2332 Tutorial Sheet
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Useful facts:
• Laplace’s equation 4φ = 0 with Dirichlet or Neumann boundary conditions has a
unique solution. This is usually proved by considering the energy-like integral
Z
dV ∇φ · ∇φ
(1)
E=
D
and assuming there are two solutions, φ1 and φ2 .
Questions
1. Prove uniqueness for solutions of the Klein-Gordon or Helmholz equation
4φ = m2 φ
(2)
on a region D and with Dirichlet or Neumann boundary conditions on δD.
2. Prove uniqueness for solutions to the heat equation
4u = k
∂u
∂t
(3)
on a region D × [0, ∞) and with Dirichlet or Neumann boundary conditions on
δD × [0, ∞), initial condition u(x, 0) = f (x) on D at time t = 0 and decay condition
u(x, t) → 0 exponential fast as t goes to infinity, k is a constant.
3. The function φ(x, y) is harmonic in the square 0 ≤ x ≤ π, 0 ≤ y ≤ π. On three sides
φ is zero and on the lower side
φ(x, 0) = cos x.
Determine φ(x, y) within the square.
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Sinéad Ryan, ryan@maths.tcd.ie
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