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.
which we may rewrite as
Set up a linear system to find the temperature at these six points inside the plate. Then use Sage to solve the linear system. If all the entries of the matrix are integers, Sage will compute the reduced row echelon form using rational numbers. To view a decimal approximation of the results, you may use
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.