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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.

Re: Diffusion equation with cylindrical coordinates - Bessel function

Hey hannah89.

Can you post the derivation included in your book?

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 :)

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.