On An Illustrative Examples of a Studied Noetherity Dirac-Delta Extensions for a Noether Operator

Abstract

The purpose of this work is to illustrate by clear examples the noetherity nature of a finite Dirac-delta Extensions of a studied noether operator. Previously in our published papers, we have investigated in different two cases, the noetherization of a Dirac-delta extensions of a noether linear integro-differential operator defined by a third kind integral equation in some specific well chosen functional spaces. Our various already published researches were connected with such topic widely studied and clearly presenting different specific approaches, applied when establishing fundamentaly noether theory for some kind of integro-differential operators to reach the noetherization. The initial considered noether operator <I>A</I> has been extended with some finite dimensional spaces of Dirac-delta functions, and the noetherization of the two cases of extensions has been established depending with the parameters of the third kind integral equation defining <I>A</I>. The previous lead us to set the problem of the construction of practical examples clearly illustrating the relationship between theory and practise. For this aim, we based on an established wellknown noether theory and, we construct in this work step by step, two illustrative examples to show the interconnexion between the theory and pratise related to the investigation of the construction of noether theory for the considered extended noether operator denoted <img width="8" height="10" src="http://article.sciencepublishinggroup.com/journal/347/3471192/image001.png" /> defined by a third kind linear singular integral equation in some generalized functional spaces. The extended operator <I>A</I> of the initial noether operator <I>A</I> is verified being also noether and therefore we deduce the index of the extended operator <img width="8" height="10" src="http://article.sciencepublishinggroup.com/journal/347/3471192/image001.png" />.