On Distributional Solutions of a Singular Differential Equation of 2-order in the Space K’

Abdourahman

doi:doi:10.11648/j.ijtam.20220802.13

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On Distributional Solutions of a Singular Differential Equation of 2-order in the Space K’

Abdourahman

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

Received: 9 March 2022 Accepted: 1 April 2022 Published: 9 April 2022

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Abstract

The main purpose of this work is to describe all the zero-centered solutions of the second order linear singular differential equation with Dirac delta function (or it derivatives of some order) in the second right hand side in the space K’. All the coefficients and the exponents of the polynomials under the unknown function and it derivatives up to second order respectively, are real and natural numbers in the considered equation. We conduct investigations for both the euler case and left euler case situations of this equation, when it is fulfilled some particular conditions in the relationships between the parameters A, B, C, m, n and r. In each of these cases, we look for the zero-centered solutions and substitute the form of the particular solution into the equation. We then after, determinate the unknown coefficients and formulate the related theorems to describe all the solutions depending of the cases to be investigated.

Published in	International Journal of Theoretical and Applied Mathematics (Volume 8, Issue 2)
DOI	10.11648/j.ijtam.20220802.13
Page(s)	45-50
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, Fourier Transform, Zero-Centered Solutions

References

[1]	Kananthai, A. Distribution solutions of third order Euler equation. Southeast Asian Bull. Math. 1999, 23, 627-631.
[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]	Hernandez-urena, L. G.; Estrada, R. Solution of ordinary differential equations by series of delta functions. J. Math. Anal. Appl. 1995, 191, 40–55. [CrossRef].
[8]	Kananthai, A. The distributional solutions of ordinary differential equation with polynomial coefficients. Southeast Asian Bull. Math. 2001, 25, 129-134.
[9]	I. M. GEL’FAND AND G. E. SHILOV. “Generalized Functions,” Vol. II, Spaces of Fundamental and Generalized Functions. Academic Press. New York. 1968.
[10]	Kanwal, R. P. Generalized functions: Theory and Technique, 3rd ed.; Springer: New York, NY, USA, 2004.
[11]	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.
[12]	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.
[13]	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.
[14]	JOSEPH WIENER. Generalized-Function Solutions of Differential and Functional Differential Equations. Journal of Mathematical Analysis and Applications 88, 170-182 (1982). Edinburg, Texas 78539.
[15]	Cooke, K. L.; Wiener, J. Distributional and analytic solutions of functional differential equations. J. Math. Anal. Appl. 1984, 98, 111-129. [CrossRef].
[16]	Wiener, J.; Cooke, K. L.; Shah, S. M. Coexistence of analytic and distributional solutions for linear differential equations, II. J. Math. Anal. Appl. 1991, 159, 271-289. [CrossRef].
[17]	K Nonlaopon, K.; Orankitjaroen, S; Kanathai, A. The generalized Solutions of a certain n order differential equations with polynomial coefficients. Integr. transf. Spec. Funct. 2015, 26, 1015-1024. [CrossRef].
[18]	Opio, I.; Mirumbe, G. I.; Ssebuliba, J.; Mango, J. M. On the solution space of ordinary differential equations with polynomial coefficients. FJMS 2017, 101, 103-118. [CrossRef].
[19]	Abdourahman; E. Djeutcha,; A. P. Yatchet. On an n-order linear singular differential equation in the space of generalized function K' over K. Rostov-On-Don. 26 April – 1 May 2015. Modern Methods, Problems and Applications of Operator Theory and Harmonic Analysis-V. Proceedings. Page 90. OTHA 2015.

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Abdourahman. (2022). On Distributional Solutions of a Singular Differential Equation of 2-order in the Space K’. International Journal of Theoretical and Applied Mathematics, 8(2), 45-50. https://doi.org/10.11648/j.ijtam.20220802.13

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ACS Style

Abdourahman. On Distributional Solutions of a Singular Differential Equation of 2-order in the Space K’. Int. J. Theor. Appl. Math. 2022, 8(2), 45-50. doi: 10.11648/j.ijtam.20220802.13

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AMA Style

Abdourahman. On Distributional Solutions of a Singular Differential Equation of 2-order in the Space K’. Int J Theor Appl Math. 2022;8(2):45-50. doi: 10.11648/j.ijtam.20220802.13

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@article{10.11648/j.ijtam.20220802.13,
  author = {Abdourahman},
  title = {On Distributional Solutions of a Singular Differential Equation of 2-order in the Space K’},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {8},
  number = {2},
  pages = {45-50},
  doi = {10.11648/j.ijtam.20220802.13},
  url = {https://doi.org/10.11648/j.ijtam.20220802.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20220802.13},
  abstract = {The main purpose of this work is to describe all the zero-centered solutions of the second order linear singular differential equation with Dirac delta function (or it derivatives of some order) in the second right hand side in the space K’. All the coefficients and the exponents of the polynomials under the unknown function and it derivatives up to second order respectively, are real and natural numbers in the considered equation. We conduct investigations for both the euler case and left euler case situations of this equation, when it is fulfilled some particular conditions in the relationships between the parameters A, B, C, m, n and r. In each of these cases, we look for the zero-centered solutions and substitute the form of the particular solution into the equation. We then after, determinate the unknown coefficients and formulate the related theorems to describe all the solutions depending of the cases to be investigated.},
 year = {2022}
}

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TY  - JOUR
T1  - On Distributional Solutions of a Singular Differential Equation of 2-order in the Space K’
AU  - Abdourahman
Y1  - 2022/04/09
PY  - 2022
N1  - https://doi.org/10.11648/j.ijtam.20220802.13
DO  - 10.11648/j.ijtam.20220802.13
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
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SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20220802.13
AB  - The main purpose of this work is to describe all the zero-centered solutions of the second order linear singular differential equation with Dirac delta function (or it derivatives of some order) in the second right hand side in the space K’. All the coefficients and the exponents of the polynomials under the unknown function and it derivatives up to second order respectively, are real and natural numbers in the considered equation. We conduct investigations for both the euler case and left euler case situations of this equation, when it is fulfilled some particular conditions in the relationships between the parameters A, B, C, m, n and r. In each of these cases, we look for the zero-centered solutions and substitute the form of the particular solution into the equation. We then after, determinate the unknown coefficients and formulate the related theorems to describe all the solutions depending of the cases to be investigated.
VL  - 8
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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). On Distributional Solutions of a Singular Differential Equation of 2-order in the Space K’. International Journal of Theoretical and Applied Mathematics, 8(2), 45-50. https://doi.org/10.11648/j.ijtam.20220802.13

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ACS Style

Abdourahman. On Distributional Solutions of a Singular Differential Equation of 2-order in the Space K’. Int. J. Theor. Appl. Math. 2022, 8(2), 45-50. doi: 10.11648/j.ijtam.20220802.13

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AMA Style

Abdourahman. On Distributional Solutions of a Singular Differential Equation of 2-order in the Space K’. Int J Theor Appl Math. 2022;8(2):45-50. doi: 10.11648/j.ijtam.20220802.13

Copy | Download

@article{10.11648/j.ijtam.20220802.13,
  author = {Abdourahman},
  title = {On Distributional Solutions of a Singular Differential Equation of 2-order in the Space K’},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {8},
  number = {2},
  pages = {45-50},
  doi = {10.11648/j.ijtam.20220802.13},
  url = {https://doi.org/10.11648/j.ijtam.20220802.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20220802.13},
  abstract = {The main purpose of this work is to describe all the zero-centered solutions of the second order linear singular differential equation with Dirac delta function (or it derivatives of some order) in the second right hand side in the space K’. All the coefficients and the exponents of the polynomials under the unknown function and it derivatives up to second order respectively, are real and natural numbers in the considered equation. We conduct investigations for both the euler case and left euler case situations of this equation, when it is fulfilled some particular conditions in the relationships between the parameters A, B, C, m, n and r. In each of these cases, we look for the zero-centered solutions and substitute the form of the particular solution into the equation. We then after, determinate the unknown coefficients and formulate the related theorems to describe all the solutions depending of the cases to be investigated.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - On Distributional Solutions of a Singular Differential Equation of 2-order in the Space K’
AU  - Abdourahman
Y1  - 2022/04/09
PY  - 2022
N1  - https://doi.org/10.11648/j.ijtam.20220802.13
DO  - 10.11648/j.ijtam.20220802.13
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  - 45
EP  - 50
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20220802.13
AB  - The main purpose of this work is to describe all the zero-centered solutions of the second order linear singular differential equation with Dirac delta function (or it derivatives of some order) in the second right hand side in the space K’. All the coefficients and the exponents of the polynomials under the unknown function and it derivatives up to second order respectively, are real and natural numbers in the considered equation. We conduct investigations for both the euler case and left euler case situations of this equation, when it is fulfilled some particular conditions in the relationships between the parameters A, B, C, m, n and r. In each of these cases, we look for the zero-centered solutions and substitute the form of the particular solution into the equation. We then after, determinate the unknown coefficients and formulate the related theorems to describe all the solutions depending of the cases to be investigated.
VL  - 8
IS  - 2
ER  -

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