The accompanying figure shows a metal plate whose edges are held at the temperatures

Chapter 9, Problem T2

(choose chapter or problem)

The accompanying figure shows a metal plate whose edges are held at the temperatures shown. It follows from thermodynamic principles that the temperature at each of the six interior nodes will eventually stabilize at a value that is approximately the average of the temperatures at the four neighboring nodes. These are called the steady-state temperatures at the nodes. Thus, for example, if we denote the steady-state temperatures at the interior nodes in the accompanying figure as T1, T2, T3, T4, T5, and T6, then at the node labeled T1 that temperature will be T1 = 1 4 (0 + 5 + T2 + T3) or, equivalently, 4T1 T2 T3 = 5 Find a linear system whose solution gives the steady-state temperatures at the nodes, and use your technology utility to solve that system by LU-decomposition.