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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