This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: CAAM 336 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 [0 , 1 / 3]; , x (1 / 3 , 2 / 3); 1 , x [2 / 3 , 1] . Recall that the eigenvalues of the operator L : C 2 N [0 , 1] C [0 , 1], Lu = u 00 are n = n 2 2 for n = 0 , 1 ,... with associated (normalized) eigenfunctions ( x ) = 1 and n ( x ) = 2cos( nx ) , n = 1 , 2 ,.... We wish to write u ( x ) as a series of the form u ( x ) = X n =0 a n (0) n ( x ) , where a n (0) = ( u , n ). Compute these inner products a n (0) = ( u , 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) 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 with Neumann boundary conditions...
View
Full
Document
 Fall '09
 Tompson

Click to edit the document details