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hwk12 - (b Construct a set of basis functions Φ n r such...

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Physics 604 Homework #12 Fall ‘11 Dr. Drake 1. The temperature in a one-dimensional medium satisfies a diffusion equation ∂T ∂t - κ 2 ∂x 2 T = 0 with diffusion coefficient κ over the interval ( a, b ), where T ( a, t ) = T ( b, t ) = T 0 . At t = 0, T ( x, 0) = 0 for x ( a, b ). Solve for T ( x, t ) by expanding in a series of sin() and/or cos() functions. At late time what is the approximate (non-trivial) time dependence? Hint: Choose your basis functions to match your B.C.’s. 2. Consider a solid sphere of radius “ a ” in a world with four spatial dimensions . At t = 0 the sphere, with initial temperature of T 0 is immersed in a heat bath with T = 0. The temperature inside the sphere satisfies a diffusion equation. ∂T ∂t - κ 2 T = 0 where 2 = 1 r 3 ∂r r 3 ∂r . (a) Estimate how long it will take for the temperature of the center of the sphere to change significantly.
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Unformatted text preview: (b) Construct a set of basis functions Φ n ( r ) such that Φ n ( a ) = 0 which are appropriate for solving for T ( r,t ). Write the basis functions as a linear combination of the two solutions of Bessel’s equation Y ν ( kr ) and J ν ( kr ). Write down expressions for the eigenvalues of your basis functions and normalize Φ n ( r ) so that Z a dr w Φ 2 n ( r ) = 1 . You do not have to prove that the eigenfunctions are orthogonal, but state why you are conﬁdent that they are. Sketch the lowest three eigenfunc-tions. (c) Write the space/time dependence of T as T ( r,t ) = ∞ X n =1 c n ( t )Φ n ( r ) and solve for c n ( t ). At late time ﬁnd an approximate expression for T ( r,t ). How does T decay at late time?...
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