Steady state heat transfer

A square metal plate measures 10 meters on each side. Exactly one side is heated to 50 K. The remaining three sides are held at 0 K. A long time has passed (plate surface temperature is no longer changing over time). What is the temperature at the center of the square plate , $u(5,5)$ , to the nearest decimal place?

Assumptions:

• $u(x,y)$ is the temperature of the plate at $(x,y)$
• $0 \leq x \leq 10$
• $0 \leq y \leq 10$
• $u(0,y) = 0$
• $u(x,0) = 0$
• $u(10,y) = 0$
• $u(x,10) = 50$ $\frac {\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2} = 0$ (steady-state)