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.^[1]

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

${\displaystyle GX=Y}$

with ${\displaystyle Y}$ representing boundary conditions and inhomogeneities, ${\displaystyle X}$ representing the field of interest, and ${\displaystyle G}$ the integral operator describing how Y is given from X based on the physics of the problem. Next, ${\displaystyle G}$ is split into ${\displaystyle G_{1}+G_{2}}$, where ${\displaystyle G_{1}}$ is invertible and contains all the singularities of ${\displaystyle G}$ and ${\displaystyle G_{2}}$ is regular. After splitting the operator and multiplying by the inverse of ${\displaystyle G_{1}}$, the equation becomes

${\displaystyle X+G_{1}^{-1}G_{2}X=G_{1}^{-1}Y}$

or

${\displaystyle X+AX=B}$

which is now a Fredholm equation of the second type because by construction ${\displaystyle A}$ is compact on the Hilbert space of which ${\displaystyle B}$ is a member.

In general, several choices for ${\displaystyle \mathbf {G} _{1}}$ will be possible for each problem.^[1]

• Santos, F C; Tort, A C; Elizalde, E (10 May 2006). "Analytical regularization for confined quantum fields between parallel surfaces". Journal of Physics A: Mathematical and General. 39 (21). IOP Publishing: 6725–6732. arXiv:quant-ph/0511230. Bibcode:2006JPhA...39.6725S. doi:10.1088/0305-4470/39/21/s73. ISSN 0305-4470. S2CID 18855340.
• Panin, Sergey B.; Smith, Paul D.; Vinogradova, Elena D.; Tuchkin, Yury A.; Vinogradov, Sergey S. (5 January 2009). "Regularization of the Dirichlet Problem for Laplace's Equation: Surfaces of Revolution". Electromagnetics. 29 (1). Informa UK Limited: 53–76. doi:10.1080/02726340802529775. ISSN 0272-6343. S2CID 121978722.
• Kleinert, H.; Schulte-Frohlinde, V. (2001), Critical Properties of φ⁴-Theories, pp. 1–474, ISBN 978-981-02-4659-4, archived from the original on 2008-02-26, retrieved 2011-02-24, Paperpack ISBN 978-981-02-4659-4 (also available online). Read Chapter 8 for Analytic Regularization.

• E-Polarized Wave Scattering from Infinitely Thin and Finitely Width Strip Systems
• Tuchkin, Yu. A. (2002). "Analytical Regularization Method for Wave Diffraction by Bowl-Shaped Screen of Revolution". Ultra-Wideband, Short-Pulse Electromagnetics 5. Boston: Kluwer Academic Publishers. pp. 153–157. doi:10.1007/0-306-47948-6_18. ISBN 0-306-47338-0.