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 that the plate 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 ; T ₂ ; T ₃ ; T ₄ denote the temperature at the four interior nodes of the mesh in the figure. The temperature at a node is equal to the average of the four nearest nodes- the left, right, above and below. For instance,

T ₁ = (10 + 20 +T ₂ + T ₄ ) = 4.

(a) Find a system of four equations whose solution will giveT ₁ ; T ₂ ; T ₃ ; T ₄ .

(b) Solve the above system.

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