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q3sols18085f04

q3sols18085f04 - MIT OpenCourseWare http/ocw.mit.edu 18.085...

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MIT OpenCourseWare http://ocw.mit.edu 18.085 Computational Science and Engineering I Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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18.085 Quiz 3 December 2, 2004 Professor Strang Your name is: SOLUTIONS Grading 1. 2. 3. Thank you for taking 18.085 ! I hope to see you in 18.086 !! 1) (40 pts.) This question is about 2 π -periodic functions . d l e ilx (a) Suppose f ( x ) = c k e ikx and g ( x ) = . Substitute for f and g and integrate to find the coeﬃcients q n in this convolution: 2 π 2 π h ( x ) = f ( t ) g ( x t ) dt = f ( x t ) g ( t ) dt = q n e inx . 0 0 (b) Compute the coeﬃcients c k for the function 1 for 0 x 1 f ( x ) = 0 for 1 x 2 π 2 ? What is the decay rate of these c k ? What is | c k | (c) Keep that f ( x ) in parts (c)–(d). If g ( x ) also has a jump, will the convolution h ( x ) have a jump ? Compare the decay rates of the d ’s and q ’s to find the behavior of h ( x ): delta function, jump, corner, or what ? (d) Find the derivative dh/dx at x = 0 in terms of two values of g ( x ). (You could take the x derivative in the convolution integral.)
Solution 1. (a) This is really proving the convolution rule (periodic case).

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