Exercise 1.4.26

The steady state temperature, $u$, of a plate solves Laplace’s equation, $\Delta u=0$. One way to approximate the solution is to divide the plate into a square mesh and require the temperature at each node to equal the average of the temperature at the four adjacent nodes. In the following picture, the numbers represent the observed temperature at the indicated nodes. Find the temperature at the interior nodes, indicated by $x$, $y$, $z$, and $w$. One of the equations is $z=\frac{1}{4}(10+0+w+x)$.