Generalized-function Solutions of a Differential Equation of L<i>-</i>order in the Space K’

Abdourahman

doi:doi:10.11648/j.ijtam.20220802.12

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Generalized-function Solutions of a Differential Equation of L-order in the Space K’

Abdourahman

Published in International Journal of Theoretical and Applied Mathematics (Volume 8, Issue 2)

Received: 28 February 2022 Accepted: 24 March 2022 Published: 31 March 2022

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Abstract

The main purpose of this work is to study the existence of solutions (zero-centered solutions), in the sense of distributions, of the non-homogeneous l-order linear singular differential equation of the following type , where l is a natural number not equal to zero, s a natural number, (a_i)_0≤i≤l real numbers and more a_l≠0, is the Dirac distribution centered at 0 and δ^(s)(x) is the sth-order derivative of the Dirac delta function. For this aim, we apply some theorems and lemmas from the general concepts of theory of generalized functions in the work. Namely, we replace the general form of the particular solution (as linear combination of Dirac delta functions and it derivatives) into the considered equation. This leads us to release the conditions of its solvency, formulated into a theorem and, let us analyze the algebraic system obtained for the determination of the unknown coefficients C_j. By this, we undertake the description of all zero-centered solutions of the considered equation into a theorem.

Published in	International Journal of Theoretical and Applied Mathematics (Volume 8, Issue 2)
DOI	10.11648/j.ijtam.20220802.12
Page(s)	40-44
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

Test Functions, Generalized Functions, Dirac Delta Function, Zero-Centered Solutions

References

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[2]	Gelfand. I. M., SHYLOV G. E: Space of Test and generalized functions. V. 1. M.: Phyzmatgiz, 1958.307 Pages. Russian Edition.
[3]	L. Schwartz, Théorie des distributions, Tomes I et II, Hermann, Paris, 1957.
[4]	Liangprom, A.; Nonlaopon, K. On the generalized solutions of a certain fourth order Euler equations. J. Nonlinear Sci. Appl. 2017, 10, 4077- 4084.
[5]	Fuchs, Zur theorie der linearen differentialgleichungen mit veränderlichen coefficienten, Journal für die reine und angewandte mathematik, t. 66, 1866, p. 121-160.
[6]	Abdourahman, On a linear differential equation in the spaces of generalized functions. Rostov on Don. Preprint at VINITI 02.08.2000. No. 2035, 2000. 27 pages.
[7]	Kananthai, A. The distributional solutions of ordinary differential equation with polynomial coefficients. Southeast Asian Bull. Math. 2001, 25, 129-134.
[8]	I. M. GEL’FAND AND G. E. SHILOV. “Generalized Functions,” Vol. II, Spaces of Fundamental and Generalized Functions. Academic Press. New York. 1968.
[9]	Kanwal, R. P. Generalized functions: Theory and Technique, 3rd ed.; Springer: New York, NY, USA, 2004.
[10]	Abdourahman. On a linear differential equation with singular coefficients. Rostov-On-Don. 1999. Collection of papers. «Integro-differential operators and their Applications» Rostov-Na-Donu: Izdat. DGTU, 1999. N°.5. PP. 4-7. Available at http://chat.ru/volume1999.
[11]	S Jhanthanam, K Nonlaopon, and S Orankitjaroen, Generalized Solutions of the Third-Order Cauchy-Euler Equation in the Space of Right-Sided Distributions via Laplace Transform, Mathematics 2019, 7, 376; doi: 10.3390/math7040376.
[12]	Abdourahman. On a linear singular differential equation with singular coefficients, Collection of papers.”Integro- differential operators and their applications”, Rostov-na-Donu: Izdat. DGTU, 2001. No. 5. P. 4-10.
[13]	JOSEPH WIENER. Generalized-Function Solutions of Differential and Functional Differential Equations. Journal of Mathematical Analysis and Applications 88, 170-182 (1982). Edinburg, Texas 78539.
[14]	Seksan Jhanthanam, Kamsing Nonlaopon, Somsak Orankitjaroen. Generalized solutions of the Third-Order Cauchy-Euler Equation in the Space of Right-Sided Distributions via Laplace Transform. Mathematics 2019, 7 (4), 376.

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Abdourahman. (2022). Generalized-function Solutions of a Differential Equation of L-order in the Space K’. International Journal of Theoretical and Applied Mathematics, 8(2), 40-44. https://doi.org/10.11648/j.ijtam.20220802.12

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ACS Style

Abdourahman. Generalized-function Solutions of a Differential Equation of L-order in the Space K’. Int. J. Theor. Appl. Math. 2022, 8(2), 40-44. doi: 10.11648/j.ijtam.20220802.12

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AMA Style

Abdourahman. Generalized-function Solutions of a Differential Equation of L-order in the Space K’. Int J Theor Appl Math. 2022;8(2):40-44. doi: 10.11648/j.ijtam.20220802.12

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@article{10.11648/j.ijtam.20220802.12,
  author = {Abdourahman},
  title = {Generalized-function Solutions of a Differential Equation of L-order in the Space K’},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {8},
  number = {2},
  pages = {40-44},
  doi = {10.11648/j.ijtam.20220802.12},
  url = {https://doi.org/10.11648/j.ijtam.20220802.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20220802.12},
  abstract = {The main purpose of this work is to study the existence of solutions (zero-centered solutions), in the sense of distributions, of the non-homogeneous l-order linear singular differential equation of the following type , where l is a natural number not equal to zero, s a natural number, (ai)0≤i≤l real numbers and more al≠0,  is the Dirac distribution centered at 0 and δ(s)(x) is the sth-order derivative of the Dirac delta function. For this aim, we apply some theorems and lemmas from the general concepts of theory of generalized functions in the work. Namely, we replace the general form of the particular solution  (as linear combination of Dirac delta functions and it derivatives) into the considered equation. This leads us to release the conditions of its solvency, formulated into a theorem and, let us analyze the algebraic system obtained for the determination of the unknown coefficients Cj. By this, we undertake the description of all zero-centered solutions of the considered equation into a theorem.},
 year = {2022}
}

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TY  - JOUR
T1  - Generalized-function Solutions of a Differential Equation of L-order in the Space K’
AU  - Abdourahman
Y1  - 2022/03/31
PY  - 2022
N1  - https://doi.org/10.11648/j.ijtam.20220802.12
DO  - 10.11648/j.ijtam.20220802.12
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  - 40
EP  - 44
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20220802.12
AB  - The main purpose of this work is to study the existence of solutions (zero-centered solutions), in the sense of distributions, of the non-homogeneous l-order linear singular differential equation of the following type , where l is a natural number not equal to zero, s a natural number, (ai)0≤i≤l real numbers and more al≠0,  is the Dirac distribution centered at 0 and δ(s)(x) is the sth-order derivative of the Dirac delta function. For this aim, we apply some theorems and lemmas from the general concepts of theory of generalized functions in the work. Namely, we replace the general form of the particular solution  (as linear combination of Dirac delta functions and it derivatives) into the considered equation. This leads us to release the conditions of its solvency, formulated into a theorem and, let us analyze the algebraic system obtained for the determination of the unknown coefficients Cj. By this, we undertake the description of all zero-centered solutions of the considered equation into a theorem.
VL  - 8
IS  - 2
ER  -

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Author Information

Abdourahman

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

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APA Style

Abdourahman. (2022). Generalized-function Solutions of a Differential Equation of L-order in the Space K’. International Journal of Theoretical and Applied Mathematics, 8(2), 40-44. https://doi.org/10.11648/j.ijtam.20220802.12

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ACS Style

Abdourahman. Generalized-function Solutions of a Differential Equation of L-order in the Space K’. Int. J. Theor. Appl. Math. 2022, 8(2), 40-44. doi: 10.11648/j.ijtam.20220802.12

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AMA Style

Abdourahman. Generalized-function Solutions of a Differential Equation of L-order in the Space K’. Int J Theor Appl Math. 2022;8(2):40-44. doi: 10.11648/j.ijtam.20220802.12

Copy | Download

@article{10.11648/j.ijtam.20220802.12,
  author = {Abdourahman},
  title = {Generalized-function Solutions of a Differential Equation of L-order in the Space K’},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {8},
  number = {2},
  pages = {40-44},
  doi = {10.11648/j.ijtam.20220802.12},
  url = {https://doi.org/10.11648/j.ijtam.20220802.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20220802.12},
  abstract = {The main purpose of this work is to study the existence of solutions (zero-centered solutions), in the sense of distributions, of the non-homogeneous l-order linear singular differential equation of the following type , where l is a natural number not equal to zero, s a natural number, (ai)0≤i≤l real numbers and more al≠0,  is the Dirac distribution centered at 0 and δ(s)(x) is the sth-order derivative of the Dirac delta function. For this aim, we apply some theorems and lemmas from the general concepts of theory of generalized functions in the work. Namely, we replace the general form of the particular solution  (as linear combination of Dirac delta functions and it derivatives) into the considered equation. This leads us to release the conditions of its solvency, formulated into a theorem and, let us analyze the algebraic system obtained for the determination of the unknown coefficients Cj. By this, we undertake the description of all zero-centered solutions of the considered equation into a theorem.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - Generalized-function Solutions of a Differential Equation of L-order in the Space K’
AU  - Abdourahman
Y1  - 2022/03/31
PY  - 2022
N1  - https://doi.org/10.11648/j.ijtam.20220802.12
DO  - 10.11648/j.ijtam.20220802.12
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  - 40
EP  - 44
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20220802.12
AB  - The main purpose of this work is to study the existence of solutions (zero-centered solutions), in the sense of distributions, of the non-homogeneous l-order linear singular differential equation of the following type , where l is a natural number not equal to zero, s a natural number, (ai)0≤i≤l real numbers and more al≠0,  is the Dirac distribution centered at 0 and δ(s)(x) is the sth-order derivative of the Dirac delta function. For this aim, we apply some theorems and lemmas from the general concepts of theory of generalized functions in the work. Namely, we replace the general form of the particular solution  (as linear combination of Dirac delta functions and it derivatives) into the considered equation. This leads us to release the conditions of its solvency, formulated into a theorem and, let us analyze the algebraic system obtained for the determination of the unknown coefficients Cj. By this, we undertake the description of all zero-centered solutions of the considered equation into a theorem.
VL  - 8
IS  - 2
ER  -

Copy | Download