- #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/ρc1)⋅[AJk(ρs√c1) + BYk(ρs√c1)]⋅[Ccos(kΦ) + Dsin(kΦ)] + (2Ω/ρc1) + (c0/c1)
Where ρ, Ω, c0, c1 are known constants.
I am not sure what to do from here. I want to apply Dirichlet boundary conditions on an inner boundary s0 and outer boundary s1. 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.
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/ρc1)⋅[AJk(ρs√c1) + BYk(ρs√c1)]⋅[Ccos(kΦ) + Dsin(kΦ)] + (2Ω/ρc1) + (c0/c1)
Where ρ, Ω, c0, c1 are known constants.
I am not sure what to do from here. I want to apply Dirichlet boundary conditions on an inner boundary s0 and outer boundary s1. 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.