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# PDE Solved Problem 01002

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### Question

• Find the most general solution  $$u(r,\theta)$$ of Laplace  equation in two dimensions in the semicircular  region $$r< a, 0< \theta< \pi$$ subject to the boundary conditions
\begin{equation*}
u(r,\theta)  =
\begin{cases} 0  & \text{ if  $r=a$, and $0<\theta < \pi$}\\
0 & \text{ if  $\theta=0$,  and $r< a$}\\
A & \text{ if  $\theta=\pi$, and  $r< a$}.
\end{cases}
\end{equation*}

•  When $$f(r)$$ is constant, $$f(r)=A$$, show that the solution is given  by  $u(r,\theta)= \frac{A}{\pi}\left[\theta + 2 \sum _{n=1}^\infty \Big(\frac{r}{a}\Big)^n \Big(\frac{\sin n \theta}{n}\Big) .\right]$

### For Teachers

• This is an interesting example.
• Unlike most other cases, sinlge valuedness property should not be imposed as the range of $$\theta$$ is restricted.
• Also this requires including $$\log r$$ solution. These points are likely to be missed by students.
• May not be given in examination in the present form.
• Best use is as an example in the lecture
• It has no soluton if the last boundary condition is replaced by $f(r) , \theta=\pi,\quad 0<r<a$ where  $$f(r)$$ is arbitraryfucntion of $$r$$ .

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