s2_95c

# 1 r u r q r t where q r t is the internal heat

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+ 1 r ∂u ∂r = Q ( r, t ) where Q ( r, t ) is the internal heat production produced by radioactive decay. We wish to determine the temperature distribution in the bar for t > 0. (a) Show that the relevent Sturm-Lioville problem to be solved in order to determine the Green’s function for the solution of the heat transfer equation is x d 2 φ n dx 2 + n dx + λ x φ n = 0 where x = r/a , φ n = φ n ( x ), with φ n (0) bounded and φ n (1) = 0. (b) If Q ( r, t ) = 0 and u ( r, t = 0) = T 0 , where T 0 is constant, show that the temperature distribution in the bar for t > 0 is u ( r, t ) = n =1 a n exp - γ 2 n κ t a 2 J 0 ( γ n r/a ) where a n = 2 T 0 γ n J 1 ( γ n ) To evaluate integrals you may use the following standard results 1 z d dz [ z J 1 ( z )] = J 0 ( z ) , γ n 0 z J 2 0 ( z ) dz = γ 2 n 2 J 2 1 ( γ n ) . Due April 16, 3:00pm.
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