- #1

Daniel Sellers

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- TL;DR Summary
- I have found general solutions to a PDE which I need help applying Dirichlet conditions to so I can plot a final solution.

I have a PDE which I have solved numerically using a guass-seidel method, but I want to compare it to the analytical solution. I have used separation of variables to get the general solution, but I need help applying it.

The PDE is

(1/s)⋅(∂/∂s)[(s/ρ)(∂ψ/∂s)] + (1/s2)⋅(∂/∂Φ)[(1/ρ)(∂ψ/∂Φ)] - 2Ω + ρ(c0 + c1ψ) = 0

and applying separation of variables gives (I think):

ψ = (1/ρc

Where ρ, Ω, c

I am not sure what to do from here. I want to apply Dirichlet boundary conditions on an inner boundary s

The domain is a polar annulus, so k must be an integer, but that's all I can readily figure out.

The PDE is

(1/s)⋅(∂/∂s)[(s/ρ)(∂ψ/∂s)] + (1/s2)⋅(∂/∂Φ)[(1/ρ)(∂ψ/∂Φ)] - 2Ω + ρ(c0 + c1ψ) = 0

and applying separation of variables gives (I think):

ψ = (1/ρc

_{1})⋅[AJ_{k}(ρs√c_{1}) + BY_{k}(ρs√c_{1})]⋅[Ccos(kΦ) + Dsin(kΦ)] + (2Ω/ρc_{1}) + (c_{0}/c_{1})Where ρ, Ω, c

_{0}, c_{1}are known constants.I am not sure what to do from here. I want to apply Dirichlet boundary conditions on an inner boundary s

_{0}and outer boundary s_{1}. Should these be series solutions? Like summations over k =1,2 ... and if so then how do I handle the coefficients (A,B,C,D)?The domain is a polar annulus, so k must be an integer, but that's all I can readily figure out.