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# 10.6we - 10.6 OTHER HEAT CONDUCTION PROBLEMS KIAM HEONG KWA...

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10.6: OTHER HEAT CONDUCTION PROBLEMS KIAM HEONG KWA 1. Non-homogeneous Boundary Conditions Recall that the solution to the heat conduction problem α 2 u xx = u t , 0 < x < L, t > 0 , (1.1a) with the initial condition u ( x, 0) = f ( x ) , 0 x L, (1.1b) where f ( x ) and f 0 ( x ) are piecewise continuous, and the homogeneous boundary conditions u (0 ,t ) = 0 , u ( L,t ) = 0 , t > 0 , (1.1c) is given by u ( x,t ) = X n =1 c n e - α 2 n 2 π 2 t/L 2 sin nπx L , (1.2a) where c n = 2 L Z L 0 f ( x ) sin nπx L dx. (1.2b) Since f ( x ) is piecewise continuous in [0 ,L ], it is bounded. Hence so are c n , n = 1 , 2 , ··· , uniformly bounded in view of (1.2b). It follows that (1.3) lim t →∞ u ( x,t ) = 0 , 0 x L, because of the negative exponential factor in every term of the series (1.2a). Explicitly, let | f ( x ) | ≤ M 2 for all x [0 ,L ]. Then (1.4) | c n | ≤ 2 L Z L 0 M 2 ± ± ± sin nπx L ± ± ± dx = M Date : February 19, 2011. 1

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2 KIAM HEONG KWA for all n , n = 1 , 2 , ··· . So for any t > 0, (1.5) | u ( x,t ) | ≤ X n =1 Me - α 2 2 t/L 2 = Me - α 2 π 2 t/L 2 1 - e - α 2 π 2 t/L 2 for all x [0 ,L ]. Since lim t →∞ Me - α 2 π 2 t/L 2 1 - e - α 2 π 2 t/L 2 = 0, (1.3) follows. This allows us to solve the following heat conduction problem with
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10.6we - 10.6 OTHER HEAT CONDUCTION PROBLEMS KIAM HEONG KWA...

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