- Method
- References
- External links
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.[1]
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 
representing boundary conditions and inhomogeneities, 
representing the field of interest, and 
the integral operator describing how Y is given from X based on the physics of the problem.
Next, is split into 
, where 
is invertible and contains all the singularities of and 
is regular. After splitting the operator and multiplying by the inverse of , the equation becomes
or
which is now a Fredholm equation of the second type because by construction 
is compact on the Hilbert space of which 
is a member.
In general, several choices for 
will be possible for each problem.[1]
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: [https://dx.doi.org/10.1088/0305-4470/39/21/S73 10.1088/0305-4470/39/21/S73.]
- 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=Kleinert| first1=H.| last2=Schulte-Frohlinde| first2=V. | title=Critical Properties of φ4-Theories| url=http://www.worldscibooks.com/physics/4733.html|isbn= 978-981-02-4659-4| date=2001| pages=1–474}}, Paperpack {{ISBN|978-981-02-4659-4}} (also available 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]
- Yu. A. Tuchkin Analytical Regularization Method for Wave Diffraction by Bowl-Shaped Screen of Revolution
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