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Homework 7 - BMED 2200 Modeling Bio medical Systems I...

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BMED 2200 Modeling Biomedical Systems I Spring 2008 Homework 7 Solution (1) First derivative of y in terms of forward finite differences with ) ( 3 h O We want ) ( 3 h O at the first derivative so m=3 and n=1 ) ( ) ( ) ( 4 3 1 h O h O h O m n = = + + , so we need 3 equations ) ( ! 3 ) ( 2 ) ( ) ( 4 3 1 2 1 1 1 h O y x x y x x y x x y y i i i i i i i i i i i + ! ! ! " + ! ! " + ! " + = + + + + ) ( ! 3 ) ( 2 ) ( ) ( 4 3 2 2 2 2 2 h O y x x y x x y x x y y i i i i i i i i i i i + ! ! ! " + ! ! " + ! " + = + + + + ) ( ! 3 ) ( 2 ) ( ) ( 4 3 3 2 3 3 3 h O y x x y x x y x x y y i i i i i i i i i i i + ! ! ! " + ! ! " + ! " + = + + + + where, h x x i i = ! + ) ( 1 , h x x i i 2 ) ( 2 = ! + , h x x i i 3 ) ( 3 = ! + , substituting we get: ) ( ! 3 2 4 3 2 1 h O y h y h y h y y i i i i i + ! ! ! + ! ! + ! + = + ) ( ! 3 ) 2 ( 2 ) 2 ( 2 4 3 2 2 h O y h y h y h y y i i i i i + ! ! ! + ! ! + ! + = + ) ( ! 3 ) 3 ( 2 ) 3 ( ) 3 ( 4 3 2 3 h O y h y h y h y y i i i i i + ! ! ! + ! ! + ! + = + Need a linear combination that will eliminate i y ! ! and i y ! ! ! + ! ! + + + ! + + + + + = + + + + + i i i i i i y h C B A y h C B A y C B A Cy By Ay 2 ) 9 4 ( ) 3 2 ( ) ( 2 3 2 1 ) ( ! 3 ) 27 8 ( 4 3 h O y h C B A i + ! ! ! + +

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Since we want to eliminate i y ! ! and i y ! ! ! set the coefficients of these equal to zero: 0 9 4 = + + C B A 0 27 8 = + + C B A Let C=1, and solve to find A=9 and B=-9/2 Plug this back into the linear combination equation ) ( ! 3 ) 0 ( 2 ) 0 ( )) 1 ( 3 ) 2 9 )( 2 ( 9 ( ) 1 2 9 9 ( 2 9 9 4 3 2 3 2 1 h O y h
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