# D thus if f and g lie in v f g f gdxdy g f d in

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ion of Wλ is called the multiplicity of the eigenvalue λ. The eigenvalue problem consist of ﬁnding the eigenvalues λ, and a basis for each nonzero eigenspace. Once we have solved the eigenvalue problem for a given region D in the (x, y )-plane, it is easy to solve the initial value problem for the heat equation or the wave equation on this region. To do so requires only that we substitute the values of λ into the equations for g . In the case of the heat equation, 2 g (t) = c2 λg (t) ⇒ g (t) = (constant)ec λt , while in the case of the wave equation, √ g (t) = c2 λg (t) ⇒ g (t) = (constant) sin(c −λ(t − t0 )). In the second case, the eigenvalues determine the frequencies of a vibrating drum which has the shape of D. Theorem. All of the eigenvalues of ∆ are negative, and each eigenvalue has ﬁnite multiplicity. The eigenvalues can be arranged in a sequence 0 > λ1 > λ2 > · · · > λ n > · · · , 142 with λn → −∞. Every well-behaved function can be represented as a convergent sum of eigenfunctions.2 Although the theorem is reassuring, it is...
View Full Document

## This document was uploaded on 01/12/2014.

Ask a homework question - tutors are online