\int_0^\pi \sin n\theta\,d\theta=\left\{\begin{array} &\frac{4a^n}{n\pi}&\text{n odd}\\0&\text{n even}\end{array}\right.\label{b}$$

Combining $(1),(2)$ we have the final solutoin
$$\frac{4}{\pi}\sum_k (\frac{a}{r})^{2k+1} \frac{\sin(2k+1)\theta}{2k+1}\label{c}$$