Consider the square plate shown in FIGURE 8.6.2, with the temperatures as indicated on
Chapter 8, Problem 58(choose chapter or problem)
Consider the square plate shown in FIGURE 8.6.2, with the temperatures as indicated on each side. Under some circumstances it can be shown that the approximate temperatures ui. Ui, UJ, and u4 at the points Pi. P2, P3, and P4, respectively, are given by U2 + U4 + 100 + 100 Ui = 4 200 + U3 + Ui + 100 U2 = 4 200 + 100 + U4 + U2 U3 = 4 U3 + 100 + 100 + Ui U4 = 4 (a) Show that the above system can be written as the matrix equation ( -; - ) ( :: ) ( = ) 0 1 -4 1 U3 -300 . 1 0 1 -4 U4 -200 (b) Solve the system in part (a) by finding the inverse of the coefficient matrix. u=200 T T I I P2+- P3 / ---- u= 100 ! --- ! -----/ ----._ ___ _. ____ _ u = 100 Pi I I P4 I I l l u= 100 FIGURE 8.6.2 Plate in 58
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