Solved: Consider the square plate shown in FIGURE 8.6.2, with the temperatures as

Chapter 8, Problem 60

(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 \(u_1,\ u_2,\ u_3\), and \(u_4\) at the points \(P_1,\ P_2,\ P_3\), and \(P_4\), respectively, are given by

\(u_{1}=\frac{u_{2}+u_{4}+100+100}{4}\)

\(u_{2}=\frac{200+u_{3}+u_{1}+100}{4}\)

\(u_{3}=\frac{200+100+u_{4}+u_{2}}{4}\)

\(u_{4}=\frac{u_{3}+100+100+u_{1}}{4}\).

(a) Show that the above system can be written as the matrix equation

\(\left(\begin{array}{rrrr} -4 & 1 & 0 & 1 \\ 1 & -4 & 1 & 0 \\ 0 & 1 & -4 & 1 \\ 1 & 0 & 1 & -4 \end{array}\right)\left(\begin{array}{l} u_{1} \\ u_{2} \\ u_{3} \\ u_{4} \end{array}\right)=\left(\begin{array}{c} -200 \\ -300 \\ -300 \\ -200 \end{array}\right)\).

(b) Solve the system in part (a) by finding the inverse of the coefficient matrix.