# Analytical regularization

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In [physics](/source/physics) and [applied mathematics](/source/applied_mathematics), '''analytical regularization''' is a technique used to convert [boundary value problem](/source/boundary_value_problem)s which can be written as [Fredholm integral equation](/source/Fredholm_integral_equation)s of the first kind involving [singular operators](/source/singular_integral_operator) into equivalent Fredholm integral equations of the second kind. The latter may be easier to solve analytically and can be studied with [discretization](/source/discretization) schemes like the [finite element method](/source/finite_element_method) or the [finite difference method](/source/finite_difference_method) because they are [pointwise convergent](/source/pointwise_convergence). In [computational electromagnetics](/source/computational_electromagnetics), it is known as the '''method of analytical regularization'''. It was first used in mathematics during the development of [operator theory](/source/operator_theory) before acquiring a name.<ref name=nosich>{{cite journal | last=Nosich | first=A.I. | title=The method of analytical regularization in wave-scattering and eigenvalue problems: foundations and review of solutions | journal=IEEE Antennas and Propagation Magazine | publisher=Institute of Electrical and Electronics Engineers (IEEE) | volume=41 | issue=3 | year=1999 | issn=1045-9243 | doi=10.1109/74.775246 | pages=34–49| bibcode=1999IAPM...41...34N }}</ref>

== 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
:<math> G X= Y </math>
with <math> Y </math> representing boundary conditions and [inhomogeneities](/source/Inhomogeneous_differential_equation), <math> X </math> representing the field of interest, and <math> G </math> the integral operator describing how Y is given from X based on the physics of the problem.  
Next, <math> G </math> is split into  <math>G_1 + G_2</math>, where  <math>G_1</math> is invertible and contains all the singularities of <math>G </math> and <math> G_2</math> is regular. After splitting the operator and multiplying by the inverse of  <math> G_1 </math>, the equation becomes
:<math> X + G_1^{-1}  G_2 X=  G_1^{-1} Y </math>
or 
:<math> X + A X =  B </math>
which is now a Fredholm equation of the second type because by construction <math> A </math> is [compact](/source/compact_operator) on the [Hilbert space](/source/Hilbert_space) of which <math> B </math> is a member.

In general, several choices for <math>\mathbf{G}_1</math> will be possible for each problem.<ref name=nosich />

==See also==
* [Method of moments (electromagnetics)](/source/Method_of_moments_(electromagnetics))

== References ==
{{Reflist}}
* {{cite journal | last1=Santos | first1=F C | last2=Tort | first2=A C | last3=Elizalde | first3=E | title=Analytical regularization for confined quantum fields between parallel surfaces | journal=Journal of Physics A: Mathematical and General | publisher=IOP Publishing | volume=39 | issue=21 | date=10 May 2006 | issn=0305-4470 | doi=10.1088/0305-4470/39/21/s73 | pages=6725–6732| arxiv=quant-ph/0511230 | bibcode=2006JPhA...39.6725S | s2cid=18855340 }}
* {{cite journal | last1=Panin | first1=Sergey B. | last2=Smith | first2=Paul D. | last3=Vinogradova | first3=Elena D. | last4=Tuchkin | first4=Yury A. | last5=Vinogradov | first5=Sergey S. | title=Regularization of the Dirichlet Problem for Laplace's Equation: Surfaces of Revolution | journal=Electromagnetics | publisher=Informa UK Limited | volume=29 | issue=1 | date=5 January 2009 | issn=0272-6343 | doi=10.1080/02726340802529775 | pages=53–76| s2cid=121978722 }}
*{{Citation| last1=Kleinert| first1=H.| last2=Schulte-Frohlinde| first2=V.| title=Critical Properties of φ<sup>4</sup>-Theories| url=http://www.worldscibooks.com/physics/4733.html| isbn=978-981-02-4659-4| date=2001| pages=1–474| access-date=2011-02-24| archive-url=https://web.archive.org/web/20080226151023/http://www.worldscibooks.com/physics/4733.html| archive-date=2008-02-26| url-status=dead| author1-link=Hagen Kleinert}}, Paperpack {{ISBN|978-981-02-4659-4}} (also available [http://users.physik.fu-berlin.de/~kleinert/kleinert/?p=booklist&details=6 online]). Read Chapter 8 for Analytic Regularization.

== External links ==
* [https://web.archive.org/web/20110416110353/http://www.circuitechs.com/electromagnatics/electromagnetics.html E-Polarized Wave Scattering from Infinitely Thin and Finitely Width Strip Systems]
* {{cite book | last=Tuchkin | first=Yu. A. | title=Ultra-Wideband, Short-Pulse Electromagnetics 5 | chapter=Analytical Regularization Method for Wave Diffraction by Bowl-Shaped Screen of Revolution | year=2002 | publisher=Kluwer Academic Publishers | location=Boston | isbn=0-306-47338-0 | doi=10.1007/0-306-47948-6_18 | pages=153–157}}

{{DEFAULTSORT:Analytical Regularization}}
Category:Diffraction
Category:Electromagnetism
Category:Applied mathematics
Category:Computational electromagnetics
Category:Fredholm theory

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