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International Journal of Latest Research in Science and Technology

DOI:10.29111/ijlrst   ISRA Impact Factor:3.35,  Peer-reviewed, Open-access Journal

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AN UNIFIED DEFINITION FOR ANTI-INTEGRAL AND ANTI-DERIVATIVE OPERATORS FOR ANY ORDER

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International Journal of Latest Research in Science and Technology Vol.2 Issue 2, pp 46-54,Year 2013

AN UNIFIED DEFINITION FOR ANTI-INTEGRAL AND ANTI-DERIVATIVE OPERATORS FOR ANY ORDER

Raoelina Andriambololona

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Received : 06 April 2013; Accepted : 17 April 2013 ; Published : 30 April 2013

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Article No. 10156
Abstract

Our study is based on operators approach. First, positive integer s order fractional integral is defined by iterating s-times one order integral using Euler’s gamma functions properties. Then, the definition is extended to any value of s (positive , negative, fractional, transcendental, real, complex numbers) using the extension of Euler’s gamma and beta functions. Many properties (limits, linearity, semi-group) are given. The most important and useful property is the semi-group one Then , the definition of fractional derivative is derived from this property. There are two types of fractional derivatives, left handed and right handed ones. The left handed fractional derivative applied to a constant function gives a non null result whereas the right handed one leads to a null result. The latter one is then better than the first one. In a previous work, we have studied the case of fractional operators applied to the set of causal functions. In the present one, we look for the set of anti-causal functions. We introduce anti-integral and anti-derivatives operators. Though properties obtained for the cases of causal and anti- causal functions are similar, there are some differences. We obtain formulae given by many authors (Liouville, Riemann, Caputo, Liouville-Caputo) as particular cases of ours.

Key Words   
Operators, Fractional anti-integrals, Fractional anti-derivatives, Gamma functions, Beta f
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References
  1. Raoelina Andriambololona,“Real and complex order integrals and derivatives operators over the set of causal functions”, INSTN-Madagascar, 2012, arXiv:1207.0409 [math.GM], International Journal of Latest Research in Science and Technology, Volume 2,Issue 1,pp.470-477,January-February, ISSN(online) 2278-5299, 2013
  2. Kenneth S. Miller, “An introduction to the fractional calculus and the fractional differential equations” Bertram Ross (Editor). Publisher: John Wiley and Sons 1st edition, ISBN 0471588849, 1993
  3. B.Oldham Keith, J. Spanier, “The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order”,
    Acad. Press, N. York and London, ISBN: 978-0486450018, 1974
  4. P. Závada, “Operator of fractional derivatives in the complex plane”, Communications in Mathematical Physics, Volume 192, Issue 2 , pp 261-285 , doi: 10.1007/s002200050299, 1998
  5. Raoelina Andriambololona, “Algèbre linéaire et multilinéaire. Applications,” 3 Tomes. Collection LIRA, INSTN Madagascar, Antananarivo, Madagascar, Tome I pp 2-59, 1986
  6. Raoelina Andriambololona,“Definition of real order integrals and derivatives using operator approach”, INSTN-Madagascar, May 2012, arXiv:1207.0409. Pure and Applied Mathematics Journal, Vol.2, No 1, pp1-9.doi:10.11-648/j.pamj.20130201.11, 2013
  7. Raoelina Andriambololona, Tokiniaina Ranaivoson, Rakotoson Hanitriarivo, “Definitions of complex order integral and derivatives using operators approach ”, INSTN-Madagascar, arXiv:1409.400. International Journal of Latest Research in Science and Technology, Volume1, Issue 4:Page No.317-323, November-December, 2012.
  8. R. Herrmann. Fractional calculus. “An introduction for physicists”, World Scientific Publishing, Singapore, 2011.
To cite this article

Raoelina Andriambololona , " An Unified Definition For Anti-integral And Anti-derivative Operators For Any Order ", International Journal of Latest Research in Science and Technology . Vol. 2, Issue 2, pp 46-54 , 2013


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