fd9e8b61-b92e-4aab-9637-f9a53c469b7720250516062643183naun:naunmdt@crossref.orgMDT DepositInternational Journal of Pure Mathematics2313-057110.46300/91019http://www.naun.org/cms.action?id=6985516202551620251210.46300/91019.2025.12https://npublications.com/journals/puremath/2025.phpAbdourahman HamanAdjiDepartment of Mathematics and Computer Sciences. University of Ngaoundere, Ngaoundere, CAMEROONShankishvili LamaraDmitrievnaDepartment of Mathematics, Georgian Technical University, Kostova Street, Tbilissi, GEORGIAThe objective set in this research work is the construction of an illustrative, precise, and concise example of the construction of the noetherian theory of an integro-differential operator defined by an integral equation of third kind in certain types of functional spaces well defined. The motivations for carrying out such work lie in the aim of linking theoretical results to those obtained in practice by finding all the aspects of the theory in the construction of the said noetherian theory obtained in my previous research, relative to an integro-differential operator defined by an integral equation of the third kind in an analytical manner with well-defined concrete parameters. The illustration of the construction of the noetherian theory through the investigation of an integro-differential operator, defined by an integral equation of the third kind in a well-chosen functional space is the focal point of the work to be undertaken following the obtaining of our recent research results widely published in various references. To achieve this, we link the theoretical recipes to the practical ones by applying the different concepts and notions of the noetherian theory to a particular case of an operator A concretely defined analytically. 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