U1 t 0 583 for students with access to mathematica

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Unformatted text preview: bounded region D in the (x, y )-plane, the boundary ∂D being a well-behaved 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 left-hand 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...
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This document was uploaded on 01/12/2014.

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