s3_95c - ACM 95/100c April 16, 2004 Problem Set III 0.- (2...

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ACM 95/100c April 16, 2004 Problem Set III 0.- (2 pts) Write down your grading-section number. 1.- (40 pts) (a) Solve the initial value problem for the heat equation with time-dependent sources ± ∂u ∂t = k 2 u ∂x 2 + Q ( x, t ) u ( x, 0) = f ( x ) (1) subject to the following boundary conditions: (i) (5 pts) u (0 , t ) = A ( t ), ∂u ∂x ( L, t ) = 0 (ii) (5 pts) ∂u ∂x (0 , t ) = A ( t ), ∂u ∂x ( L, t ) = B ( t ) (iii) (5 pts) ∂u ∂x (0 , t ) = 0, ∂u ∂x ( L, t ) = 0 (iv) (5 pts) u (0 , t ) = 0, u ( L, t ) = 0. (b) (10 pts) Specialize part (iii) to the case Q ( x, t ) = Q ( x ) (independent of t ) such that R L 0 Q ( x ) dx 6 = 0. Show that, in this case, there are no time-independent solutions. What happens to the time-dependent solution as t → ∞ ? Explain! (c) (10 pts) Specialize part (iv) to the case Q ( x, t ) = Q ( x ) (independent of t ). Show that, in this case, the solution approaches a steady-state solution. Explain! 2.- (30 pts) Let
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s3_95c - ACM 95/100c April 16, 2004 Problem Set III 0.- (2...

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