# The dirichlet boundary conditions 435 yield f 0 0

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Unformatted text preview: sin θ − cos 3θ, 4.7 Sturm-Liouville 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 x-axis 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.

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