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Math Help - Solving Diffusion PDE

  1. #1
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    Solving Diffusion PDE

    Dear all,

    I'm trying to solve the diffusion PDE for my system, shown below:

     \frac{\partial C}{\partial t} = D (\frac{\partial^2 C}{\partial r^2} + \frac{1}{r} \frac{\partial C}{\partial r})

    where C is the concentration, changing with time t and radius r. D is the diffusion coefficient.

    I'm solving this using seperation of variables, giving me two ODE.

    One of the ODE is:

     \frac{d^2R}{dr^2} + \frac{1}{r} \frac{dR}{dr} + lamda^2 R = 0

    where lamba is a constant.

    Does any one have any ideas on how to solve this ODE tro obtain a general solution? I tried using a change in variables.

    Any help would be appreciated. Thanks in advance!

    Regards,

    Billy
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  2. #2
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    First multiply through by r^2 to get r^2\frac{d^2R}{dr^2}+ r\frac{dR}{dr}+ \lambda^2r^2 R= 0. That's Bessel's equation of order 0. Look up "Bessel's function".
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  3. #3
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    Thanks for the quicky reply HallsofIvy.

    I'm not too sure how i am to solve Bessel's function. Could you give me an example?

    Cheers.
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  4. #4
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    Also, am i solving the Bessel function for integer v, or non integer v? As of now, i do not have a value for my lamda.
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  5. #5
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    Well, I can't give an entire course on Bessel's equation. Doesn't your text book deal with it? Here is a link to the Wikipedia article:
    Bessel function - Wikipedia, the free encyclopedia

    Typically, you determine what values the eigenvalue can take by looking at the boundary conditions.
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  6. #6
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    Thanks again for your direction.

    After going through the derivation, i know what Jo(x) is, which is for Bessel's equation of order 0. So is it true to say that R(r) = Jo(x)? Or is it more complicated than this?
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  7. #7
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    Quote Originally Posted by wenzhe2092 View Post
    Thanks again for your direction.

    After going through the derivation, i know what Jo(x) is, which is for Bessel's equation of order 0. So is it true to say that R(r) = Jo(x)? Or is it more complicated than this?
    There's actually a second solution Y_0 but it has a singularity at r = 0 so yes, you are right.
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  8. #8
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    Quote Originally Posted by wenzhe2092 View Post
    Thanks for the quicky reply HallsofIvy.

    I'm not too sure how i am to solve Bessel's function. Could you give me an example?

    Cheers.
    Quote Originally Posted by Danny View Post
    There's actually a second solution Y_0 but it has a singularity at r = 0 so yes, you are right.
    Thanks for the reply. Just to clarify, why is it that we can just get rid of Y_0? What do you mean by singularity?
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  9. #9
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    Update:

    So far i have established my principal solution for my ODE, as shown below:

     <br />
u(r,t)=\sum_{n=1}^{\infty}A_n \cdot J_0(\lambda_n^(0.5) r) exp (-\lambda_n Dt)
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  10. #10
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    Update:

    So far i have established my principal solution for my ODE, as shown below:

     <br />
c(r,t)=\sum_{n=1}^{\infty}A_n \cdot J_0(\lambda_n^{0.5} r) exp (-\lambda_n Dt)

    Now to solve it. The initial condition is given as c(r,0) = 0.0398 mol/m^3 for a<r<b where a and b are the inner and outer radius respectively.

    Any tips on how to apply this condition?

    Next, the boundary condition is such that:

    c= o at r = b
    c= 0.433 at r = a

    Any help appreciated! Thanks.
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  11. #11
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    bump
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