527ass12

527ass12 - 642:527 ASSIGNMENT 12: REVISED FALL 2007...

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642:527 ASSIGNMENT 12: REVISED FALL 2007 Multiple-page homework must be STAPLED when handed in. Turn in starred problems Thursday 12/6/2007. Problems marked with two stars will be treated as extra credit.problems Section 17.7: 8 Section 18.3: 10 (a), (b), (c)*, 14, 19*, 28**, 29* 12.A ** Consider the following problem for the function u ( x, t ): 9 u xx = u t , 0 < x < 1 , t > 0; (12.1) u (0 , t ) = 0 , γu (1 , t ) + u x (1 , t ) = 0 , t > 0; (12.2) u ( x, 0) = 1 , 0 < x < 1 . (12.3) (a) Separate variables and investigate the eigenvalues of the resulting Sturm-Liouville problem. In particular, show that (i) if γ > - 1 then all eigenvalues are positive, (ii) if γ = - 1 then zero is an eigenvalue and all other eigenvalues are positive, and (iii) if γ < - 1 then there is one negative eigenvalue and all other eigenvalues are positive. You will not be able to ±nd the eigenvalues analytically. NOTE: We discussed the case γ = - 1 in class, and Example 3 of Section 17.7 of our text is a model for the case
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This note was uploaded on 09/29/2009 for the course 642 527 taught by Professor Speer during the Fall '07 term at Rutgers.

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