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Unformatted text preview: CAAM 336 Â· DIFFERENTIAL EQUATIONS Problem Set 9 Posted Thursday 28 October 2010. Due Wednesday 3 November 2010, 5pm. 1. [40 points] (a) Consider the function u ( x ) = 1 , x âˆˆ [0 , 1 / 3]; , x âˆˆ (1 / 3 , 2 / 3); 1 , x âˆˆ [2 / 3 , 1] . Recall that the eigenvalues of the operator L : C 2 N [0 , 1] â†’ C [0 , 1], Lu = u 00 are Î» n = n 2 Ï€ 2 for n = 0 , 1 ,... with associated (normalized) eigenfunctions Ïˆ ( x ) = 1 and Ïˆ n ( x ) = âˆš 2cos( nÏ€x ) , n = 1 , 2 ,.... We wish to write u ( x ) as a series of the form u ( x ) = âˆž X n =0 a n (0) Ïˆ n ( x ) , where a n (0) = ( u ,Ïˆ n ). Compute these inner products a n (0) = ( u ,Ïˆ n ) by hand and simplify as much as possible. For m = 0 , 2 , 4 , 80, plot the partial sums u ,m ( x ) = m X n =0 a n (0) Ïˆ n ( x ) . (You may superimpose these on one single, welllabeled plot if you like.) (b) Write down a series solution to the homogeneous heat equation u t ( x,t ) = u xx ( x,t ) , < x < 1 , t â‰¥ with Neumann boundary conditions...
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 Fall '09
 Tompson
 Summation, Boundary value problem, Boundary conditions, Neumann boundary condition

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