Regularization of electromagnetic scattering problems via the Abel integral transform

The Abel integral transform is a powerful mathematical tool for solving mixed boundary value problems for the Helmholtz and Maxwell equations. It is particularly effective for treating two- and three-dimensional electromagnetic wave scattering from cavity backed apertures. Such scattering problems give rise to dual, triple (and higher order) series and integral equations. These equations are inherently ill-posed and discretization results in ill-conditioned systems that resist stable numerical solution. Their regularization commences by representing the basis functions occurring in the equations in terms of Jacobi polynomials of a particular class. A sequence of Abel integral transforms is applied to each member of the series equations producing Jacobi polynomials of a different class. The transforms are arranged so that the resulting system is well-posed and may be converted to a well-conditioned Fredholm matrix system of second kind. Its numerical solution provides stable and convergent results of guaranteed accuracy. This paper discusses the treatment of three typical examples of dual and triple series arising in electromagnetic wave scattering from ideally conducting arbitrary slotted cylinders and axisymmetric thin-walled shells with one or two apertures. These are among the examples most commonly encountered in scattering problems of this nature.This article is part of the theme issue 'Analytically grounded full-wave methods for advances in computational electromagnetics'.