quiz1_18085_f06

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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 1 October 2, 2006 Professor Strang Your PRINTED name is: Grading 1 2 3 1) (36 pts.) (a) Suppose u ( x ) is linear on each side of x = 0, with slopes u ( x ) = A on the left and u ( x ) = B on the right: Ax Bx for x 0 u ( x ) = for x 0 What is the second derivative u �� ( x ) ? Give the answer at every x . (b) Take discrete values U n at all the whole numbers x = n : U n = An for n 0 Bn for n 0 For each n , what is the second difference 2 U n ? Using coeﬃcients 1 , 2 , 1 (notice signs !) give the answer 2 U n at every n . (c) Solve the differential equation u �� ( x ) = ( x ) from x = 2 to x = 3 with boundary values u ( 2) = 0 and u (3) = 0. (d) Approximate problem (c) by a difference equation with h = � x = 1. What is the matrix in the equation K U = F ? What is the solution U ?
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