55555555 - x=0(boundary condition C p = C p,2 t>0...

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BIM 107 Winter 2011 Problem Set 5 (Partial Differential Equations) Due date: 3/7/11. 1. Consider the problem of diffusion of drugs through a planar polymeric membrane as shown below. The diffusion of drugs across the membrane is described by the equation ! C p ! t = D p " 2 C p , where C p is the concentration of the drug and D p is the diffusion coefficient of the drug within the polymer material. We are trying to solve the above PDE to calculate the concentration C p of the drug within the membrane as a function of time subject to the following boundary conditions: C p = C p,0 t=0 0<x<L (initial condition) C p = C p,1 t>0
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Unformatted text preview: x=0 (boundary condition) C p = C p,2 t>0 x=L (boundary condition) (a) (1 pt) Consider the diffusion of the drug in the x direction only. Show that the above diffusion equation takes the form ! 2 C p ! x 2 = 1 D p ! C p ! t . (b) (9 pts) Solve the equation in (a) using the method of separation of variables. Show that the general solution (using the principle of superposition) is given by C p = C p,1 + (C p,2 ! C p,1 ) x L + B n Sin( n " x L )e ! D p n 2 2 t L 2 n = 1 # $ n = 1,2, 3. ... Provide an equation that can be used to estimate the constants B n . C p,1 x=0 x=L C p,2 L x...
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This note was uploaded on 04/03/2011 for the course BIM 107 taught by Professor Raychaudhari during the Winter '11 term at UC Davis.

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