Using Poisson kernel to solve heat equation I came across this exercise in my study of PDE:
Find the equilibrium temperature on a half-disk of radius 1 when the temperature is held to 1◦ on the curved edge and 0◦ on the straight edge.
It is possible to obtain a solution in terms of Fourier series but I believe the aim of this exercise is to make use of the formula with the Poisson kernel. From the exercise, I deduce the following initial conditions (using polar coordinates):
\begin{align*}
\Delta u = 0,\ & \theta \in [0,\pi], r \in (0,1) \\
u(1,\theta) = 1,\ & \theta \in (0,\pi) \\
u(r,0) = u(r,\pi) = 0,\ & r \in (0,1)
\end{align*}
I was thinking of using Poisson kernel to get
$u(r,\theta) = \int^{\pi}_0 \frac{1-r^2}{1+r^2-2r\cos(t-\theta)} dt$ which should give $1-\frac{1}{\pi}\tan^{-1}\left(\frac{1-r^2}{2r\sin(\theta)}\right)$ but the solution given in the textbook is $1-\frac{\textbf{2}}{\pi}\tan^{-1}\left(\frac{1-r^2}{2r\sin(\theta)}\right)$. How do I account for the initial conditions at $\theta = 0,\pi$?
That you in advance!
 A: The question asks for a solution $u$ to $\Delta  u = 0$ on the upper half disk, with boundary condition $u= 1$ on the semicircular part of the boundary, and boundary condition $ u = 0$ on the straight part of the boundary that coincides with the $x$-axis.
Although you can't solve this problem directly using the Poisson kernel, you can instead try to find a solution $\tilde u$ to $\Delta \tilde u = 0$ on the whole disk, with boundary condition $\tilde u = 1$ on the upper semicircular part of the boundary, and boundary condition $\tilde u = -1$ on the lower semicircular part of the boundary.
If you think about it, you'll realise that this $\tilde u$ is odd under reflection in the $x$-axis. (See if you can prove this, using the uniqueness theorem for the Laplace equation.) And since $\tilde u(x, y)$ is odd under reflection in the $x$-axis, it must necessarily be zero on the $x$-axis. In other words, the restriction of $\tilde u$ to the upper half disk solves the original problem in your question.
The rest is algebra. I hope it works out!
