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Unformatted text preview: with variable coefficients: ∂u ∂ =x ∂t ∂x x ∂u ∂x − 3u, u(1, t) = u(eπ , t) = 0, u(x, 0) = 3 sin(log x) + 7 sin(2 log x) − 2 sin(3 log x). 4.8 Numerical solutions to the eigenvalue problem* We can also apply Sturm-Liouville theory to study the motion of a string of variable mass density. We can imagine a violin string stretched out along the x-axis with endpoints at x = 0 and x = 1 covered with a layer of varnish which causes its mass density to vary from point to point. We could let ρ(x) = the mass density of the string at x for 0 ≤ x ≤ 1. If the string is under constant tension T , its motion might be governed by the partial differential equation ∂2u T ∂2u = , ∂t2 ρ(x) ∂x2 (4.34) which would be subject to the Dirichlet boundary conditions u(0, t) = 0 = u(1, t), for all t ≥ 0. (4.35) It is natural to try to find the general solution to (4.34) and (4.35) by separation of variables, letting u(x, t) = f (x)g (t) as usual. Substituting into (4.34) yields T g (t) T f...
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This document was uploaded on 01/12/2014.

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