527ass13 - the rod). (b) In each case above, nd the...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
642:527 ASSIGNMENT 13 FALL 2009 No problems from this assignment will be collected. Section 19.2: 5, 6, 8 Section 19.4: 4, 6 (a), (c) 13.A Consider the following problem for the function u ( x, t ): 9 u xx = u t , 0 < x < 1 , t > 0; (1.1) u (0 , t ) = 0 , γu (1 , t ) + u x (1 , t ) = 0 , t > 0; (1.2) u ( x, 0) = 1 , 0 < x < 1 . (1.3) (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 γ > - 1 (although in the text example the Robin boundary condition is imposed at the left, not the right, end of
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: the rod). (b) In each case above, nd the solution of the problem as an innite series. Express the coecients as ratios of integrals, but do not attempt to evaluate them. The series and integrals will involve the eigenvalues from (a), so you wont be able to be too specic. (c) Discuss the behavior of u ( x, t ) as t . You should nd, in the various cases of (a), that (i) u ( x, t ) approaches zero as t ; (ii) u ( x, t ) approaches a non-zero steady state as t ; (iii) u ( x, t ) becomes innite (blows up) as t . (d) What is the physical interpretation of the boundary condition at x = L when > 1, and why, on physical grounds, does the solution blow up in that case?...
View Full Document

This note was uploaded on 12/15/2009 for the course MATH 527 taught by Professor Staff during the Fall '08 term at Rutgers.

Ask a homework question - tutors are online