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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.
 Winter '14
 Equations

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