Preface

**Chapter 5 63**

Electro-magnetic Simulation Based on the Integral Form of

Maxwell's Equations *by Naofumi Kitsunezaki*

**II**

Today, research into integral equations is fundamental to the increasing development of functional analysis, function theory, functional equations, and other areas in mathematics. Integral equations can solve problems of any nature in dynamical systems, partial differential equations, representation theory and its realizations through integral transforms, integration on infinite dimension spaces and their operators, algebra, inverse problems in control theory, and system analysis. This book includes five chapters written by prestigious experts and researchers who present their work on integral equations and related themes considering the wide content of concepts and methodologies. Some of these important aspects are the functional resolution of integral equations, operator theories to nonsymmetric and symmetric kernels, extension problems of Banach algebras to kernels of integral equations, and probabilistic methods to integral equations among others. The book has been divided into five sections to facilitate the search for research information.

> **Dr. Francisco Bulnes** Professor, Research Department in Mathematics and Engineering, TESCHA, Mexico IINAMEI, Mexico

Section 1

Introduction

1

Section 1 Introduction

Chapter 1

Francisco Bulnes

world is one country"

1. General discussion

Introductory Chapter: Frontier

Research on Integral Equations

"Mathematics knows no races or geographic boundaries; for mathematics, the cultural

The themes of recent research are focused on nonlinear integral equations [1], the new numerical and adaptive methods of resolution of integral equations [2], the generalization of Fredholm integral equations [3] of second kind, integral equations in time scales and the spectral densities [3, 4], operator theories for nonsymmetric and symmetric kernels [1, 5], extension problems to Banach algebras to kernels of integral equations [5–7], singular integral equations [10], special treatments to solve Fredholm integral equations of first and second kinds, nondegenerate kernels [3, 6] and symbols of integral equations [7], topological methods for the resolution of integral equations and representation problems of operators of integral equations. Now, well, the field of the integral equations is not finished yet, not much less with the integral equations for which the Fredholm theorem is worth [fredholm], nor with the completely continuous operators, since there exist other integral equations developed of the Hilbert theory respect to the Fredholm discussion, and studies on singular integral equations, also by Hilbert, Wiener and others [8]. Arise numerical and approximate methods on the big vastness that give the Banach algebras, even using probabilistic measures to solve some integral equations in the ambit of distributions and stochastic process. Likewise, there arise integral equations in which the proper values are corresponded to linearly independent infinite proper functions. Such is the


and Recent Results

case, for example, of the Lalesco-Picard integral equation:

ωðÞ�t λ

in which the kernel e�j j <sup>t</sup>�<sup>s</sup> is not of L<sup>2</sup>

ple, to singular integral equations.

3

ðþ<sup>∞</sup> �∞ e

even, we consider nonlinear integral equations, etc., that represent the last and recent studies on integral equations after of their study considering extensions of the Banach algebras to integral operators that can define to this proposit, for exam-

Likewise, as special case, for their important theory, we can treat the singular integral equations of Cauchy. This theory was created almost immediately after the Fredholm theory, and their beginning is given in the "Lecons de Mécanique

�j j <sup>t</sup>�<sup>s</sup> <sup>ω</sup>ð Þ<sup>s</sup> ds <sup>¼</sup> f tð Þ, (1)

class and gives a continuous spectra, or
