Noetherization Theory for a Singular Linear Differential Operator of Higher Order


Author: Abdourahman Haman Adji

Abstract: The main objective set in this research is the construction of noetherian theory for a singular linear integro-differential operator L defined by a linear singular differential equation of higher order in a specific functional space well chosen to achieve the goal. It should be emphasized that the case where $$n=1$$ has been completely studied in the two situations separately when $$p=1$$ and $$p \geq 2$$. Our previous various published research was related to this topic. The methodology adopted on a case-by-case basis, and depending on the values and sign of the parameter $$ γ \in \mathbb{R} $$, leads us to solve the linear differential equation studied with a well-known second specific right-hand side $$ f(x)\in C_0^{\lbrace p \rbrace} [-1,1] $$, systematically identifying the conditions solvency. One of the major difficulties arising in this work, apart from those related to the quantitative techniques of solving the differential equation defining the considered integrodifferential operator, is the construction of the continuity of the regularizers by starting from the smallest segment [0,1] to completely cover the whole closed interval [-1,1]. We rely on the approaches and methods built by the researcher Yurko V.A to achieve the noetherity of the considered operator. This takes us straight to the investigation and construction of the noetherity of the operator L. Finally, depending on each case, we evaluate and calculate the deficient numbers and the index of the operator considered in various situations, relative to the parameter $$ γ \in \mathbb{R} $$.

Pages: 1-10

DOI: 10.46300/91019.2024.11.1

International Journal of Pure Mathematics, E-ISSN: 2313-0571, Volume 11, 2024, Art. #1

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