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Laplace's Equation and Steady State
Find the solution to the following system consisting of Laplace's equation with boundary conditions:
$\nabla^2u=0, u(0)=-1,u(2)=3$
Your solution should be quite a simple algebraic expression. Graph the solution and interpret it in the context of steady state heat flow.
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The equation is as follows.
$$u=2x-1$$
Notice that $u(0)=-1$, $u(2)=3$, and $\nabla^2u=0$.
The graph is quite simple
If we think of the object as a bar, the $u$ value on the graph show the temperature of the bar at every point on the bar. This is a steady state heat flow equation because the temperature will not change with time.