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Unformatted text preview: y . (5.8) 2. u satisﬁes the “Dirichlet boundary condition” u(x, y, t) = 0, for (x, y ) ∈
∂D.
3. u satisﬁes the initial condition u(x, y, 0) = h(x, y ).
The ﬁrst two of these conditions are homogeneous and linear, so our strategy is
to treat them ﬁrst by separation of variables, and then use Fourier analysis to
satisfy the last condition.
In this section, we consider the special case where
D = {(x, y ) ∈ R 2 : 0 ≤ x ≤ a, 0 ≤ y ≤ b},
so that the Dirichlet boundary condition becomes
u(0, y, t) = u(a, y, t) = u(x, 0, t) = u(x, b, t) = 0.
In this case, separation of variables is done in two stages. First, we write
u(x, y, t) = f (x, y )g (t),
and substitute into (5.8) to obtain
f (x, y )g (t) = c2 ∂2f
∂2f
+2
∂x2
∂y g (t). Then we divide by c2 f (x, y )g (t),
1
1
g (t) =
2 g (t)
c
f (x, y ) ∂2f
∂2f
+2
∂x2
∂y . The lefthand side of this equation does not depend on x or y while the right
hand side does not depend on t. Hence neither side can depend on x, y , or...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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