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Unformatted text preview: CAAM 336 DIFFERENTIAL EQUATIONS Problem Set 5 Posted Thursday 23 September 2010. Corrected 28 Sept 2010. Due Wednesday 29 September 2010, 5pm. All of the problems on this set use the inner product ( u,v ) = Z 1 u ( x ) v ( x ) dx. 1. [30 points] Consider the linear operator L b : C 2 b [0 , 1] C [0 , 1] defined by L b u = d 2 u dx 2 , where C 2 b [0 , 1] = n u C 2 [0 , 1] : du dx (0) = u (1) = 0 o . (a) Is L b symmetric? (b) What is the null space of L b ? That is, find all u C 2 b [0 , 1] such that L b u ( x ) = 0 for all x [0 , 1]. (c) Show that ( L b u,u ) 0 for all nonzero u C 2 b [0 , 1] and explain why this implies that 0 for all eigenvalues . (d) Find the eigenvalues and eigenfunctions of L b . 2. [35 points] Consider the operator L D : C 2 D [0 , 1] C [0 , 1] defined by L D u = d 2 u dx 2 , with homogeneous Dirichlet boundary conditions imposed by the domain C 2 D [0 , 1] = { u C 2 [0 , 1] : u (0) = u (1) = 0 } . Recall that the eigenvalues of L D are n = n 2 2 with associated normalized eigenfunctions n ( x ) = 2sin( nx ) , n = 1 , 2 ,.......
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This note was uploaded on 01/21/2012 for the course CAAM 330 taught by Professor Tompson during the Fall '09 term at UVA.
 Fall '09
 Tompson

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