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s3_95c

# s3_95c - ACM 95/100c Problem Set III 0(2 pts Write down...

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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 L 0 Q ( x ) dx = 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 u ( x, y ) and v ( r, θ ) be two functions which, for x = r cos( θ ) and y = r sin( θ ), satisfy u ( x, y ) = v ( r, θ ) . (2) (a) (10 pts) Show that, for x = r cos( θ ) and y = r sin( θ ) we have ∂u ∂x ( x, y ) = cos( θ ) ∂v ∂r ( r, θ ) - sin( θ ) r ∂v ∂θ ( r, θ ) (3) and

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s3_95c - ACM 95/100c Problem Set III 0(2 pts Write down...

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