[QUE/PDE-01002]

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  \[u(r,\theta)= \frac{A}{\pi}\left[\theta + 2 \sum _{n=1}^\infty   \Big(\frac{r}{a}\Big)^n \Big(\frac{\sin n \theta}{n}\Big) .\right]\]

Remarks

• 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 \(r\) .