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Unformatted text preview: Solutions to Test 2 Test 2 Introduction to PDE MATH 3363-25820 (Fall 2009) This exam has 2 questions, for a total of 20 points. Please answer the questions in the spaces provided on the question sheets. If you run out of room for an answer, continue on the back of the page. Upon finishing PLEASE write and sign your pledge below: On my honor I have neither given nor received any aid on this exam. 1 Rules You may only use pencils, pens, erasers, and straight edges. No calculators, notes, books or other aides are permitted. Scrap paper will be provided. Be sure to show a few key intermediate steps when deriving results - answers only will not get full marks. 2 Given The solution to the 1d wave equation u tt = c 2 u xx ,-∞ < x < ∞ , t > , subject to the ICs u ( x, 0) = f ( x ) ,-∞ < x < ∞ , u t ( x, 0) = g ( x ) ,-∞ < x < ∞ , is u ( x, t ) = F ( x- ct ) + G ( x + ct ) where G ( x ) = 1 2 f ( x ) + 1 2 c Z x g ( s ) ds F ( x ) = 1 2 f ( x )- 1 2 c Z x g ( s ) ds Page 1 of 8 Please go to the next page. . . Solutions to Test 2 3 Questions 1. 8 points Suppose that an “infinite string” is initially streched into the shape of a single rectangular pulse and is let go from the rest. We model the problem using the 1D wave equation u tt = u xx ,-∞ < x < ∞ , t > , subject to the initial conditions u ( x, 0) = f ( x ) = 1 , | x | ≤ 1 , | x | > 1 = , x <- 1 1 ,- 1 ≤ x ≤ 1 , x > 1 u t ( x, 0) = 0 . (a) Plot the four characteristics: x- t =- 1, x + t =- 1, x- t = 1 and x + t = 1 in the space-time plane ( xt ). Show that these four characteristics divide the space-time plane ( xt ) into six distinct regions. In each region, show that the solution u is constant, and give its value....
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- Spring '08
- Math, Constant of integration, Picard–Lindelöf theorem, Solutoin, space-time plane