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527ass13 - the rod(b In each case above ±nd the solution...

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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 find 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
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Unformatted text preview: the rod). (b) In each case above, ±nd the solution of the problem as an in±nite series. Express the coe²cients as ratios of integrals, but do not attempt to evaluate them. The series and integrals will involve the eigenvalues from (a), so you won’t be able to be too speci±c. (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 in±nite (“blows up”) as t →∞ . (d) What is the physical interpretation of the boundary condition at x = L when γ > 1, and why, on physical grounds, does the solution blow up in that case?...
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