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

SOLUTION WILL BE ADDED SOON

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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