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

Abdourahman; Ecclésiaste Tompé Weimbapou; Emmanuel Kengne; Shankishvili Lamara Dmitrievna

doi:doi:10.11648/j.ijtam.20220806.12

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

Abdourahman, Ecclésiaste Tompé Weimbapou, Emmanuel Kengne, Shankishvili Lamara Dmitrievna

Published in International Journal of Theoretical and Applied Mathematics (Volume 8, Issue 6)

Received: 21 December 2022 Accepted: 10 January 2023 Published: 17 January 2023

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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 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 .

Published in	International Journal of Theoretical and Applied Mathematics (Volume 8, Issue 6)
DOI	10.11648/j.ijtam.20220806.12
Page(s)	121-127
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

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

References

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[4]	Bart G. R. “Three theorems on third kind linear integral equations”, J. Math. Anal. Appl. 79, 48–57 (1981).
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[6]	Sukavanam N. “A Fredholm-Type theory for third kind linear integral equations”, J. Math. Analysis Appl. 100, 478–484 (1984).
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[8]	Shulaia D. “Solution of a linear integral equation of third kind”, Georgian Math. J. 9, 179–196 (2002).
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[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).
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[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.
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[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.
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[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.
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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

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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

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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

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@article{10.11648/j.ijtam.20220806.12,
  author = {Abdourahman and Ecclésiaste Tompé Weimbapou and Emmanuel Kengne and Shankishvili Lamara Dmitrievna},
  title = {On An Illustrative Examples of a Studied Noetherity Dirac-Delta Extensions for a Noether Operator},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {8},
  number = {6},
  pages = {121-127},
  doi = {10.11648/j.ijtam.20220806.12},
  url = {https://doi.org/10.11648/j.ijtam.20220806.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20220806.12},
  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 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 .},
 year = {2023}
}

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TY  - JOUR
T1  - On An Illustrative Examples of a Studied Noetherity Dirac-Delta Extensions for a Noether Operator
AU  - Abdourahman
AU  - Ecclésiaste Tompé Weimbapou
AU  - Emmanuel Kengne
AU  - Shankishvili Lamara Dmitrievna
Y1  - 2023/01/17
PY  - 2023
N1  - https://doi.org/10.11648/j.ijtam.20220806.12
DO  - 10.11648/j.ijtam.20220806.12
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 121
EP  - 127
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20220806.12
AB  - 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 .
VL  - 8
IS  - 6
ER  -

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Author Information

Abdourahman

Department of Mathematics, Higher Teachers’ Training College, University of Maroua, Maroua, Cameroon
Ecclésiaste Tompé Weimbapou

Department of Mathematics, Higher Teachers’ Training College, University of Maroua, Maroua, Cameroon
Emmanuel Kengne

School of Physics and Electronic Information Engineering, Zhejiang Normal University, Jinhua, China
Shankishvili Lamara Dmitrievna

Department of Mathematics, Georgian Technical University, Tbilissi, Georgia

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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

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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

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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

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@article{10.11648/j.ijtam.20220806.12,
  author = {Abdourahman and Ecclésiaste Tompé Weimbapou and Emmanuel Kengne and Shankishvili Lamara Dmitrievna},
  title = {On An Illustrative Examples of a Studied Noetherity Dirac-Delta Extensions for a Noether Operator},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {8},
  number = {6},
  pages = {121-127},
  doi = {10.11648/j.ijtam.20220806.12},
  url = {https://doi.org/10.11648/j.ijtam.20220806.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20220806.12},
  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 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 .},
 year = {2023}
}

Copy | Download

TY  - JOUR
T1  - On An Illustrative Examples of a Studied Noetherity Dirac-Delta Extensions for a Noether Operator
AU  - Abdourahman
AU  - Ecclésiaste Tompé Weimbapou
AU  - Emmanuel Kengne
AU  - Shankishvili Lamara Dmitrievna
Y1  - 2023/01/17
PY  - 2023
N1  - https://doi.org/10.11648/j.ijtam.20220806.12
DO  - 10.11648/j.ijtam.20220806.12
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 121
EP  - 127
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20220806.12
AB  - 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 .
VL  - 8
IS  - 6
ER  -

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