A typical problem in thermodynamics is to find the steady-state temperature distribution inside a thin plate if we know the temperature around the boundary. Let \(T_1, T_2, \ldots, T_6\) be the temperatures at the six nodes inside the plate as shown below.

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The temperature at a node is approximately the average of the four nearest nodes: for instance,

\begin{equation*} T_1 = (10 + 15 + T_2 + T_4)/4 \text{,} \end{equation*}

which we may rewrite as

\begin{equation*} 4T_1 - T_2 - T_4 = 25 \text{.} \end{equation*}

Set up a linear system to find the temperature at these six points inside the plate. Then use Sage to solve the linear system. You may wish to enter one of the numbers with a decimal so that Sage displays the results as decimals rather than fractions.

In the real world, the approximation becomes better the closer the points are together, which happens as we add more and more points into the grid. This is a situtation where we would want to solve linear systems having millions of equations and millions of unknowns.