An important concern in the study of heat transfer is to determine the steady-state temperature distribution of a thin plate when the temperature around the boundary is known. Assume the plate shown in the figure represents a cross section of a metal beam, with negligible heat flow in the direction perpendicular to the plate. Let T₁,.....,T₄ denote the temperatures at the four interior nodes of the mesh in the figure. The temperature at a node is approximately equal to the average of the four nearest nodes—to the left, above, to the right, and below. For instance,

Write a system of four equations whose solution gives estimates for the temperatures T₁,.........,T₄.

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T₁ (10+20+ T₂+ T4)/4, = or 4T₁-T₂-T4 = 30