Some Separate Quasi-Asymptotics Properties of Multidimensional Distributions and Application

Nenad Stojanovic

doi:doi:10.11648/j.pamj.20200903.13

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Some Separate Quasi-Asymptotics Properties of Multidimensional Distributions and Application

Nenad Stojanovic

Published in Pure and Applied Mathematics Journal (Volume 9, Issue 3)

Received: 17 June 2020 Accepted: 3 July 2020 Published: 13 July 2020

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Abstract

Quasi-asymptotic behavior of functions as a method has its application in observing many physical phenomena which are expressed by differential equations. The aim of the asymptotic method is to allow one to present the solution of a problem depending on the large (or small) parameter. One application of asymptotic methods in describing physical phenomena is the quasi-asymptotic approximation. The aim of this paper is to look at the quasi-asymptotic properties of multidimensional distributions by extracted variable. Distribution T(x₀,x) from S'(Ṝ₊¹×Rⁿ) has the property of the separability of variables, if it can be represented in form T(x₀,x)=∑φ_i(x₀)ψ_i (x) where distributions, φ_i(x₀) from S'(Ṝ¹) and ψ_i from S(Rⁿ), x₀ from Ṝ¹₊ and x is element Rⁿ different values of do not depend on each other. Distribution T(x₀,x) the element S'(Ṝ₊¹×Rⁿ) is homogeneous and of order α at variable x₀ is element Ṝ¹₊ and x=x₁,x₂,…,x_n from Rⁿ if for k>0 it applies that T(kx₀,kx)=k^α T(x₀,x). The method of separating variables is one of the most widespread methods for solving linear differential equations in mathematical physics. In this paper, the results by V. S Vladimirov are used to present the proof of the basic theorems, regarding the quasi-asymptotic behavior of multidimensional distributions by a singular variable, with the application of quasi-asymptotics to the solution of differential equations.

Published in	Pure and Applied Mathematics Journal (Volume 9, Issue 3)
DOI	10.11648/j.pamj.20200903.13
Page(s)	64-69
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), 2020. Published by Science Publishing Group

Keywords

Distribution Spaces, Asymptotics, Separate Quasi-Asymptotics, Multidimensional Distributions

References

[1]	V. S. Vladimirov, N. Drozinov, B. I. Zavjalov, Mnogomernie Tauberovi Teoremi dlja Obobosce- nie funkcii, Nauka, Moskva, 1986.
[2]	N. Drozinov, B. I, Zavjalov, Vvedenie v teoriju oboboscenih funkcii, Mathematiceskij institut im V. A. Steklov, RAN (MIAN), Moskva. 2006.
[3]	S. Pilipović, B. Stanković, Vindas, J., Asymptotic behavior of generalizedfunctions, Series on Analysis, Applications and Computation, 5, World Sci-entific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012.
[4]	S. Pilipović, J. Toft, P seudo-Differential Operators and Genera -lized Functions, Birkhäuser; 2015.
[5]	V. S. Vladimirov, Oboboscenie funkcii u matematice skoj fizici, Nauka, 1979.
[6]	N. Y. Drozhzhinov, B. I. Zav’yalov, Asymptotically homogeneous generalized functions and boundary properties of functions holomorphic in tubularcones. Izv. Math., 70, (2006), 1117–1164.
[7]	S. Pilipović, B. Stanković, Prostori distribucija, SANU, Novi Sad, 2000.
[8]	S. Pilipović, On the Quasiasymptotic of Schwartz distributions. Math. Nachr. 141 (1988), 19-25.
[9]	J. Vindas, S. Pilipović, Structural theorems for quasiasympto- tics of distributions at the origin, Math. Nachr. 282 (2.11), (2009), 1584–1599.
[10]	J. Vindas, The structure of quasiasymptotics of Schwartz distributions, Banach Center Publ. 88 (2010), 297-31.
[11]	S. Pilipović, Quasiasymptotic and the translation asymptotic behavior of distributions, Acta Math. Hungarica, 55 (3-4) (1990), 239-243.
[12]	V. S. Vladimirov, Uravnenija mathematiceskoj fiziki, Nauke, Moskva, 1981.
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[14]	S. Pilipović, B. Stanković, Asymptotic Behavior or Generalized Function, Novi Sad, 2008.
[15]	N. Stojanović, Separirana kvaziasimptotika više-dimenzionih distribucija, PMF, Novi Sad, 2009-magistarska teza.
[16]	M. Tomić, 'Jovan Karamata 1902-1967', Bulletin T. CXXII de l’Acad´emie Serbe des Sciences et des Arts, No 26, 2001.

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Nenad Stojanovic. (2020). Some Separate Quasi-Asymptotics Properties of Multidimensional Distributions and Application. Pure and Applied Mathematics Journal, 9(3), 64-69. https://doi.org/10.11648/j.pamj.20200903.13

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ACS Style

Nenad Stojanovic. Some Separate Quasi-Asymptotics Properties of Multidimensional Distributions and Application. Pure Appl. Math. J. 2020, 9(3), 64-69. doi: 10.11648/j.pamj.20200903.13

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AMA Style

Nenad Stojanovic. Some Separate Quasi-Asymptotics Properties of Multidimensional Distributions and Application. Pure Appl Math J. 2020;9(3):64-69. doi: 10.11648/j.pamj.20200903.13

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@article{10.11648/j.pamj.20200903.13,
  author = {Nenad Stojanovic},
  title = {Some Separate Quasi-Asymptotics Properties of Multidimensional Distributions and Application},
  journal = {Pure and Applied Mathematics Journal},
  volume = {9},
  number = {3},
  pages = {64-69},
  doi = {10.11648/j.pamj.20200903.13},
  url = {https://doi.org/10.11648/j.pamj.20200903.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20200903.13},
  abstract = {Quasi-asymptotic behavior of functions as a method has its application in observing many physical phenomena which are expressed by differential equations. The aim of the asymptotic method is to allow one to present the solution of a problem depending on the large (or small) parameter. One application of asymptotic methods in describing physical phenomena is the quasi-asymptotic approximation. The aim of this paper is to look at the quasi-asymptotic properties of multidimensional distributions by extracted variable. Distribution T(x0,x) from S'(Ṝ+1×Rn) has the property of the separability of variables, if it can be represented in form T(x0,x)=∑φi(x0)ψi (x) where distributions, φi(x0) from S'(Ṝ1) and ψi  from S(Rn), x0 from Ṝ1+ and  x is element Rn different values of do not depend on each other. Distribution T(x0,x) the element S'(Ṝ+1×Rn) is homogeneous and of order α at variable  x0 is element Ṝ1+ and x=x1,x2,…,xn  from Rn if for k>0 it applies that T(kx0,kx)=kα T(x0,x). The method of separating variables is one of the most widespread methods for solving linear differential equations in mathematical physics. In this paper, the results by V. S Vladimirov are used to present the proof of the basic theorems, regarding the quasi-asymptotic behavior of multidimensional distributions by a singular variable, with the application of quasi-asymptotics to the solution of differential equations.},
 year = {2020}
}

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TY  - JOUR
T1  - Some Separate Quasi-Asymptotics Properties of Multidimensional Distributions and Application
AU  - Nenad Stojanovic
Y1  - 2020/07/13
PY  - 2020
N1  - https://doi.org/10.11648/j.pamj.20200903.13
DO  - 10.11648/j.pamj.20200903.13
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 64
EP  - 69
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20200903.13
AB  - Quasi-asymptotic behavior of functions as a method has its application in observing many physical phenomena which are expressed by differential equations. The aim of the asymptotic method is to allow one to present the solution of a problem depending on the large (or small) parameter. One application of asymptotic methods in describing physical phenomena is the quasi-asymptotic approximation. The aim of this paper is to look at the quasi-asymptotic properties of multidimensional distributions by extracted variable. Distribution T(x0,x) from S'(Ṝ+1×Rn) has the property of the separability of variables, if it can be represented in form T(x0,x)=∑φi(x0)ψi (x) where distributions, φi(x0) from S'(Ṝ1) and ψi  from S(Rn), x0 from Ṝ1+ and  x is element Rn different values of do not depend on each other. Distribution T(x0,x) the element S'(Ṝ+1×Rn) is homogeneous and of order α at variable  x0 is element Ṝ1+ and x=x1,x2,…,xn  from Rn if for k>0 it applies that T(kx0,kx)=kα T(x0,x). The method of separating variables is one of the most widespread methods for solving linear differential equations in mathematical physics. In this paper, the results by V. S Vladimirov are used to present the proof of the basic theorems, regarding the quasi-asymptotic behavior of multidimensional distributions by a singular variable, with the application of quasi-asymptotics to the solution of differential equations.
VL  - 9
IS  - 3
ER  -

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Author Information

Nenad Stojanovic

Department of Mathematics, Faculty of Agriculture, University of Banja Luka, Banja Luka, Bosnia and Herzegovina

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Nenad Stojanovic. (2020). Some Separate Quasi-Asymptotics Properties of Multidimensional Distributions and Application. Pure and Applied Mathematics Journal, 9(3), 64-69. https://doi.org/10.11648/j.pamj.20200903.13

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ACS Style

Nenad Stojanovic. Some Separate Quasi-Asymptotics Properties of Multidimensional Distributions and Application. Pure Appl. Math. J. 2020, 9(3), 64-69. doi: 10.11648/j.pamj.20200903.13

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AMA Style

Nenad Stojanovic. Some Separate Quasi-Asymptotics Properties of Multidimensional Distributions and Application. Pure Appl Math J. 2020;9(3):64-69. doi: 10.11648/j.pamj.20200903.13

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@article{10.11648/j.pamj.20200903.13,
  author = {Nenad Stojanovic},
  title = {Some Separate Quasi-Asymptotics Properties of Multidimensional Distributions and Application},
  journal = {Pure and Applied Mathematics Journal},
  volume = {9},
  number = {3},
  pages = {64-69},
  doi = {10.11648/j.pamj.20200903.13},
  url = {https://doi.org/10.11648/j.pamj.20200903.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20200903.13},
  abstract = {Quasi-asymptotic behavior of functions as a method has its application in observing many physical phenomena which are expressed by differential equations. The aim of the asymptotic method is to allow one to present the solution of a problem depending on the large (or small) parameter. One application of asymptotic methods in describing physical phenomena is the quasi-asymptotic approximation. The aim of this paper is to look at the quasi-asymptotic properties of multidimensional distributions by extracted variable. Distribution T(x0,x) from S'(Ṝ+1×Rn) has the property of the separability of variables, if it can be represented in form T(x0,x)=∑φi(x0)ψi (x) where distributions, φi(x0) from S'(Ṝ1) and ψi  from S(Rn), x0 from Ṝ1+ and  x is element Rn different values of do not depend on each other. Distribution T(x0,x) the element S'(Ṝ+1×Rn) is homogeneous and of order α at variable  x0 is element Ṝ1+ and x=x1,x2,…,xn  from Rn if for k>0 it applies that T(kx0,kx)=kα T(x0,x). The method of separating variables is one of the most widespread methods for solving linear differential equations in mathematical physics. In this paper, the results by V. S Vladimirov are used to present the proof of the basic theorems, regarding the quasi-asymptotic behavior of multidimensional distributions by a singular variable, with the application of quasi-asymptotics to the solution of differential equations.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Some Separate Quasi-Asymptotics Properties of Multidimensional Distributions and Application
AU  - Nenad Stojanovic
Y1  - 2020/07/13
PY  - 2020
N1  - https://doi.org/10.11648/j.pamj.20200903.13
DO  - 10.11648/j.pamj.20200903.13
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 64
EP  - 69
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20200903.13
AB  - Quasi-asymptotic behavior of functions as a method has its application in observing many physical phenomena which are expressed by differential equations. The aim of the asymptotic method is to allow one to present the solution of a problem depending on the large (or small) parameter. One application of asymptotic methods in describing physical phenomena is the quasi-asymptotic approximation. The aim of this paper is to look at the quasi-asymptotic properties of multidimensional distributions by extracted variable. Distribution T(x0,x) from S'(Ṝ+1×Rn) has the property of the separability of variables, if it can be represented in form T(x0,x)=∑φi(x0)ψi (x) where distributions, φi(x0) from S'(Ṝ1) and ψi  from S(Rn), x0 from Ṝ1+ and  x is element Rn different values of do not depend on each other. Distribution T(x0,x) the element S'(Ṝ+1×Rn) is homogeneous and of order α at variable  x0 is element Ṝ1+ and x=x1,x2,…,xn  from Rn if for k>0 it applies that T(kx0,kx)=kα T(x0,x). The method of separating variables is one of the most widespread methods for solving linear differential equations in mathematical physics. In this paper, the results by V. S Vladimirov are used to present the proof of the basic theorems, regarding the quasi-asymptotic behavior of multidimensional distributions by a singular variable, with the application of quasi-asymptotics to the solution of differential equations.
VL  - 9
IS  - 3
ER  -

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