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Unformatted text preview: sin θ − cos 3θ, 4.7 SturmLiouville Theory* We would like to be able to analyze heat ﬂow in a bar even if the speciﬁc
heat σ (x), the density ρ(x) and the thermal conductivity κ(x) vary from point
to point. As we saw in Section 4.1, this leads to consideration of the partial
diﬀerential equation
∂u
1
∂
=
∂t
ρ(x)σ (x) ∂x κ(x) ∂u
∂x , (4.28) where we make the standing assumption that ρ(x), σ (x) and κ(x) are positive.
We imagine that the bar is situated along the xaxis with its endpoints
situated at x = a and x = b. As in the constant coeﬃcient case, we expect that
there should exist a unique function u(x, t), deﬁned for a ≤ x ≤ b and t ≥ 0
such that
1. u(x, t) satisﬁes the heat equation (4.28).
2. u(x, t) satisﬁes the boundary condition u(a, t) = u(b, t) = 0.
3. u(x, t) satisﬁes the initial condition u(x, 0) = h(x), where h(x) is a given
function, deﬁned for x ∈ [a, b], the initial temperature of the bar.
Just as before, we substitute u(x, t) = f (x)g (t) into (4.28) and obtain
f (x)g (t) = 1
d
ρ(x)σ (x) dx κ(x) df
(x) g (t).
dx...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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