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Unformatted text preview: for (x, y ) ∈ ∂D. The same method can be used to treat the wave equation
∂2u
= c2 ∆u,
∂t2
141 (5.27) with the Dirichlet boundary condition,
u(x, y, t) = 0 for (x, y ) ∈ ∂D. This time, substitution of u(x, y, t) = f (x, y )g (t) into (5.27) yields
f (x, y )g (t) = c2 (∆f )(x, y )g (t), or 1
1
g (t) =
(∆f )(x, y ).
c2 g (t)
f (x, y ) Once again, both sides must equal a constant λ, and we obtain
1
1
g (t) = λ =
(∆f )(x, y ).
c2 g (t)
f (x, y )
This separates into
g (t) = λg (t) and (∆f )(x, y ) = λf (x, y ), in which f is once again subject to the boundary condition,
f (x, y ) = 0 for (x, y ) ∈ ∂D. In both cases, we must solve the “eigenvalue problem” for the Laplace operator with Dirichlet boundary conditions. If λ is a real number, let
Wλ = {smooth functions f (x, y ) : ∆f = λf, f ∂D = 0}.
We say that λ is an eigenvalue of the Laplace operator ∆ on D if Wλ = 0.
Nonzero elements of Wλ are called eigenfunctions and Wλ itself is called the
eigenspace for eigenvalue λ. The dimens...
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 Winter '14
 Equations

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