Solving the heat equation

Calculus Level 4

A two-dimensional non-transient rectangular system follows the heat equation \[ \nabla^2 T = 0 \] where \(T(y,x)\) is the temperature at \((y,x)\).

  • The system is 5 units wide (vertical) and 10 units (horizontal).
  • The temperatures of north, east, west and south walls are maintained at 100, 40, 80 and 20 respectively.

What is the temperature of a point located 4 units to the right of the west wall and 2 units to the bottom of the north wall?

Note: This problem might require a numerical approach.

Inspired by Himadri Chattopadhyay.

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