Analytical solution of the second-order radial Heun-type ODE for the RDSW stream equation

Derive an analytical solution for the non-homogeneous Heun-type second-order ordinary differential equation that governs the second-order radial function R(x) in the Blandford–Znajek perturbative stream-equation expansion for the Rotating Damour–Solodukhin wormhole spacetime. The equation arises after separation of variables for the magnetic flux correction Y2(r,θ)=R2(r)Θ2(θ), features regular singular points at x=0,1,ε,∞ with ε=1+λ^2 (λ the wormhole deformation parameter), and the solution must be regular at the throat (x=1) and at spatial infinity.

Background

The authors solve the force-free Maxwell equations in the Rotating Damour–Solodukhin wormhole (RDSW) background using the Blandford–Znajek perturbative method. After showing that the first-order magnetic flux correction vanishes, they obtain at second order a separated form Y2(r,θ)=R2(r)Θ2(θ). The radial function R2(r), upon a change of variables, satisfies a non-homogeneous Heun-type ODE with singularities at x=0,1,ε,∞ (ε=1+λ^2).

Seeking a closed-form solution consistent with regularity at the throat and at infinity, the authors report that they could not find an analytical solution and therefore proceed numerically for selected values of the deformation parameter λ. An analytic solution would provide explicit control over the second-order magnetic flux correction and improve analytic estimates of the Poynting flux in RDSW spacetimes and their comparison with the Kerr case.