BP1 - 1.0 2 4 6 8 10(b Solve the “blowtorch” problem by...

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Bonus Problem #1 15 points, expires Wednesday 15 September Consider the heat equation problem whose initial condition is achieved by concentrating a unit of thermal energy at a single point on the rod. (Think of a blowtorch.) We begin with the approximate problem. (a) Given 0 < c < c + h < L , solve u t = ku xx on [0 ,L ] with boundary conditions u (0 ,t ) = 0 = u ( L,t ) and initial condition u ( x, 0) = 1 /h for c x c + h and 0 otherwise. Here’s the temperature evolution with c = 0 . 3 ,h = 0 . 1 ,L = 1 ,k = 1 2 Plot @8 u @ x, .001 D , u @ x, .01 D , u @ x, .04 D , u @ x, .1 D , u @ x, .2 D , u @ x, .3 D , If @ .3 < x < .4, 10, 0 D< , 8 x, 0, 1 < , PlotRange Ø 8 0, 11 <D 0.0 0.2 0.4 0.6 0.8
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Unformatted text preview: 1.0 2 4 6 8 10 (b) Solve the “blowtorch” problem by letting h → 0 in solution for part (a). You have my permission to take the limit inside the sum of the series. Fix a value of c and make a plot of the temperature distributions at several times. Submit a well-drawn sketch or attach printout. (c) Solve the “blowtorch” problem for the boundary conditions u (0 ,t ) = 0 = u x ( L,t ), keeping everything else the same, and make a plot as in part (b) to show how the temperature distribution evolves....
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This note was uploaded on 10/11/2010 for the course MATH 647 taught by Professor Stanislavova,m during the Spring '08 term at Kansas.

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