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

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 A 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 A. 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 defined by a third kind linear singular integral equation in some generalized functional spaces. The extended operator A of the initial noether operator A is verified being also noether and therefore we deduce the index of the extended operator .

Noether Theory, Noetherization, Third Kind Integral Equation, Singular Linear Integro-Differential Operator, Deficient Numbers, Index of the Operator

APA Style

Abdourahman, Ecclésiaste Tompé Weimbapou, Emmanuel Kengne, Shankishvili Lamara Dmitrievna. (2023). On An Illustrative Examples of a Studied Noetherity Dirac-Delta Extensions for a Noether Operator. International Journal of Theoretical and Applied Mathematics, 8(6), 121-127. https://doi.org/10.11648/j.ijtam.20220806.12

ACS Style

Abdourahman; Ecclésiaste Tompé Weimbapou; Emmanuel Kengne; Shankishvili Lamara Dmitrievna. On An Illustrative Examples of a Studied Noetherity Dirac-Delta Extensions for a Noether Operator. Int. J. Theor. Appl. Math. 2023, 8(6), 121-127. doi: 10.11648/j.ijtam.20220806.12

AMA Style

Abdourahman, Ecclésiaste Tompé Weimbapou, Emmanuel Kengne, Shankishvili Lamara Dmitrievna. On An Illustrative Examples of a Studied Noetherity Dirac-Delta Extensions for a Noether Operator. Int J Theor Appl Math. 2023;8(6):121-127. doi: 10.11648/j.ijtam.20220806.12

Copyright © 2022 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Ferziger J. H., Kaper H. G. Mathematical theory of Transport Processes in Gases (North-Holland Publ. Company, Amsterdam–London, 1972).
2. Hilbert D. Grundzüge einer allgemeinen Theorie der linear Integralgleichungen (Chelsea Publ. Company, New York, 1953).
3. Picard E. “Un théorème général sur certaines équations intégrales de troisième espèce”, Comptes Rendus 150, 489–491 (1910).
4. Bart G. R. “Three theorems on third kind linear integral equations”, J. Math. Anal. Appl. 79, 48–57 (1981).
5. Bart G. R., Warnock R. L. “Linear integral equations of the third kind”, SIAM J. Math. Anal. 4, 609–622 (1973).
6. Sukavanam N. “A Fredholm-Type theory for third kind linear integral equations”, J. Math. Analysis Appl. 100, 478–484 (1984).
7. Shulaia D. “On one Fredholm integral equation of third kind”, Georgian Math. J. 4, 464–476 (1997).
8. Shulaia D. “Solution of a linear integral equation of third kind”, Georgian Math. J. 9, 179–196 (2002).
9. Shulaia D. “Integral equations of third kind for the case of piecewise monotone coefficients”, Transactions of A. Razmadze Math. Institute 171, 396–410 (2017).
10. Rogozhin V. S., Raslambekov S. N. “Noether theory of integral equations of the third kind in the space of continuous and generalized functions”, Soviet Math. (Iz. VUZ) 23 (1), 48–53 (1979).
11. Abdourahman A. On a linear integral equation of the third kind with a singular differential operator inthe main part (Rostov-na-Donu, deposited in VINITI, Moscow, 28.03.2002, No. 560-B2002).
12. Abdourahman A., Karapetiants N. “Noether theory for third kind linear integral equation with a singular linear differential operator in the main part”, Proceedings of A. Razmadze Math. Institute 135, 1–26 (2004).
13. Gobbassov N. S. On direct methods of the solutions of Fredholm’s integral equations in the space of generalized functions, PhD thesis (Kazan, 1987).
14. Gobbassov N. S. “Methods for Solving an Integral Equation of the Third Kind with Fixed Singularities in the Kernel”, Diff. Equ. 45, 1341–1348 (2009).
15. Gobbassov N. S. “A Special Version of the Collocation Method for Integral Equations of the Third Kind”, Diff. Equ. 41, 1768–1774 (2005).
16. Gobbassov N. S. Metody Resheniya integral’nykh uravnenii Fredgol’ma v prostranstvakh obobshchennykh funktsii (Methods for Solving Fredholm Integral Equations in Spaces of Distributions) (Izd-vo Kazan. Un-ta, Kazan, 2006) [in Russian].
17. Karapetiants N. S., Samko S. G. Equations with Involutive Operators (Birkhauser, Boston–Basel–Berlin, 2001).
18. Prossdorf S. Some classes of singular equations (Mir, Moscow, 1979) [in Russian].
19. Bart G. R., Warnock R. L. “Solutions of a nonlinear integral equation for high energy scattering. III. Analyticity of solutions in a parameter explored numerically”, J. Math. Phys. 13, 1896–1902 (1972).
20. Bart G. R., Johnson P. W., Warnock R. L., “Continuum ambiguity in the construction of unitary analytic amplitudes from fixed-energy scattering data”, J. Math. Phys. 14, 1558–1565 (1973).
21. E. Tompé Weimbapou1*, Abdourahman1**, and E. Kengne2***. «On Delta-Extension for a Noether Operator». ISSN 1066-369X, Russian Mathematics, 2021, Vol. 65, No. 11, pp. 34–45. c Allerton Press, Inc., 2021. Russian Text c The Author (s), 2021, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2021, No. 11, pp. 40–53.
22. Rogozhin V. S. Noether theory of operators. 2nd edition. Rostov-na- Donu: Izdat. Rostov Univ., 1982. 99 p.
23. Abdourahman. "Linear integral equation of the third kind with a singular differential operator in the main part." Ph.D Thesis. Rostov State University. 142 Pages. 2003. [in Russian].
24. Abdourahman. "Integral equation of the third kind with singularity in the main part." Abstracts of reports, International conference. Analytic methods of analysis and differential equations. AMADE. 15-19th of February 2001, Minsk Belarus. Page 13.
25. Abdourahman. «On a linear integral equation of the third kind with singularity in the main part». Abstract of reports. International School-seminar in Geometry and analysis dedicated to the 90th year N. V Efimov. Abrao-Diurso. Rostov State university, 5-11th September 2000. pp 86-87.
26. Duduchava R. V. Singular integral equations in the Holder spaces with weight. I. Holder coefficients. Mathematics Researches. T. V, 2nd Edition. (1970) Pp 104-124.
27. Tsalyuk Z. B. Volterra Integral Equations//Itogi Nauki i Techniki. Mathematical analysis. V. 15. Moscow: VINITI AN SSSR. P. 131-199.
28. Abdourahman, Ecclésiaste Tompé Weimbapou, Emmanuel Kengne. Noetherity of a Dirac Delta-Extension for a Noether Operator. International Journal of Theoretical and Applied Mathematics. Vol. 8, No. 3, 2022, pp. 51-57. doi: 10.11648/j.ijtam.20220803.11.
29. Shulaia. D. A SOLUTION OF A LINEAR INTEGRAL EQUATION OF THIRD KIND. Georgian Mathematical Journal. Volume 9 (2002), Number 1, 179 -196.
30. Yurko V. A. Integral transforms connected with differential operators having singularities inside the interval//Integral transforms and special functions.1997. V.5 N° 3-4 P.309-322.
31. Yurko V. A. On a differential operators of higher order with singularities inside the interval. Kratkie sochenie// Mathematicheskie Zamietkie, 2002. T.71, N° 1 P152-156.
32. Abdourahman. Construction of Noether Theory for a Singular Linear Differential Operator. International Journal of Innovative Research in Sciences and Engineering Studies (IJIRSES) www.ijirses.com ISSN: 2583-1658 | Volume: 2 Issue: 7 | 2022.
33. Abdourahman. Noetherization of a Singular Linear Differential Operator. International Journal of Innovative Research in Sciences and Engineering Studies (IJIRSES) www.ijirses.com ISSN: 2583-1658 | Volume: 2 Issue: 12 | 2022 © 2022, IJIRSES Page 9-17.