PRODUCT OF THE GENERALIZED FUNCTIONS x-- k AND
δ(p)(x) IN COLOMBEAU ALGEBRA
Marija Miteva1, Biljana Jolevska-Tuneska2,
Limonka Koceva Lazarova3
1Faculty of Computer Science
Goce Delcev University - Stip
2000 Stip, MACEDONIA
e-mail: marija.miteva@ugd.edu.mk
2Faculty of Electrical Engineering
and Information Technologies
Ss. Cyril and Methodius University
1000 Skopje, MACEDONIA
biljanaj@feit.ukim.edu.mk
3Faculty of Computer Science
Goce Delcev University - Stip
2000 Stip, MACEDONIA
e-mail: limonka.lazarova@ugd.edu.mk

Abstract

In this paper the product of the distributions ${{x_ - }^{ - k}}$ and ${\delta ^{\left( p \right)}}\left( x \right)$ is derived. The result is obtained in Colombeau algebra of generalized functions which contains the space of Schwartz distributions as a subspace and has a notion of 'association' that allows us to evaluate the results in terms of distributions.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 35
Issue: 1
Year: 2022

DOI: 10.12732/ijam.v35i1.11

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References

  1. [1] A. Gsponer, A concise introduction to Colombeau generalized functions and their applications in classical electrodynamics, European Journal of Physics, 30, No 1 (2009), 109-126.
  2. [2] A.H. Zemanian, Distribution Theory and Transform Analysis, Dover (1965).
  3. [3] B. Damyanov, Results on Colombeau product of distributions, Commentationes Mathematicae Universitatis Carolinae, 38, No 4 (1997), 627-634.
  4. [4] B. Damyanov, Multiplication of Schwartz distributions and Colombeau generalized functions, Journal of Applied Analysis, 5, No 2 (1999), 249260.
  5. [5] B. Damyanov, Balanced Colombeau products of the distributions x−p ± and x−p, Czechoslovak Mathematical Journal, 55, No 1 (2005), 189-201.
  6. [6] B. Damyanov, Results on balanced products of the distributions xa ± in Colombeau algebra G (R), Integral Transforms and Special Functions, 17, No 9 (2006), 623-635.
  7. [7] B. Fisher, The product of distributions, The Quarterly Journal of Mathematics, 22, No 2 (1971), 291-298.
  8. [8] B. Fisher, On defining the product of distributions, Mathematische Nachrichten, 99, No 1 (1980), 239-249.
  9. [9] B.J. Tuneska, A. Takaci, E. Ozcag, On differential equations with nonstandard coefficients, Applicable Analysis and Discrete Mathematics, 1 (2007), 276-283.
  10. [10] B.J. Tuneska, T.A. Pacemska, Further results on Colombeau product of distributions, International Journal of Mathematics and Mathematical Sciences (2013); doi: 10.1155/2013/918905.
  11. [11] C.L. Zhi, B. Fisher, Several products of distributions on Rm, Proceedings of the Royal Society of London, London (1989), A426: 425-439.
  12. [12] E. Nigsch, C. Samman, Global algebras of nonlinear generalized functions with applications in general relativity, Sao Paulo Journal of Mathematical Sciences, 7, No 2 (2013), 143-171.
  13. [13] F. Farassat, Introduction to Generalized Functions with Applications in Aerodynamics and Aeroacoustics, NASA Technical Paper 3428.
  14. [14] I.M. Gelfand, G.E. Shilov, Generalized Functions, Academic Press, New York, USA (1964).
  15. [15] J.F. Colombeau, New Generalized Functions and Multiplication of Distributions, North Holland Math. Studies, 84 (1984).
  16. [16] J.F. Colombeau, Elementary Introduction New Generalized Functions, North Holland Math. Studies, 113 (1985).
  17. [17] J. Aragona, J.F. Colombeau, S.O. Juriaans, Nonlinear generalized functions and jump conditions for a standard one pressure liquid-gas model, Journal of Mathematical Analysis and Applications, 418, No 2 (2014), 964-977.
  18. [18] K. Ohgitani, M. Dowker, Burges equation with a passive scalar: dissipation anomaly and Colombeau calculus, Journal of Mathematical Physics, 51, No 3 (2010), 1-7.
  19. [19] M. Alimohammady, F. Fariba, Existence/uniqueness of solutions to heat equation in extended Colombeau algebra, Sahand Communications in Mathematical Analysis, 1, No 1 (2014), 21-28.
  20. [20] M. Miteva, B.J. Tuneska, Some results on Colombeau product of distributions, Advances in Mathematics: Scientific Journal, 1, No 2 (2012), 121-126.
  21. [21] M. Miteva, B.J. Tuneska, T.A. Pacemska, On products of distributions in Colombeau algebra, Mathematical Problems in Engineering (2014); doi: 10.1155/2014/910510.
  22. [22] M. Miteva, B.J. Tuneska, T.A. Pacemska, Colombeau products of distributions, SpringerPlus 5 (2016); doi: 10.1186/s40064-016-3742-8.
  23. [23] M. Miteva, B.J. Tuneska, T.A. Pacemska, Results on the Colombeau products of the distribution x−r−1/2 + with the distributions x−k−1/2 − and xk−1/2 − , Functional Analysis and its Applications, 52, No 1 (2018), 9-20.
  24. [24] M. Oberguggenberger, T. Todorov, An embedding of Schwartz distributions in the algebra of asymptotic functions, International Journal of Mathematics and Mathematical Science, 21, No 3 (1998), 417-428.
  25. [25] M. Stojanovic, Singular fractional evolution differential equations, Central Europian Journal of Physics, 11, No 10 (2013), 1337-1349.
  26. [26] R. Steinbauer, The ultrarelativistic Reissner-Nordstrom field in the Colombeau algebra, Journal of Mathematical Physics, 38, No 3 (1997), 1-8.
  27. [27] R. Steinbauer, J.A. Vickers, The use of generalized functions and distributions in general relativity, Classical and Quantum Gravity, 23, No 10 (2006).
  28. [28] U. Capar, Colombeau solutions of a nonlinear stochastic predator - pray equation, Turkish Journal of Mathematics, 34 (2013), 1048-1060.
  29. [29] V. Prusa, K.R. Rajagopal, On the response of physical systems governed by non-linear ordinary differential equations to step input, International Journal of Non-Linear Mechanics, 81 (2016), 207-221.