# Homework Help: Bessel equation

1. May 15, 2005

### JohanL

Im trying to solve Helmholtz equation

$$\nabla ^2u(r,\phi,z) + k^2u(r,\phi,z) = 0$$

in a hollow cylinder with length L and a < r < b
and the boundary conditions:

$$u(a,\phi,z) = F(\phi,z)$$
$$u(b,\phi,z) = G(\phi,z)$$
$$u(r,\phi,0) = P(\phi,z)$$
$$u(r,\phi,L) = Q(\phi,z)$$
$$u(r,\phi,z) = u(r,\phi + 2\pi,z)$$

Solution:

With
$$u(r,\phi,z) = v_1(r,\phi,z) + v_2(r,\phi,z) + v_3(r,\phi,z)$$

i get three problems which i can solve separately.
Separation of variables gives 9 d.e. Three of them are bessel equations.

$$r\frac {d} {dr}(r\frac {dR_i(r)} {dr}) + (\mu_i^2r^2-m_i^2)R_i(r) = 0$$

i = 1,2,3. and $$\mu, m$$ are separation constants.

The boundary conditions are

$$R_1(a,\phi,z) = F(\phi,z), R_1(b,\phi,z) = G(\phi,z)$$

$$R_2(a,\phi,z) = 0, R_2(b,\phi,z) = 0$$

$$R_3(a,\phi,z) = 0, R_3(b,\phi,z) = 0$$

The general solutions of Bessels equation are

$$R = C_1 J_m(nr) + C_2 N_m(nr)$$

where J_m is the mth bessel function of the first kind and N_m is the mth neumann function (or bessel function of the second kind)

I dont know how to continue with the boundary conditions.
Any ideas?

2. May 15, 2005

### dextercioby

There are some theorems which under certain conditions allow you expand any function (namely the functions which appear as boundary conditions) in terms of Bessel functions.Note that these functions are not orthonormal polynomials (so no Hilbert space here),but that still doesn't prevent this from happening.

So i suggest you read more on the Bessel functions.Gray & Matthews wrote a monography.And there are tons of other useful books.

Daniel.