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quiz3f07

# quiz3f07 - 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 7, 2007 Professor Strang Your PRINTED name is: Grading 1 2 3 ** NOTE AT NOON A BIG CHEMISTRY CLASS IS COMING !!! 1) (30 pts.) (a) Solve by a Fourier sine series u ( x ) = b k sin kx : 1 0 < x < π u �� + 4 u ( x ) = f ( x ) = with u ( π ) = u ( π ) = 0 . 1 π < x < 0 That right side f ( x ) is the square wave SW( x ) on page 318. (b) What is the decay rate of the coeﬃcients b k ? What is the smoothness of u ( x ) which derivative jumps ? 1
1 3 1 x n 1 + x n + x = y < n < 5 5 5 n +1 n − ∞ 2) (30 pts.) This problem is about the equation (a) Suppose the vector x = ( . . . , x 1 , x 0 , x 1 , . . . ) is known. The equation is a non-cyclic convolution a x = y . What is the infinite vector a ? Transform the equation into the frequency domain using X ( ω ) = x n e inω and Y ( ω ) and A ( ω ). What is A ( ω ) in this problem ?

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