Unformatted text preview: CAAM 336 · DIFFERENTIAL EQUATIONS Problem Set 2 Posted January 17. Due Wednesday January 24, 2007 in class. 1. [35 points] Consider the temperature function u ( x, t ) = e κθ 2 t/ ( ρc ) sin( θx ) for constant κ , ρ , c , and θ . (a) Show that this function u ( x, t ) is a solution of the homogeneous heat equation ρc ∂u ∂t = κ ∂ 2 u ∂x 2 , for 0 < x < and all t . (b) For which values of θ will u satisfy homogeneous Dirichlet boundary conditions at x = 0 and x = ? (c) Suppose κ = 0 . 80 W/(cm K), ρ = 7 . 6 g/cm 3 , and c = 0 . 45 J/(g K) (approximate values for iron found in the CRC Handbook of Chemistry and Physics , and on the web), and that the bar has length = 10 cm. Let θ be such that u ( x, t ) satisfies homogeneous Dirichlet boundary conditions as in part (b) and u ( x, t ) ≥ 0 for all x and t . Use MATLAB to plot the solution u ( x, t ) for 0 ≤ x ≤ and time 0 ≤ t ≤ 60 sec ....
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 Fall '09
 Tompson
 Thermodynamics, Energy, Heat, Dirichlet boundary conditions

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