Differential Equations Solutions 177

Differential Equations Solutions 177 - Note that as the...

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Chapter 31 Solutions: Case Study: Eigenvalues: Valuable Principles CHALLENGE 31.1. We can verify by direct computation that w m satisFes the boundary condition and that 2 w /∂x 2 2 w /∂y 2 is ( m 2 + ± 2 ) π 2 /b 2 times w ,so λ =( m 2 + ± 2 ) π 2 /b 2 . CHALLENGE 31.2. (a) The eigenvalues are λ jk = j 2 + k 2 4 π 2 for j, k =1 , 2 ,.... One expression for the eigenfunction is v jk =sin( ( x +1) / 2) sin( ( y +1) / 2) . This is not unique, since some of the eigenvalues are multiple. So, for example, λ 12 = λ 21 , and any function av 12 + bv 21 , for arbitrary scalars a and b , is an eigenfunction. Even for simple eigenvalues, the function v jj can be multiplied by an arbitrary constant, positive or negative. The Frst six v jk are plotted in ±igure 31.1, and it is an interesting exercise to describe them in words.
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Unformatted text preview: Note that as the eigenvalue increases, the number of oscillations in the eigenfunction increases. In order to capture this behavior in a piecewise linear approximation, we need a Fner mesh for the eigenfunctions corresponding to larger eigenvalues than we do for those corresponding to smaller eigenvalues. (b) When using piecewise linear Fnite elements, the j th computed eigenvalue lies in an interval [ λ j , λ j + C j h 2 ], where h is the mesh size used in the triangulation. This is observed in our computation using the program problem1b.m , found on the website. The error plots are shown in ±igure 31.2. The horizontal axis is the 187...
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This note was uploaded on 01/21/2012 for the course MAP 3302 taught by Professor Dr.robin during the Fall '11 term at University of Florida.

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