# The eigenvalues of l are the constants for which 431

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Unformatted text preview: brating string, ∂2u ∂2u = , 2 ∂t ∂x2 u(0, t) = u(π, t) = 0, u(x, 0) = x − x4 , ∂u (x, 0) = 0. ∂t c. Construct a sequence of sketches of the positions of the vibrating string at the times ti = ih, where h = .1 by running the Mathematica program: vibstring = Table[ 102 Plot[ Sum[ b[n] Sin[n Pi x] Cos[n Pi t], {n,1,10}], {x,0,1}, PlotRange -> {-1,1} ], {t,0,1.,.1} d. Select the cell containing vibstring and animate the sequence of graphics by running “Animate selected graphics,” from the Cell menu. 4.6 Heat ﬂow in a circular wire The theory of Fourier series can also be used to solve the initial value problem for the heat equation in a circular wire of radius 1 which is insulated along the sides. In this case, we seek a function u(θ, t), deﬁned for θ ∈ R and t ≥ 0 such that 1. u(θ, t) satisﬁes the heat equation ∂u ∂2u = c2 2 , ∂t ∂θ (4.23) where c is a constant. 2. u(θ, t) is periodic of period 2π in the variable θ; in other words, u(θ + 2π, t) = u(θ, t), for all θ and t. 3. u(θ, t) satisﬁes the initial condition u(θ, 0) = h(θ),...
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## This document was uploaded on 01/12/2014.

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