533 n1 k1 we need to determine the ank s and the bnk

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: usually quite difficult to determine the eigenvalues λ1 , λ2 , · · · , λn , · · · explicitly for a given region in the plane. There are two important cases in which the eigenvalues can be explicitly determined. The first is the case where D is the rectangle, D = {(x, y ) : 0 ≤ x ≤ a, 0 ≤ y ≤ b}. We saw how to solve the eigenvalue problem ∂2f ∂2f + 2 = λf, 2 ∂x ∂y f (x, y ) = 0 for (x, y ) ∈ ∂D, when we discussed the heat and wave equations for a rectangular region. The nontrivial solutions are λmn = − πm a 2 − πn b 2 , fmn (x, y ) = sin π ny π mx sin , a b where m and n are positive integers. A second case in which the eigenvalue problem can be solved explicitly is that of the disk, D = {(x, y ) : x2 + y 2 ≤ a2 }, where a is a positive number, as we will see in the next section.3 Exercises: 5.7.1. Show that among all rectangular vibrating membranes of area one, the square has the lowest fundamental frequency of vibration by minimizing the function π2 π2 f (a, b) = + a b subject to the constraints...
View Full Document

This document was uploaded on 01/12/2014.

Ask a homework question - tutors are online