Integral Operators Acting as Variables of the Matrix Polynomial‎: ‎Application to System of Integral Equations

Abstract

‎The aim of this work is to clarify a new viewpoint of connection between system of integral‎ ‎equations and matrix polynomials‎. ‎A procedure is described for transforming a‎ ‎linear system of integral equations to an independent‎ ‎system‎. ‎The latter is converted to equalities by considering its‎ ‎equivalent matrix polynomial equation which employs the integral operator as its variable‎, ‎and admits a normal form‎ ‎for simplifying the system‎. ‎We will show that under certain suitable‎ ‎conditions‎, ‎an independent reduced system is obtained‎, ‎which can be shown to have the same unknowns as the main system‎, ‎and‎ ‎has only one unknown in each equation‎. ‎In fact‎, ‎the basic idea enables us to develop a methodology to solve general systems of linear integral equations‎.