Noetherity of a Dirac Delta-Extension for a Noether Operator

Abdourahman; Ecclésiaste Tompé Weimbapou; Emmanuel Kengne

doi:doi:10.11648/j.ijtam.20220803.11

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Noetherity of a Dirac Delta-Extension for a Noether Operator

Abdourahman, Ecclésiaste Tompé Weimbapou, Emmanuel Kengne

Published in International Journal of Theoretical and Applied Mathematics (Volume 8, Issue 3)

Received: 27 April 2022 Accepted: 17 May 2022 Published: 26 May 2022

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Abstract

The main goal of this work is to establish the extension of a noether operator defined by a third kind singular integral equation in a special class of generalized functions and to investigate the noetherity of the extended operator. The initial considered operator has been investigated for its noetherity nature in our previous works. Special approach has been developed and applied when constructing noether theory for the mentioned operator by using the concept of Taylor derivatives for continuous functions to achieve noetherity. We realize the extension of the initial noether operator defined onto the class of coninuous functions, having first continous derivative and taking the value zero at the left boundary point of the closed interval by adding a finite number of Dirac delta functions and it Taylor derivatives of some order. Then, we investigate the noetherity of the extended initial noether operator when we realize the finite dimensional extension, taking the unknown function from a special space of generalized functions. For this aim, we will use the direct formal calculations and apply the principle of the conservation of the index of noether operator after it finite dimensional extension in the new constructed functional space. The noetherity of the extended operator is established and the deficient numbers with the corresponding index are calculated in various cases investigated.

Published in	International Journal of Theoretical and Applied Mathematics (Volume 8, Issue 3)
DOI	10.11648/j.ijtam.20220803.11
Page(s)	51-57
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), 2022. Published by Science Publishing Group

Keywords

Integral Equation of Third Kind, Deficients Numbers, Noether Operator, Fundamental Functions, Singular Integral Operator

References

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[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 in the main part (Rostov-na-Donu, deposited in VINITI, Moscow, 28. 03. 2002, No. 560-B (2002).
[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]	Gabbassov N. S. On direct methods of the solutions of Fredholm’s integral equations in the space of generalized functions, PhD thesis (Kazan, 1987).
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[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].
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[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 Weimbapou, Abdourahman, and E. Kengne. «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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APA Style

Abdourahman, Ecclésiaste Tompé Weimbapou, Emmanuel Kengne. (2022). Noetherity of a Dirac Delta-Extension for a Noether Operator. International Journal of Theoretical and Applied Mathematics, 8(3), 51-57. https://doi.org/10.11648/j.ijtam.20220803.11

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ACS Style

Abdourahman; Ecclésiaste Tompé Weimbapou; Emmanuel Kengne. Noetherity of a Dirac Delta-Extension for a Noether Operator. Int. J. Theor. Appl. Math. 2022, 8(3), 51-57. doi: 10.11648/j.ijtam.20220803.11

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AMA Style

Abdourahman, Ecclésiaste Tompé Weimbapou, Emmanuel Kengne. Noetherity of a Dirac Delta-Extension for a Noether Operator. Int J Theor Appl Math. 2022;8(3):51-57. doi: 10.11648/j.ijtam.20220803.11

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@article{10.11648/j.ijtam.20220803.11,
  author = {Abdourahman and Ecclésiaste Tompé Weimbapou and Emmanuel Kengne},
  title = {Noetherity of a Dirac Delta-Extension for a Noether Operator},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {8},
  number = {3},
  pages = {51-57},
  doi = {10.11648/j.ijtam.20220803.11},
  url = {https://doi.org/10.11648/j.ijtam.20220803.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20220803.11},
  abstract = {The main goal of this work is to establish the extension of a noether operator defined by a third kind singular integral equation in a special class of generalized functions and to investigate the noetherity of the extended operator. The initial considered operator has been investigated for its noetherity nature in our previous works. Special approach has been developed and applied when constructing noether theory for the mentioned operator by using the concept of Taylor derivatives for continuous functions to achieve noetherity. We realize the extension of the initial noether operator defined onto the class of coninuous functions, having first continous derivative and taking the value zero at the left boundary point of the closed interval by adding a finite number of Dirac delta functions and it Taylor derivatives of some order. Then, we investigate the noetherity of the extended initial noether operator when we realize the finite dimensional extension, taking the unknown function from a special space of generalized functions. For this aim, we will use the direct formal calculations and apply the principle of the conservation of the index of noether operator after it finite dimensional extension in the new constructed functional space. The noetherity of the extended operator is established and the deficient numbers with the corresponding index are calculated in various cases investigated.},
 year = {2022}
}

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TY  - JOUR
T1  - Noetherity of a Dirac Delta-Extension for a Noether Operator
AU  - Abdourahman
AU  - Ecclésiaste Tompé Weimbapou
AU  - Emmanuel Kengne
Y1  - 2022/05/26
PY  - 2022
N1  - https://doi.org/10.11648/j.ijtam.20220803.11
DO  - 10.11648/j.ijtam.20220803.11
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  - 51
EP  - 57
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20220803.11
AB  - The main goal of this work is to establish the extension of a noether operator defined by a third kind singular integral equation in a special class of generalized functions and to investigate the noetherity of the extended operator. The initial considered operator has been investigated for its noetherity nature in our previous works. Special approach has been developed and applied when constructing noether theory for the mentioned operator by using the concept of Taylor derivatives for continuous functions to achieve noetherity. We realize the extension of the initial noether operator defined onto the class of coninuous functions, having first continous derivative and taking the value zero at the left boundary point of the closed interval by adding a finite number of Dirac delta functions and it Taylor derivatives of some order. Then, we investigate the noetherity of the extended initial noether operator when we realize the finite dimensional extension, taking the unknown function from a special space of generalized functions. For this aim, we will use the direct formal calculations and apply the principle of the conservation of the index of noether operator after it finite dimensional extension in the new constructed functional space. The noetherity of the extended operator is established and the deficient numbers with the corresponding index are calculated in various cases investigated.
VL  - 8
IS  - 3
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

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Plain Text BibTeX RIS

APA Style

Abdourahman, Ecclésiaste Tompé Weimbapou, Emmanuel Kengne. (2022). Noetherity of a Dirac Delta-Extension for a Noether Operator. International Journal of Theoretical and Applied Mathematics, 8(3), 51-57. https://doi.org/10.11648/j.ijtam.20220803.11

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ACS Style

Abdourahman; Ecclésiaste Tompé Weimbapou; Emmanuel Kengne. Noetherity of a Dirac Delta-Extension for a Noether Operator. Int. J. Theor. Appl. Math. 2022, 8(3), 51-57. doi: 10.11648/j.ijtam.20220803.11

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AMA Style

Abdourahman, Ecclésiaste Tompé Weimbapou, Emmanuel Kengne. Noetherity of a Dirac Delta-Extension for a Noether Operator. Int J Theor Appl Math. 2022;8(3):51-57. doi: 10.11648/j.ijtam.20220803.11

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@article{10.11648/j.ijtam.20220803.11,
  author = {Abdourahman and Ecclésiaste Tompé Weimbapou and Emmanuel Kengne},
  title = {Noetherity of a Dirac Delta-Extension for a Noether Operator},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {8},
  number = {3},
  pages = {51-57},
  doi = {10.11648/j.ijtam.20220803.11},
  url = {https://doi.org/10.11648/j.ijtam.20220803.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20220803.11},
  abstract = {The main goal of this work is to establish the extension of a noether operator defined by a third kind singular integral equation in a special class of generalized functions and to investigate the noetherity of the extended operator. The initial considered operator has been investigated for its noetherity nature in our previous works. Special approach has been developed and applied when constructing noether theory for the mentioned operator by using the concept of Taylor derivatives for continuous functions to achieve noetherity. We realize the extension of the initial noether operator defined onto the class of coninuous functions, having first continous derivative and taking the value zero at the left boundary point of the closed interval by adding a finite number of Dirac delta functions and it Taylor derivatives of some order. Then, we investigate the noetherity of the extended initial noether operator when we realize the finite dimensional extension, taking the unknown function from a special space of generalized functions. For this aim, we will use the direct formal calculations and apply the principle of the conservation of the index of noether operator after it finite dimensional extension in the new constructed functional space. The noetherity of the extended operator is established and the deficient numbers with the corresponding index are calculated in various cases investigated.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - Noetherity of a Dirac Delta-Extension for a Noether Operator
AU  - Abdourahman
AU  - Ecclésiaste Tompé Weimbapou
AU  - Emmanuel Kengne
Y1  - 2022/05/26
PY  - 2022
N1  - https://doi.org/10.11648/j.ijtam.20220803.11
DO  - 10.11648/j.ijtam.20220803.11
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  - 51
EP  - 57
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20220803.11
AB  - The main goal of this work is to establish the extension of a noether operator defined by a third kind singular integral equation in a special class of generalized functions and to investigate the noetherity of the extended operator. The initial considered operator has been investigated for its noetherity nature in our previous works. Special approach has been developed and applied when constructing noether theory for the mentioned operator by using the concept of Taylor derivatives for continuous functions to achieve noetherity. We realize the extension of the initial noether operator defined onto the class of coninuous functions, having first continous derivative and taking the value zero at the left boundary point of the closed interval by adding a finite number of Dirac delta functions and it Taylor derivatives of some order. Then, we investigate the noetherity of the extended initial noether operator when we realize the finite dimensional extension, taking the unknown function from a special space of generalized functions. For this aim, we will use the direct formal calculations and apply the principle of the conservation of the index of noether operator after it finite dimensional extension in the new constructed functional space. The noetherity of the extended operator is established and the deficient numbers with the corresponding index are calculated in various cases investigated.
VL  - 8
IS  - 3
ER  -

Copy | Download