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Math Help - Diffusion equation with cylindrical coordinates - Bessel function

  1. #1
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    Diffusion equation with cylindrical coordinates - Bessel function

    Hi,

    I'm trying to describe diffusion through a solid cylinder by following Crank's "The Mathematics of Diffusion". I more or less follow the method of adapting the diffusion equation for a cylinder, and using separation of variables to get the general equation. But when it comes time to introduce the boundary conditions we see Bessel's functions appear and this is where I get lost. I have never seen these before (I am not a mathematician!) and I was wondering if someone could give me some tips for how to use an equation like the one attached (sorry, I don't know how to write it in the post). Specifically, what are the alpha terms? Crank's book says they are the roots of the bessel function, but I feel like this means you go around in circles? Any help would be greatly appreciated.
    Attached Files Attached Files
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  2. #2
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    Re: Diffusion equation with cylindrical coordinates - Bessel function

    Hey hannah89.

    Can you post the derivation included in your book?
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  3. #3
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    Re: Diffusion equation with cylindrical coordinates - Bessel function

    Hi chiro,

    Thanks for your reply.

    I think the easiest way might be to look at this online version:

    The Mathematics of Diffusion

    and head to page 79, eqn 5.22.

    Thanks
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  4. #4
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    Re: Diffusion equation with cylindrical coordinates - Bessel function

    Unfortunately I don't think I have the experience to give advice on this subject: I'm not familiar enough with it.

    If I had to guess it would have to do with the roots of the function satisfying a zero that define the actual function (i.e. re-arranging all the terms to get a zero on the RHS).

    You can expand the Bessel functions using a series expansion or you can relate the Bessel function J1 to J0 (or vice versa).

    I'm sorry I can't help you out much more than this.
    Thanks from hannah89
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