hw2Jan2007

hw2Jan2007 - CAAM 336 DIFFERENTIAL EQUATIONS Problem Set 2...

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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- 2 t/ ( c ) sin( x ) for constant , , c , and . (a) Show that this function u ( x, t ) is a solution of the homogeneous heat equation c u t = 2 u x 2 , for 0 < x < and all t . (b) For which values of will u satisfy homogeneous Dirichlet boundary conditions at x = 0 and x = ? (c) Suppose = 0 . 80 W/(cm K), = 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 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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