Generalized Fractional Order Derivatives for Products and Quotients

Ali Karci

doi:doi:10.11648/j.si.20150305.13

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Generalized Fractional Order Derivatives for Products and Quotients

Ali Karci

Published in Science Innovation (Volume 3, Issue 5)

Received: 28 August 2015 Accepted: 11 September 2015 Published: 24 September 2015

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Abstract

The concept of fractional order derivative (FOD) can be found in extensive range of many different subject areas. For this reason, the concept of FOD should be examined in wide range. There are lots of methods about FOD in the literature; however, none of them are FOD methods. Since all of them are curve fitting or curve approximation methods. In fact, the methods used in the literature are not FOD methods; they are approximation methods. In this paper, we redefined FOD for product and quotient. The obtained definition is same as classical derivative definition in case of fractional order is equal to 1. FOD of products and quotients were handled in this paper with some applications. The properties of both theorems were analysed in this paper.

Published in	Science Innovation (Volume 3, Issue 5)
DOI	10.11648/j.si.20150305.13
Page(s)	58-62
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), 2015. Published by Science Publishing Group

Keywords

Fractional Order Derivatives, Derivative of Product, Derivative of Quotient

References

[1]	S. Pooseh, R. Almeida, D. F. M. Torres, Discrete direct methods in the fractional calculus of variations, Computers and Mathematics with Applications, doi:10.1016/j.camwa.2013.01.045.
[2]	S. P. Mirevski, L. Boyadjiev, R. Scherer, On the Riemann-Liouville Fractional Calculus, g-Jacobi Functions and F. Gauss Functions, Applied Mathematics and Computation, 187; 315-325 (2007).
[3]	S. E. Schiavone, W. Lamb, A Fractional Power Approach to Fractional Calculus, Journal of Mathematical Analysis and Applications, 149; 377-401 (1990).
[4]	A. S. Bataineh A. K. Alomari M. S. M. Noorani I. Hashim R. Nazar, Series Solutions of Systems of Nonlinear Fractional Differential Equations, Acta Applied Mathematics, 105; 189-198 (2009).
[5]	K. Diethelm, N. J. Ford, A. D. Freed, Yu Luchko, Algorithms for the Fractional Calculus: A Selection of Numerical Methdos, Computer Methods in Applied Mechanics and Engineering, 194; 743-773 (2005).
[6]	C. Li, A. Chen, J. Ye, Numerical Approaches to Fractional Calculus and Fractional Ordinary Differential Equation, Journal of Computational Physics, 230; 3352-3368 (2011).
[7]	Y. Li, Y. Q. Chen, H.-S. Ahn, G. Tian, A Survey on Fractional-Order Iterative Learning Control, Journal of Optimal Theory and Applications, 156; 127-140 (2013).
[8]	C. Li, D. Qian, Y.-Q. Chen, On Riemann-Liouville and Caputo Derivatives, Discrete Dynamics in Nature and Society, 2011, DOI: 10.1155/2011/562494.
[9]	M. Ö. Efe, Fractional Order Sliding Mode Control with Reaching Law Approach, Turkish Journal of Electrical Engineering & Computer Science, 18; 731-747 (2010).
[10]	E. A. Swokowski, Calculus with Analytic Geometry, Prindle-Weber & Schmidt, p: 87-110, (1983).
[11]	A. Karcı, Kesirli Türev için Yapılan Tanımlamaların Eksiklikleri ve Yeni Yaklaşım, TOK-2013 Turkish Automatic Control National Meeting and Exhibition.
[12]	A. Karcı, Generalized Fractional Order Derivatives, Its Properties and Applications, ArXiv: 1306. 5672 General Mathematics.
[13]	A. Karcı, A.Karadoğan, “Fractional Order Derivative and Relationship between Derivative and Complex Functions”, IECMSA-2013:2^nd International Eurasian Conference on Mathematical Sciences and Applications, Sarajevo, Bosnia and Herzogovina, p: 55-56, Aug. 26-29, 2013.
[14]	A. Karcı, “A New Approach for Fractional Order Derivative and Its Applications”, Universal Journal of Engineering Sciences, 1(3): 110-117, 2013.

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Ali Karci. (2015). Generalized Fractional Order Derivatives for Products and Quotients. Science Innovation, 3(5), 58-62. https://doi.org/10.11648/j.si.20150305.13

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ACS Style

Ali Karci. Generalized Fractional Order Derivatives for Products and Quotients. Sci. Innov. 2015, 3(5), 58-62. doi: 10.11648/j.si.20150305.13

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AMA Style

Ali Karci. Generalized Fractional Order Derivatives for Products and Quotients. Sci Innov. 2015;3(5):58-62. doi: 10.11648/j.si.20150305.13

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@article{10.11648/j.si.20150305.13,
  author = {Ali Karci},
  title = {Generalized Fractional Order Derivatives for Products and Quotients},
  journal = {Science Innovation},
  volume = {3},
  number = {5},
  pages = {58-62},
  doi = {10.11648/j.si.20150305.13},
  url = {https://doi.org/10.11648/j.si.20150305.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.si.20150305.13},
  abstract = {The concept of fractional order derivative (FOD) can be found in extensive range of many different subject areas. For this reason, the concept of FOD should be examined in wide range. There are lots of methods about FOD in the literature; however, none of them are FOD methods. Since all of them are curve fitting or curve approximation methods. In fact, the methods used in the literature are not FOD methods; they are approximation methods. In this paper, we redefined FOD for product and quotient. The obtained definition is same as classical derivative definition in case of fractional order is equal to 1. FOD of products and quotients were handled in this paper with some applications. The properties of both theorems were analysed in this paper.},
 year = {2015}
}

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TY  - JOUR
T1  - Generalized Fractional Order Derivatives for Products and Quotients
AU  - Ali Karci
Y1  - 2015/09/24
PY  - 2015
N1  - https://doi.org/10.11648/j.si.20150305.13
DO  - 10.11648/j.si.20150305.13
T2  - Science Innovation
JF  - Science Innovation
JO  - Science Innovation
SP  - 58
EP  - 62
PB  - Science Publishing Group
SN  - 2328-787X
UR  - https://doi.org/10.11648/j.si.20150305.13
AB  - The concept of fractional order derivative (FOD) can be found in extensive range of many different subject areas. For this reason, the concept of FOD should be examined in wide range. There are lots of methods about FOD in the literature; however, none of them are FOD methods. Since all of them are curve fitting or curve approximation methods. In fact, the methods used in the literature are not FOD methods; they are approximation methods. In this paper, we redefined FOD for product and quotient. The obtained definition is same as classical derivative definition in case of fractional order is equal to 1. FOD of products and quotients were handled in this paper with some applications. The properties of both theorems were analysed in this paper.
VL  - 3
IS  - 5
ER  -

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Author Information

Ali Karci

Department of Computer Engineering, Faculty of Engineering, ?n?nü University, Malatya, Turkey

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APA Style

Ali Karci. (2015). Generalized Fractional Order Derivatives for Products and Quotients. Science Innovation, 3(5), 58-62. https://doi.org/10.11648/j.si.20150305.13

Copy | Download

ACS Style

Ali Karci. Generalized Fractional Order Derivatives for Products and Quotients. Sci. Innov. 2015, 3(5), 58-62. doi: 10.11648/j.si.20150305.13

Copy | Download

AMA Style

Ali Karci. Generalized Fractional Order Derivatives for Products and Quotients. Sci Innov. 2015;3(5):58-62. doi: 10.11648/j.si.20150305.13

Copy | Download

@article{10.11648/j.si.20150305.13,
  author = {Ali Karci},
  title = {Generalized Fractional Order Derivatives for Products and Quotients},
  journal = {Science Innovation},
  volume = {3},
  number = {5},
  pages = {58-62},
  doi = {10.11648/j.si.20150305.13},
  url = {https://doi.org/10.11648/j.si.20150305.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.si.20150305.13},
  abstract = {The concept of fractional order derivative (FOD) can be found in extensive range of many different subject areas. For this reason, the concept of FOD should be examined in wide range. There are lots of methods about FOD in the literature; however, none of them are FOD methods. Since all of them are curve fitting or curve approximation methods. In fact, the methods used in the literature are not FOD methods; they are approximation methods. In this paper, we redefined FOD for product and quotient. The obtained definition is same as classical derivative definition in case of fractional order is equal to 1. FOD of products and quotients were handled in this paper with some applications. The properties of both theorems were analysed in this paper.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - Generalized Fractional Order Derivatives for Products and Quotients
AU  - Ali Karci
Y1  - 2015/09/24
PY  - 2015
N1  - https://doi.org/10.11648/j.si.20150305.13
DO  - 10.11648/j.si.20150305.13
T2  - Science Innovation
JF  - Science Innovation
JO  - Science Innovation
SP  - 58
EP  - 62
PB  - Science Publishing Group
SN  - 2328-787X
UR  - https://doi.org/10.11648/j.si.20150305.13
AB  - The concept of fractional order derivative (FOD) can be found in extensive range of many different subject areas. For this reason, the concept of FOD should be examined in wide range. There are lots of methods about FOD in the literature; however, none of them are FOD methods. Since all of them are curve fitting or curve approximation methods. In fact, the methods used in the literature are not FOD methods; they are approximation methods. In this paper, we redefined FOD for product and quotient. The obtained definition is same as classical derivative definition in case of fractional order is equal to 1. FOD of products and quotients were handled in this paper with some applications. The properties of both theorems were analysed in this paper.
VL  - 3
IS  - 5
ER  -

Copy | Download