Solving Laplace Equation ∇²u = 0

Calculus 3 · Axiom Academy

EXAMPLE Solving Laplace's Equation Verify a harmonic function, then apply boundary conditions to pin down the steady-state temperature of a plate. A thin square metal plate occupies , . Its steady-state temperature u(x,y) satisfies Laplace's equation . Three edges are held at : u(0,y)=0 , u(1,y)=0 , and u(x,0)=0 . The top edge is heated to . Show that is harmonic, then find the constant A so that it satisfies the top-edge condition. The steady-state temperature field Solutions of are harmonic : they take their max and min only on the boundary. Heat set at the top spreads smoothly inward, decaying to toward the three cold edges — no hot or cold spots form in the interior. Nicely done. You verified a harmonic function and pinned down the one free constant from a boundary condition. Verify harmonic: to check a solution of , compute u_ xx and u_ yy and confirm they sum to zero. Here the from and the from cancel exactly. Frequencies must match: is harmonic because both factors share the same . That is what makes u_ xx and u_ yy cancel — mismatched frequencies would not be harmonic. Boundary conditions fix the constants: Laplace's equation alone leaves A free; the physical edge temperatures determine it. Three zero edges are automatic here, and the top edge gives . Result: , the steady-state temperature of the plate.

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