Solution methods for integral equations - a survey

Abstract

The theory of integral equations has been an active field of research for many years and is inextricably related with other areas of Mathematics such as complex and mathematical analysis, function theory, integral transforms and functional analysis. Integral Equations arise naturally in applications, in many areas of Mathematics, Engineering, Science and Technology and have been studied extensively both at the theoretical and practical level. It is significant to note that a MathSciNet keyword search on Integral Equations returns more than eleven thousand items. In this paper, we do a brief survey of the existing literature on methods of solving integral equations of Volterra and Fredholm type of the first, second and third kind, Cauchy type singular integral equations and integral equations over an infinite interval. The objective is to classify the selected methods and evaluate their applicability while discussing challenges faced by individual researchers in this field. We also provide a rather extensive bibliography for the reader who would be interested in learning more about various theoretical and computational aspects of Integral Equations.