- #1

shreddinglicks

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- TL;DR Summary
- Using Bessel functions to solve the heat equation for hollow cylinders.

I've been studying a few books on PDE's, specifically the heat equation. I have one book that covers this topic in cylindrical coordinates. All the examples are applied to a solid cylinder and result in a general Fourier Bessel series for 3 common cases that can be found easily with an online source.

Using separation of variables I have an equation,

J(0,alpha*r) + Y(0,alpha*r)

Then it says the 2nd term is eliminated due being bounded at r = 0. The boundary condition at the outer wall of the cylinder is then evaluated for alpha.

Afterwards I'm presented with the three Fourier Bessel solutions for the boundary conditions,

J'=0

hJ + alpha*b*J' = 0

and J = 0

I want to know how would I solve this problem if I had a hollow cylinder. I've attempted it on my own with poor results.

I assume in the hollow cylinder case Y(0,alpha*r) does not go to 0. How would I obtain my solution in this case?

Using separation of variables I have an equation,

J(0,alpha*r) + Y(0,alpha*r)

Then it says the 2nd term is eliminated due being bounded at r = 0. The boundary condition at the outer wall of the cylinder is then evaluated for alpha.

Afterwards I'm presented with the three Fourier Bessel solutions for the boundary conditions,

J'=0

hJ + alpha*b*J' = 0

and J = 0

I want to know how would I solve this problem if I had a hollow cylinder. I've attempted it on my own with poor results.

I assume in the hollow cylinder case Y(0,alpha*r) does not go to 0. How would I obtain my solution in this case?