# HW04 - solve the initial value problem 3 ∂ 2 u ∂t 2 = c...

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APPM 4350/5350: Fourier Series and Boundary Value Problems Homework #4 Monday October 11, 2010 Due: Monday, October 25, 2010 1. Solve: ∂T ∂t = κ 2 T ∂x 2 + e - αt sin 2 πx L , T ( x = 0 ,t ) = T ( x = L,t ) = 0 , T ( x,t = 0) = f ( x ) α > 0 , constant, κ > 0, constant, 0 < x < L , t > 0 ,f ( x ) is piecewise smooth. Hint: look for the solution as a sum of a forced (particular) and homogeneous solution. There is a solution for α 6 = α 0 and one when α = α 0 ; you need to ﬁnd α 0 . 2. Consider 2 u ∂t 2 = c 2 0 2 u ∂x 2 , 0 < x < L where c 2 0 > 0 constant and u x ( x = 0 ,t ) = 0 ,u x ( x = L,t ) = 0 ,u ( x,t = 0) = f ( x ) ,u t ( x, 0) = g ( x ); where f ( x ) ,g ( x ) are piecewise smooth. Find the normal modes and natural frequencies and
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Unformatted text preview: solve the initial value problem. 3. ∂ 2 u ∂t 2 = c 2 ∂ 2 u ∂x 2 , | x | < ∞ where c 2 > 0. Find the solution of the following problems: a) u ( x,t = 0) = sech( x-x ) ,u t ( x, 0) = 0; sketch the solution. b) u ( x,t = 0) = 0 ,u t ( x, 0) = U H ( x ) = { x < ,U x > } i.e. H ( x ) is the Heaviside function; sketch the solution. 4. 4.4.9 5. 4.4.10 a-d; in part a there is a small misprint; use u (0 ,t ) = 0 6. 4.4.12 7. 5.3.6 a-d 8. 5.3.7 9. 5.3.9 a-e 10. 5.4.4 1...
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