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Solve Laplace's equation on a circular disk of radius a subject to the piecewise boundary condition

$$

u(a,\theta) = \begin{cases}

1, & \frac{\pi}{2} - \epsilon < \theta < \frac{\pi}{2} + \epsilon\\

0, & \text{otherwise}

\end{cases}

$$

where $\epsilon \ll 1$. Physically, this would reflect the electric potential distribution on a conducting disk whose edge is almost completely grounded except a small portion of angular extent $\Delta\theta = 2\epsilon$ around the location $\theta = \frac{\pi}{2}$. Obtain the solution to this problem and plot the solution for the case of $a = 1$ and $\epsilon = 0.05$.