This chapter describes the principal ideas and more advanced techniques of the analytical regularization method (ARM) and the semi-inversion procedure (SIP) particularly as they apply to the integral and integro-differential equations that arise in scattering and diffraction problems in electromagnetics. The purpose of this chapter is to present a clear and unified treatment of the ARM and SIP approaches that draws on some recent developments using the language of pseudo-differential operators, while clarifying its rigorous application to boundary value problems (BVP) of electromagnetics.

Chapter Contents:

• 13.1 Introduction
• 13.2 Instability in the numerical solution of infinite algebraic systems
• 13.3 The ARM: when is it necessary?
• 13.4 Potentials and their pseudodifferential representations
• 13.5 Solution of the key diffraction problems
• 13.5.1 Dirichlet BVP
• 13.5.2 Neumann BVP
• 13.6 Diffraction by a semi-transparent obstacle
• 13.6.1 The BVP description
• 13.6.2 Integral representation for us(+) and ∂nus(+)
• 13.6.3 Reduction of the BVP to a system of integral equations
• 13.6.4 Reduction of the system of integral equations to an infinite system of linear algebraic equations
• 13.7 Diffraction of waves with complex frequencies and spectral theory of open cavities
• 13.7.1 Description of the BVP
• 13.7.2 Dirichlet BVP for complex-valued wave numbers
• 13.7.3 Qualitative features of the Dirichlet BVP
• 13.7.4 Numerical calculation of complex-valued eigen-wavenumbers and eigenmodes
• 13.8 ARM: considerations for implementation
• 13.9 ARM: various applications and conclusion
• References

Inspec keywords: boundary-value problems; integro-differential equations; electromagnetic wave diffraction; integral equations; electromagnetic wave scattering

Other keywords: electromagnetic wave diffraction; integro-differential equations; boundary value problems; electromagnetic wave scaterring; BVP; semiinversion procedure; ARM; integral equations; analytical regularization method; pseudodifferential operators

Subjects: Electromagnetic waves: theory; Numerical approximation and analysis; Integral equations (numerical analysis); Differential equations (numerical analysis); Electromagnetic wave propagation; Function theory, analysis