Definitions of Real Order Integrals and Derivatives Using Operator Approach
Pure and Applied Mathematics Journal
Volume 2, Issue 1, February 2013, Pages: 1-9
Received: Dec. 11, 2012; Published: Feb. 20, 2013
Views 3166      Downloads 102
Raoelina Andriambololona, Theoretical Physics Dept., Antananarivo, Madagascar, Institut National des Sciences et Techniques Nucléaires (INSTN-Madagascar), Boite Postale 4279, Antananarivo 101, Madagascar
Article Tools
Follow on us
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.
Gamma Functions; Beta Functions; Integrals; Derivatives; Arbitrary Orders; Operators
To cite this article
Raoelina Andriambololona, Definitions of Real Order Integrals and Derivatives Using Operator Approach, Pure and Applied Mathematics Journal. Vol. 2, No. 1, 2013, pp. 1-9. doi: 10.11648/j.pamj.20130201.11
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.
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.
Zavada, "Operator of fractional derivative in the complex plane", Institute of Physics, Academy of Sciences of Czech Republic, 1997.
F .Dubois, A. C Galucio, N. Point, "Introduction à la dérivation fractionaire. Théorie et Applications",, 29 Mars 2010.
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.
Raoelina Andriambololona, "Algèbre linéaire et multilinéaire." Applications. 3 tomes. Collection LIRA-INSTN Madagascar, Antananarivo, Madagascar, 1986 Tome 1, pp 2-59.
E.T. Whittaker, and G.N. Watson, "A course of modern analysis", Cambridge University Press, Cambridge, 1965.
R. Herrmann, "Fractional Calculus. An introduction for Physicist", World Scientific Publishing, Singapore, 2011.
E. Artin, "The Gamma Function", Holt, Rinehart and Winston, New York, 1964.
S.C. Krantz, "The Gamma and beta functions" § 13.1 in handbook of complex analysis, Birkhauser, Boston, MA, 1999, pp.155-158.
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,
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
Tel: (001)347-983-5186