I have an EDP that I digitally solve on a 2D polar annular grid. I am trying to solve the axisymmetric analysis analytically in order to test my numerical solution.

The PDE:

$$ frac {1} {s} frac {} {∂s} ( frac {s} {ρ} frac {∂ψ} {} s}) + frac {1} {s ^ 2} ⋅ frac {∂} {} ( frac {1} {ρ} frac {∂ψ} {∂Φ}) – 2Ω + ρc_0 + ρc_1ψ = 0 $$

Or

$$ Ω, ρ, c_0, c_1 $$

are known constants (ρ is constant only in the axisymmetric case, normally depends on s and

Using the separation of variables, I think we get:

$$ = frac {1} {ρc_1} (AJ_k (ρs sqrt {c_1}) + BY_k (ρs sqrt {c_1})) (Ccos (kΦ) + Dsin (kΦ)) + frac {2Ω} { ρc_1} + frac {c_0} {c_1} $$

I do not know what to do here. I want to apply the boundary conditions of Dirichlet to an internal limit s0 and an external limit s1. Are these solutions in series? As the summations on k = 1,2 … and if so, how can I handle the coefficients (A, B, C, D)?

My instinct is to say that k = 0 for the dependent part to become a constant, but I'm not sure how to handle the bessel functions as this would limit them to only bessel functions of order 0. I am generally little experienced in the use of Bessel functions.

How should this problem be solved with only 2 boundary conditions (inner and outer rings of the domain)? I do not even try to apply neumann conditions yet.

Thank you!