In the next section we will see how to solve the

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Unformatted text preview: tionality being determined by the density and specific heat of the material making up the bar. More generally, if σ (x) denotes the specific heat at the point x and ρ(x) is the density of the bar at x, then the heat within the region Dx1 ,x2 between x1 and x2 is given by the formula x2 Heat within Dx1 ,x2 = ρ(x)σ (x)u(x, t)dx. x1 To calculate the rate of change of heat within Dx1 ,x2 with respect to time, we simply differentiate under the integral sign: x2 d dt x2 ρ(x)σ (x)u(x, t)dx = ρ(x)σ (x) x1 x1 ∂u (x, t)dx. ∂t Now heat is a form of energy, and conservation of energy implies that if no heat is being created or destroyed in Dx1 ,x2 , the rate of change of heat within Dx1 ,x2 is simply the rate at which heat enters Dx1 ,x2 . Hence the rate at which heat leaves Dx1 ,x2 is given by the exp x2 Rate at which heat leaves Dx1 ,x2 = − ρ(x)σ (x) x1 ∂u (x, t)dx. ∂t (4.1) (More generally, if heat is being created within Dx1 ,x2 , say by a chemical reaction, at the rate µ(x)u(x, t) + ν (x) per...
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This document was uploaded on 01/12/2014.

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