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Unformatted text preview: respectively, find the solution of (*) which satisfies the initial conditions u ( x , o ) = ~ ( x ) and u,(x,o)=y(x) forall  < x < m . b V $ & I M lo p. 40 k w e . 1 Bonus.(25 pts.) A homogeneous solid material occupying D = {(x, y, z) E B3 : 4 i x2 + y2 + z2 i 100) is completely insulated and its initial temperature at position (x, y, z ) in D is 200/,/. (a) Write (without proof or derivation) the partial differential equation and initialhoundary conditions that completely govern the temperature u (x, y, z,t ) at position (x, y, z) in D and time t 2 0. @) Use Gauss' divergence theorem to help show that the heat energy H (t) = IIIcpu (x, y, z, t)dv of D the material in D at time t is a constant function of time. Here c and p denote the (constant) specific heat and mass density, respectively, of the material in D. (c) Compute the (constant) steadystate temperature that the material in D reaches after a long time. 7....
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 Fall '10
 DavidFrow
 Math, Thermodynamics, Equations, Derivative, Vector Calculus, Partial differential equation

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