This preview shows page 1. Sign up to view the full content.
Unformatted text preview: bounded region D in the (x, y )plane,
the boundary ∂D being a wellbehaved curve. We would also like to consider
the wave equation for a vibrating drum in the shape of such a region D. Both
cases quickly lead to the eigenvalue problem for the Laplace operator
∆= ∂2
∂2
+ 2.
2
∂x
∂y Let’s start with the heat equation
∂u
= c2 ∆u,
∂t (5.26) with the Dirichlet boundary condition,
u(x, y, t) = 0 for (x, y ) ∈ ∂D. To solve this equation, we apply separation of variables as before, setting
u(x, y, t) = f (x, y )g (t),
and substitute into (5.26) to obtain
f (x, y )g (t) = c2 (∆f )(x, y )g (t).
Then we divide by c2 f (x, y )g (t),
1
1
g (t) =
(∆f )(x, y ).
c2 g (t)
f (x, y )
The lefthand side of this equation does not depend on x or y while the righthand side does not depend on t. Hence both sides equal a constant λ, and we
obtain
1
1
g (t) = λ =
(∆f )(x, y ).
2 g (t)
c
f (x, y )
This separates into
g (t) = c2 λg (t) and (∆f )(x, y ) = λf (x, y ) in which f is subject to the boundary condition,
f (x, y ) = 0...
View
Full
Document
This document was uploaded on 01/12/2014.
 Winter '14
 Equations

Click to edit the document details