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Unformatted text preview: CAAM 336 Â· 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] â†’ C [0 , 1] defined by L b u = d 2 u dx 2 , where C 2 b [0 , 1] = n u âˆˆ 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 âˆˆ C 2 b [0 , 1] such that L b u ( x ) = 0 for all x âˆˆ [0 , 1]. (c) Show that ( L b u,u ) â‰¥ 0 for all nonzero u âˆˆ C 2 b [0 , 1] and explain why this implies that Î» â‰¥ 0 for all eigenvalues Î» . (d) Find the eigenvalues and eigenfunctions of L b . 2. [35 points] Consider the operator L D : C 2 D [0 , 1] â†’ 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 âˆˆ C 2 [0 , 1] : u (0) = u (1) = 0 } . Recall that the eigenvalues of L D are Î» n = n 2 Ï€ 2 with associated normalized eigenfunctions Ïˆ n ( x ) = âˆš 2sin( nÏ€x ) , n = 1 , 2 ,.......
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 Fall '09
 Tompson
 Boundary conditions, Dirichlet boundary condition, homogeneous Dirichlet boundary, Johann Peter Gustav Lejeune Dirichlet, Ïˆn

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