Pure and Applied Mathematics Journal

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Definitions of Real Order Integrals and Derivatives Using Operator Approach

Received: 11 December 2012    Accepted:     Published: 20 February 2013
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Abstract

The set E of functions f fulfilling some conditions is taken to be the definition domain of s-order integral operator J^s (iterative integral), first for any positive integer s and after for any positive s (fractional, transcendental π and e). The definition of k-order derivative operator D^k for any positive k (fractional, transcendental π and e) is derived from the definition of J^s. Some properties of J^sand D^k are given and demonstrated. The method is based on the properties of Euler’s gamma and beta functions.

DOI 10.11648/j.pamj.20130201.11
Published in Pure and Applied Mathematics Journal (Volume 2, Issue 1, February 2013)
Page(s) 1-9
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), 2024. Published by Science Publishing Group

Keywords

Gamma Functions; Beta Functions; Integrals; Derivatives; Arbitrary Orders; Operators

References
[1] S. Miller, Kenneth, "An introduction to the fractional calculus and the fractional differential equations", Bertram Ross (Editor). Publisher: John Wiley and Sons 1st edition ,1993, ISBN 0-471-58884-9.
[2] B. Oldham Keith and Spanier Jerome, "The fractional calculus. Theory and Application of differentiation and integration to arbitrary order" (Mathematics in Science and engineering). Publisher: Academic Press, Nov 1974, ISBN 0-12-525550-0.
[3] Zavada, "Operator of fractional derivative in the complex plane", Institute of Physics, Academy of Sciences of Czech Republic, 1997.
[4] F .Dubois, A. C Galucio, N. Point, "Introduction à la dérivation fractionaire. Théorie et Applications", http://www.mathu-psud.fr/~fdubois/travaux/evolution/ananelly-07/techinge-2010/dgp-fractionnaire-mars2010.pdf, 29 Mars 2010.
[5] Raoelina Andriambololona, Tokiniaina Ranaivoson, Rakotoson Hanitriarivo, Roland Raboanary, , "Two definitions of fractional derivatives of power functions". Institut National des Sciences et Techniques Nucleaires (INSTN-Madagascar), 2012, arXiv:1204.1493.
[6] Raoelina Andriambololona, "Algèbre linéaire et multilinéaire." Applications. 3 tomes. Collection LIRA-INSTN Madagascar, Antananarivo, Madagascar, 1986 Tome 1, pp 2-59.
[7] E.T. Whittaker, and G.N. Watson, "A course of modern analysis", Cambridge University Press, Cambridge, 1965.
[8] R. Herrmann, "Fractional Calculus. An introduction for Physicist", World Scientific Publishing, Singapore, 2011.
[9] E. Artin, "The Gamma Function", Holt, Rinehart and Winston, New York, 1964.
[10] S.C. Krantz, "The Gamma and beta functions" § 13.1 in handbook of complex analysis, Birkhauser, Boston, MA, 1999, pp.155-158.
[11] Raoelina Andriambololona, Tokiniaina Ranaivoson, Rakotoson Hanitriarivo, "Definition of complex order integrals and derivatives using operator approach", INSTN preprint 120829, arXiv 1409.400. Published in IJLRST, Vol.1,Issue 4:Page No.317-323, November-December(2012), ISSN (online):2278-5299, http://www.mnkjournals.com/ijlrts.htm
Author Information
  • Theoretical Physics Dept., Antananarivo, Madagascar, Institut National des Sciences et Techniques Nucléaires (INSTN-Madagascar), Boite Postale 4279, Antananarivo 101, Madagascar

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  • APA Style

    Raoelina Andriambololona. (2013). Definitions of Real Order Integrals and Derivatives Using Operator Approach. Pure and Applied Mathematics Journal, 2(1), 1-9. https://doi.org/10.11648/j.pamj.20130201.11

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    ACS Style

    Raoelina Andriambololona. Definitions of Real Order Integrals and Derivatives Using Operator Approach. Pure Appl. Math. J. 2013, 2(1), 1-9. doi: 10.11648/j.pamj.20130201.11

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    AMA Style

    Raoelina Andriambololona. Definitions of Real Order Integrals and Derivatives Using Operator Approach. Pure Appl Math J. 2013;2(1):1-9. doi: 10.11648/j.pamj.20130201.11

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  • @article{10.11648/j.pamj.20130201.11,
      author = {Raoelina Andriambololona},
      title = {Definitions of Real Order Integrals and Derivatives Using Operator Approach},
      journal = {Pure and Applied Mathematics Journal},
      volume = {2},
      number = {1},
      pages = {1-9},
      doi = {10.11648/j.pamj.20130201.11},
      url = {https://doi.org/10.11648/j.pamj.20130201.11},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.pamj.20130201.11},
      abstract = {The set E of functions f fulfilling some conditions is taken to be the definition domain of s-order integral operator J^s (iterative integral), first for any positive integer s and after for any positive s (fractional, transcendental π and e). The definition of k-order derivative operator D^k for any positive k (fractional, transcendental π and e) is derived from the definition of J^s. Some properties of J^sand D^k are given and demonstrated. The method is based on the properties of Euler’s gamma and beta functions.},
     year = {2013}
    }
    

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