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# Homework 8 - 1 From the parabolic PDE governing equation!u...

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± .PEFMJOH #JPNFEJDBM 4ZTUFNT * 4QSJOH ²³³´ )PNFXPSL ´ 4PMVUJPO 1. From the parabolic PDE governing equation: 2 2 x u t u ! ! = ! ! " for L x ! ! 0 ············ (1) Approximation of the second-order spatial and first-order time derivative using a central difference formula is ) ( ) 2 ( ) ( 1 2 , 1 , , 1 2 , 2 2 x O u u u x x u n i n i n i n i ! + + " ! = # # " + ············ (2) ) ( ) ( ) ( 2 1 2 1 , 1 , , t O u u t t u n i n i n i ! + " ! = # # " + ············ (3) Substitute Eqs.(2)&(3) into the governing parabolic PDE, Eq(1), ) ( ) 2 ( ) ( 1 ) ( ) ( ) ( 2 1 2 , 1 , , 1 2 2 1 , 1 , x O u u u x t O u u t n i n i n i n i n i ! + " " # \$ % % + ( ! = ! + ( ! ( + ( + ) Rearranging and solving for the highest point in time and in the form: ) ( 2 2 , 1 , , 1 1 , t x O Cu Bu Au u n i n i n i n i ! + ! + + " = " + + We get ) ( ) ( 2 ) ( 4 ) ( 2 2 2 , 1 2 , 2 , 1 2 1 , 1 , t x O u x t u x t u x t u u n i n i n i n i n i ! + ! + ! ! + ! ! " ! ! + = " + " + # Stability requires that all coefficients are greater than zero (A>0, B>0, etc.) Stability: Unstable, because B will always be less than zero. 2. The flow and migration of human leukocytes on a substrate in chapter8 of text is:

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± z C v z C t C eff D ! ! " ! ! = ! ! 2 2 μ ············ (1) Solve this Eq numerically by expressing the partial derivative into finite difference approximation with second-order spatial and first-order time derivatives using a central difference formula is ) ( ) ( 2 1 2 , 1 , 1 , z O C C z z C n i n i n i ! + " ! = # # " + ············ (2) ) ( ) 2 ( 1 2 , 1 , , 1 2 , 2 2 z O C C C z z C n i n i n i n i ! + + " ! = # # " + ············ (3) Solve the time derivative into forward finite difference: ) ( ) ( 1 2 , 1 , , t O C C t t C n i n i n i ! + " ! = # # + Combine the above Eqs(2)&(3) into Eq (1), and rearrange to solve for the highest point in time and in the form ) ( 2 2 , 1 , , 1 1 , t x O Cu Bu Au u n i n i n i n i ! + ! + + " = " + + We get: !
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• Spring '08
• RobertSpilker
• Partial differential equation, Concentration Profiles, Therapeutic Concentration Profiles, central difference formula

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Homework 8 - 1 From the parabolic PDE governing equation!u...

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