Chapter 1.1, Problem 33E

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 T_1, …, 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.² For instance,

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

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