physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...

and

applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemat ...

, analytical regularization is a technique used to convert

boundary value problem In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to ...

s which can be written as

Fredholm integral equation In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The integral equation was studied by Ivar Fredholm. A useful method to ...

s 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 In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numeri ...

schemes like the

finite element method The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...

or the

finite difference method In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are dis ...

because they are pointwise convergent. In

computational electromagnetics Computational electromagnetics (CEM), computational electrodynamics or electromagnetic modeling is the process of modeling the interaction of electromagnetic fields with physical objects and the environment. It typically involves using computer ...

, it is known as the method of analytical regularization. It was first used in mathematics during the development of

operator theory In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear oper ...

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 :

G X= Y

with

Y

representing boundary conditions and inhomogeneities,

X

representing the field of interest, and

G

the integral operator describing how Y is given from X based on the physics of the problem. Next,

G

is split into

G_1 + G_2

, where

G_1

is invertible and contains all the singularities of

G

and

G_2

is regular. After splitting the operator and multiplying by the inverse of

G_1

, the equation becomes :

X + G_1^  G_2 X=  G_1^ Y

or :

X + A X =  B

which is now a Fredholm equation of the second type because by construction

A

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

on the

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...

of which

B

is a member. In general, several choices for

\mathbf_1

will be possible for each problem.

References

* * *, Paperpack (also availabl
online
. Read Chapter 8 for Analytic Regularization.

External links

* {{DEFAULTSORT:Analytical Regularization Diffraction Electromagnetism Applied mathematics Computational electromagnetics