Steady state heat transfer

Calculus Level 4

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)
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