Laplace's Equation and Steady State

SighHillDoveN

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.

NegativeOneTwelfth

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.