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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- Fall '08
- Math, Eigenvalues, Boundary value problem, Sturm–Liouville theory, boundary condition, non-zero steady state