How do I solve the 3D magnetic field of a Halbach Rotor?

[Fieldly]

Homework Statement

I'm currently trying to equate two functions represented by unequal Fourier Bessel series within a specific region. The coefficients have to be independent of any variables, as their dependency would violate the properties of the Poisson or Laplace equations.

I tried to use eigen decomposition, which requires that the functions be self-adjoint, which is contingent upon satisfying Robin boundary conditions. The eigenvalues must also be consistent for both axial and radial directions, as dictated by the separation of variables technique. In the analysis, the eigenvalue
k_n=2nπ/Wr
was selected, which ensures natural orthogonality in the axial direction. However, this choice leads to singular behaviour in the radial direction Bessel functions, resulting in a lack of self-adjointness. Consequently, there is no orthogonality in the region of interest, preventing the separation of coefficients. Is the separation of variables approach ineffective in this scenario? Would it be advisable to consider any alternative methods, such as Green's functions?

Relevant Equations

k_n=2nπ/Wr

Boundary conditions:

The derived Dirichlet and Neumann boundary conditions

In terms of the magnetic scalar potential:

 

[Fieldly]

As an add-on, can I adapt complex eigenvalues and use techniques like the Wiener-Hopf technique?