2D Laplace equation - Your turn

A rectangular plate that is insulated on its two sides and is bounded by

\[\begin{split} \begin{alignat}{2} x &= 0, \quad x &&= \pi \\ y &= 0, \quad y &&= \pi \end{alignat} \end{split}\]

The temperatures at the boundaries are

\[\begin{split} \begin{alignat}{2} T(x, 0) &= 0, \quad \dfrac{\partial T}{\partial x} \bigg|_{x=\pi,y} &&= 0 \\ \dfrac{\partial T}{\partial x} \bigg|_{x=0, y} &= 0, \quad T(x, \pi) &&= 100 \sin(x) \end{alignat} \end{split}\]

The steady-state temperature distribution is governed by \(\dfrac{\partial^2 T}{\partial x^2} + \dfrac{\partial^2 T}{\partial y^2} = 0\). The solution is:

\[ T(x,y) = \dfrac{200}{\pi^2} y - \sum_{n=\text{even}}^{\infty} \dfrac{400}{\pi} \dfrac{\sinh(n y)}{(n^2 - 1) \sinh(n \pi)} \cos(nx)\]

Plot this temperature distribution.

# TODO: Write your solution below

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