Numerical expansion-iterative method for analysis of integral equation models arising in one- and two-dimensional electromagnetic scattering

• Published in: Engineering analysis with boundary elements 2012-03, Vol.36 (3), p.416-422
• Main Authors: Hatamzadeh-Varmazyar, Saeed, Masouri, Zahra
• Format: Article
• Language: English
• Subjects: Algebra; Applied classical electromagnetism; Block-pulse functions; Electromagnetic wave propagation, radiowave propagation; Electromagnetism; electron and ion optics; Exact sciences and technology; Expansion method; First kind Fredholm integral equation; Fundamental areas of phenomenology (including applications); Ill posed problems; Integral equations; Inversions; Iterative method; Linear and multilinear algebra, matrix theory; Linear systems; Mathematical analysis; Mathematical models; Mathematics; One- and two-dimensional electromagnetic scattering; Perfectly conducting bodies; Physics; Sciences and techniques of general use; Solid dynamics (ballistics, collision, multibody system, stabilization...); Solid mechanics; Two dimensional; Vector forms
• ISSN: 0955-7997; 1873-197X
• DOI: 10.1016/j.enganabound.2011.09.008

Most integral equations of the first kind are ill-posed, and obtaining their numerical solution needs often to solve a linear system of algebraic equations of large condition number. So, solving this system may be difficult or impossible. Since many problems in one- and two-dimensional scattering from perfectly conducting bodies can be modeled by Fredholm integral equations of the first kind, this paper presents an effective numerical expansion-iterative method for solving them. This method is based on vector forms of block-pulse functions. By using this approach, solving the first kind integral equation reduces to solve a recurrence relation. The approximate solution is most easily produced iteratively via the recurrence relation. Therefore, computing the numerical solution does not need to directly solve any linear system of algebraic equations and to use any matrix inversion. Also, the method practically transforms solving of the first kind Fredholm integral equation which is inherently ill-posed into solving second kind Fredholm integral equation. Another advantage is low cost of setting up the equations without applying any projection method such as collocation, Galerkin, etc. To show convergence and stability of the method, some computable error bounds are obtained. Test problems are provided to illustrate its accuracy and computational efficiency, and some practical one- and two-dimensional scatterers are analyzed by it.