Analytical regularization

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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 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 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[edit]

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^{-1}  G_2 X=  G_1^{-1} Y

or

 X + A X =  B

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

In general, several choices for \mathbf{G}_1 will be possible for each problem.[1]

References[edit]

  1. ^ a b 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

External links[edit]