Two Definitions of Fractional Derivative of Powers Functions

Raoelina Andriambololona; Rakotoson Hanitriarivo; Tokiniaina Ranaivoson; Roland Raboanary

doi:doi:10.11648/j.pamj.20130201.12

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Two Definitions of Fractional Derivative of Powers Functions

Raoelina Andriambololona, Rakotoson Hanitriarivo, Tokiniaina Ranaivoson, Roland Raboanary

Published in Pure and Applied Mathematics Journal (Volume 2, Issue 1)

Published: 20 February 2013

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Abstract

We consider the set of powers functions defined on R_+ and their linear combinations. After recalling some properties of the gamma function, we give two general definitions of derivatives of positive and negative integers, positive and negative fractional orders. Properties of linearity and commutativity are studied and the notions of semi-equality, semi-linearity and semi-commutativity are introduced. Our approach gives a unified definition of the common derivatives and integrals and their generalization.

Published in	Pure and Applied Mathematics Journal (Volume 2, Issue 1)
DOI	10.11648/j.pamj.20130201.12
Page(s)	10-19
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), 2013. Published by Science Publishing Group

Keywords

Gamma Function; Fractional Derivatives; Fractional Integrals; Power Functions

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]	K.B.Oldham, J.Spanier, "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, S. Mengué, "Mixed Collocation for Fractional Differential Equations", Numerical Algorithms, vol. 34, p. 303-311, 2003.
[5]	A.C. Galucio, J.-F.Deü, S. Mengué, F. Dubois,"An adaptation of the Gear scheme for fractional derivatives", Comp. Methods in Applied Mech. Engineering, vol. 195, p. 6073–6085, 2006.
[6]	A. Babakhani, V. Daftardar-Gejji, "On calculus of local fractional derivatives",J. Math. Anal. Appl. 270, 66-79 (2002).
[7]	A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, "Theory and Applications of Fractional Differential Equations" (North-Holland Mathematical Studies 204, Ed.Jan van Mill, Amsterdam (2006))
[8]	E. Artin, "The Gamma Function". New York: Holt, Rinehart, and Winston, 1964.
[9]	G.E. Andrews, R. Askey, and R. Roy, "Special functions", Cambridge, 1999.
[10]	S.C. Krantz, "The Gamma and Beta functions", §13.1 in Handbook of Complex analysis, Birkhauser, Boston, MA, 1999, pp. 155-158.
[11]	G. Arfken, "The Gamma Function". Ch 10 in Mathematical Methods for Physics FL: Academic Press, pp.339-341and 539-572, 1985.
[12]	Raoelina Andriambololona,"Definition of real order integrals and derivatives using operator approach," preprint Institut National des Sciences et Techniques Nucléaires, (INSTN-Madagascar), May 2012, arXiv:1207.0409
[13]	Raoelina Andriambololona, Tokiniaina Ranaivoson, Rako-toson 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), http://www.mnkjournals.com/ijlrts.htm.

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APA Style

Raoelina Andriambololona, Rakotoson Hanitriarivo, Tokiniaina Ranaivoson, Roland Raboanary. (2013). Two Definitions of Fractional Derivative of Powers Functions. Pure and Applied Mathematics Journal, 2(1), 10-19. https://doi.org/10.11648/j.pamj.20130201.12

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ACS Style

Raoelina Andriambololona; Rakotoson Hanitriarivo; Tokiniaina Ranaivoson; Roland Raboanary. Two Definitions of Fractional Derivative of Powers Functions. Pure Appl. Math. J. 2013, 2(1), 10-19. doi: 10.11648/j.pamj.20130201.12

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AMA Style

Raoelina Andriambololona, Rakotoson Hanitriarivo, Tokiniaina Ranaivoson, Roland Raboanary. Two Definitions of Fractional Derivative of Powers Functions. Pure Appl Math J. 2013;2(1):10-19. doi: 10.11648/j.pamj.20130201.12

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@article{10.11648/j.pamj.20130201.12,
  author = {Raoelina Andriambololona and Rakotoson Hanitriarivo and Tokiniaina Ranaivoson and Roland Raboanary},
  title = {Two Definitions of Fractional Derivative of Powers Functions},
  journal = {Pure and Applied Mathematics Journal},
  volume = {2},
  number = {1},
  pages = {10-19},
  doi = {10.11648/j.pamj.20130201.12},
  url = {https://doi.org/10.11648/j.pamj.20130201.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20130201.12},
  abstract = {We consider the set of powers functions defined on R_+ and their linear combinations. After recalling some properties of the gamma function, we give two general definitions of derivatives of positive and negative integers, positive and negative fractional orders. Properties of linearity and commutativity are studied and the notions of semi-equality, semi-linearity and semi-commutativity are introduced. Our approach gives a unified definition of the common derivatives and integrals and their generalization.},
 year = {2013}
}

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TY  - JOUR
T1  - Two Definitions of Fractional Derivative of Powers Functions
AU  - Raoelina Andriambololona
AU  - Rakotoson Hanitriarivo
AU  - Tokiniaina Ranaivoson
AU  - Roland Raboanary
Y1  - 2013/02/20
PY  - 2013
N1  - https://doi.org/10.11648/j.pamj.20130201.12
DO  - 10.11648/j.pamj.20130201.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
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PB  - Science Publishing Group
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AB  - We consider the set of powers functions defined on R_+ and their linear combinations. After recalling some properties of the gamma function, we give two general definitions of derivatives of positive and negative integers, positive and negative fractional orders. Properties of linearity and commutativity are studied and the notions of semi-equality, semi-linearity and semi-commutativity are introduced. Our approach gives a unified definition of the common derivatives and integrals and their generalization.
VL  - 2
IS  - 1
ER  -

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Author Information

Raoelina Andriambololona

Theoretical Physics Dept., Antananarivo, Madagascar
Rakotoson Hanitriarivo

Theoretical Physics Dept., Antananarivo, Madagascar
Tokiniaina Ranaivoson

Theoretical Physics Dept., Antananarivo, Madagascar
Roland Raboanary

Theoretical Physics Dept., Antananarivo, Madagascar

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APA Style

Raoelina Andriambololona, Rakotoson Hanitriarivo, Tokiniaina Ranaivoson, Roland Raboanary. (2013). Two Definitions of Fractional Derivative of Powers Functions. Pure and Applied Mathematics Journal, 2(1), 10-19. https://doi.org/10.11648/j.pamj.20130201.12

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ACS Style

Raoelina Andriambololona; Rakotoson Hanitriarivo; Tokiniaina Ranaivoson; Roland Raboanary. Two Definitions of Fractional Derivative of Powers Functions. Pure Appl. Math. J. 2013, 2(1), 10-19. doi: 10.11648/j.pamj.20130201.12

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AMA Style

Raoelina Andriambololona, Rakotoson Hanitriarivo, Tokiniaina Ranaivoson, Roland Raboanary. Two Definitions of Fractional Derivative of Powers Functions. Pure Appl Math J. 2013;2(1):10-19. doi: 10.11648/j.pamj.20130201.12

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@article{10.11648/j.pamj.20130201.12,
  author = {Raoelina Andriambololona and Rakotoson Hanitriarivo and Tokiniaina Ranaivoson and Roland Raboanary},
  title = {Two Definitions of Fractional Derivative of Powers Functions},
  journal = {Pure and Applied Mathematics Journal},
  volume = {2},
  number = {1},
  pages = {10-19},
  doi = {10.11648/j.pamj.20130201.12},
  url = {https://doi.org/10.11648/j.pamj.20130201.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20130201.12},
  abstract = {We consider the set of powers functions defined on R_+ and their linear combinations. After recalling some properties of the gamma function, we give two general definitions of derivatives of positive and negative integers, positive and negative fractional orders. Properties of linearity and commutativity are studied and the notions of semi-equality, semi-linearity and semi-commutativity are introduced. Our approach gives a unified definition of the common derivatives and integrals and their generalization.},
 year = {2013}
}

Copy | Download

TY  - JOUR
T1  - Two Definitions of Fractional Derivative of Powers Functions
AU  - Raoelina Andriambololona
AU  - Rakotoson Hanitriarivo
AU  - Tokiniaina Ranaivoson
AU  - Roland Raboanary
Y1  - 2013/02/20
PY  - 2013
N1  - https://doi.org/10.11648/j.pamj.20130201.12
DO  - 10.11648/j.pamj.20130201.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 10
EP  - 19
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20130201.12
AB  - We consider the set of powers functions defined on R_+ and their linear combinations. After recalling some properties of the gamma function, we give two general definitions of derivatives of positive and negative integers, positive and negative fractional orders. Properties of linearity and commutativity are studied and the notions of semi-equality, semi-linearity and semi-commutativity are introduced. Our approach gives a unified definition of the common derivatives and integrals and their generalization.
VL  - 2
IS  - 1
ER  -

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