Note that the heat equation itself and the boundary

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Unformatted text preview: unit volume, then the rate at which heat leaves Dx1 ,x2 is − x2 ρ(x)σ (x) x1 ∂u (x, t)dx + ∂t x2 (µ(x)u(x, t) + ν (x))dx.) x1 On the other hand, the rate of heat flow F (x, t) is proportional to the partial derivative of temperature, F (x, t) = −κ(x) ∂u (x, t), ∂x (4.2) where κ(x) is the thermal conductivity of the bar at x. Thus we find that the rate at which heat leaves the region Dx1 ,x2 is also given by the formula F (x2 , t) − F (x1 , t) = x2 x1 ∂F (x, t)dx. ∂x (4.3) 1 Further reading can be found in the many excellent upper-division texts on partial differential equations. We especially recommend Mark Pinsky, Partial differential equations and boundary-value problems with applications , 2nd edition, McGraw-Hill, 1991. An excellent but much more advanced book is Michael Taylor, Partial differential equations: basic theory , Springer, New York, 1996. 82 Comparing the two formulae (4.1) and (4.3), we find that x2 x1 ∂F (x, t)dx = − ∂x x2 ρ(x)σ (x) x1 ∂u (x, t)dx. ∂t This equation is true for all choices of x1 and x2 , so the in...
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