Analytical regularization

In physics and applied mathematics, analytical regularization is a technique used to convert boundary value problems which can be written as Fredholm integral equations of the first kind involving singular operators into equivalent Fredholm integral equations of the second kind. The latter may be easier to solve analytically and can be studied with discretization schemes like the finite element method or the finite difference method because they are pointwise convergent. In computational electromagnetics, it is known as the method of analytical regularization. It was first used in mathematics during the development of operator theory before acquiring a name.
Method.
Analytical regularization proceeds as follows. First, the boundary value problem is formulated as an integral equation. Written as an operator equation, this will take the form
with formula_2 representing boundary conditions and inhomogeneities, formula_3 representing the field of interest, and formula_4 the integral operator describing how Y is given from X based on the physics of the problem. 
Next, formula_4 is split into formula_6, where formula_7 is invertible and contains all the singularities of formula_8 and formula_9 is regular. After splitting the operator and multiplying by the inverse of formula_10, the equation becomes
or 
which is now a Fredholm equation of the second type because by construction formula_13 is compact on the Hilbert space of which formula_14 is a member.
In general, several choices for formula_15 will be possible for each problem.