hw3(1) - u versus x for t = 1 2 a/c t = a/c t = 3 2 a/c t =...

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Math 425 Dr. DeTurck Assignment 3 Due Tuesday, February 9, 2010 Reading: Textbook, Chapter 1, skim through Chapter 2. 1. Take a moment to review the divergence theorem from vector calculus, then work the following problem: Suppose V ( x,y,z ) is a vector-valued function defined everywhere in 3-dimensional space. Further, suppose that V is differentiable and that k V ( x,y,z ) k ≤ 1 1 + ( x 2 + y 2 + z 2 ) 3 / 2 for all ( x,y,z ). Show that Z -∞ Z -∞ Z -∞ ∇ · V ( x,y,z ) dxdy dz = 0 . (Hint: Apply the divergence theorem on a ball of large radius) 2. Solve the wave equation (for an infinite string) u tt = c 2 u xx with initial conditions u ( x, 0) = ln(1 + x 2 ) and u t ( x, 0) = 4 + x . 3. (The dulcimer) Solve the wave equation u tt = c 2 u xx with initial conditions u ( x, 0) = 0 and u t ( x, 0) = g ( x ), where g ( x ) = 1 if | x | < a and g ( x ) = 0 for | x | ≥ a . This corresponds to hitting the string with a hammer of width 2 a . Draw sketches of snapshots of the string (i.e., plot
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Unformatted text preview: u versus x ) for t = 1 2 a/c , t = a/c , t = 3 2 a/c , t = 2 a/c and t = 5 2 a/c . 4. In the preceding problem, for each value of t , determine the maximum value of u ( x,t ) (your pictures should help!). 5. Solve u xx + u xt-20 u tt = 0, with initial conditions u ( x, 0) = φ ( x ), u t ( x, 0) = ψ ( x ). (“Factor” the operator the way we did for the wave equation.) 6. Show that, for any function u ( x,t ) that satisfies the wave equation u tt = u xx (with c = 1), we have u ( x + h,t + k ) + u ( x-h,t-k ) = u ( x + k,t + h ) + u ( x-k,t-h ) for all x , t , h , k . 7. Solve u tt = 9 u xx on 0 < x < π/ 2, with u ( x, 0) = cos x , u t ( x, 0) = 0, u x (0 ,t ) = 0, u ( π/ 2 ,t ) = 0. 8. Textbook page 17, problems 3, 10; page 27, problems 2, 5....
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