temperature system

Question

In a grid of wires, the temperature at exterior mesh points is maintained at constant values as shown in the figure. When the grid is in thermal equilibrium, the temperature at each interior mesh point is the average of the temperatures at the four adjacent points. For instance,

T1 = (T2 + T3 + 0 + 30) / 4

Find the temperatures T1, T2, T3, T4 when the grid is in thermal equilibrium.

Answer 1

Step 1: Write the average equations for the other 3 interior meshes/points.

Step 2: Rewrite the equations in standard form.

Step 3: Convert into a matrix

Step 4: Row reduce echelon form to get the solution T1 = 20, T2 = 0, T3 = 50, T4 = 30.