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In [[physics]] and [[applied mathematics]], '''analytical regularization''' is a technique used to convert [[boundary value problem]]s which can be written as [[Fredholm integral equation]]s of the first kind involving [[singular integral operator | singular operators]] into equivalent Fredholm integral equations of the second kind which may be easier to solve analytically and which can be studied with [[discretization]] schemes like the [[finite element method]] or the [[finite difference method]] because they are [[pointwise convergence | 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.<ref name=nosich> Nosich, Alexander I. 'The Method of Analytic Regularization in Wave-Scattering and Eigenvalue Problems: Foundations and Review of Solutions' ''IEEE Antennas and Propagation Magazine''. Vol 41, No. 3. June 1999 </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 [[Inhomogeneous differential equation | inhomogeneities]], <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 operator | compact]] on the [[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 /> 

== References ==
<references />
* Analytical regularization for confined quantum fields between parallel surfaces F C Santos et al. 2006 J. Phys. A: Math. Gen. 39 6725-6732 doi: [http://dx.doi.org/10.1088/0305-4470/39/21/S73 10.1088/0305-4470/39/21/S73.]
* [http://www.informaworld.com/smpp/content~content=a907381504~db=all~jumptype=rss Regularization of the Dirichlet Problem for Laplace's Equation: Surfaces of Revolution ] Authors: Sergey B. Panin ab; Paul D. Smith a; Elena D. Vinogradova a; Yury A. Tuchkin bc; Sergey S. Vinogradov d
*{{Citation | last1=[[Hagen Kleinert|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| year=2001| pages=1–474}}, 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 ==
* [http://www.circuitechs.com/electromagnatics/electromagnetics.html E-Polarized Wave Scattering from Infinitely Thin and Finitely Width Strip Systems]
* [http://www.springerlink.com/content/q6hj6637nv0815r5/ Yu. A. Tuchkin Analytical Regularization Method for Wave Diffraction by Bowl-Shaped Screen of Revolution]

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[[Category:Diffraction]]
[[Category:Electromagnetism]]