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Chain Rule for Fractional Order Derivatives

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

The concept of derivative is an old concept and there are numerous studies on this concept. Some of these studies are on fractional order derivative. In this paper, we will emphasize that the methods for fractional order derivative are not valid for chain rule, and all definitions for fractional order derivatives have some deficiencies, since the basic concepts of these definitions are based on the pseudo-continuity and gamma function derived from classical derivation. Due to this case, a new definition for chain rule in fractional order derivative was improved. The validity of definition was verified by theorems and examples.

Published in Science Innovation (Volume 3, Issue 6)
DOI 10.11648/j.si.20150306.11
Page(s) 63-67
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 Calculus, Derivative, Fractional Order Derivatives, Chain Rule

References
[1] Newton, I. Philosophiæ Naturalis Principia Mathematica, 1687.
[2] L'Hôpital, G. Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes ("Infinitesimal calculus with applications to curved lines"), Paris, 1696.
[3] L'Hôpital, G. Analyse des infinement petits, Paris, 1715.
[4] Baron, M. E. The Origin of the Infinitesimal Calculus, New York, 1969.
[5] Leibniz, G. F. Correspondence with l‘Hospital, 1695.
[6] Das, S. Functional Fractional Calculus, Springer-Verlag Berlin Heidelberg, 2011.
[7] Mandelbrot, B. B., van Ness, J. W. Fractional Brownian motion, fractional noise and applications, SIAM Rev. 10: 422, 1968.
[8] Mirevski, S. P., Boyadjiev, L., Scherer, R. On the Riemann-Liouville Fractional Calculus, g-Jacobi Functions and F. Gauss Functions, Applied Mathematics and Computation, 187:315-325, 2007.
[9] Schiavone, S. E., Lamb, W. A Fractional Power Approach to Fractional Calculus, Journal of Mathematical Analysis and Applications, 149:377-401, 1990.
[10] Bataineh, A. S., Alomari, A. K., Noorani, M. S. M., Hashim, I., Nazar, R. Series Solutions of Systems of Nonlinear Fractional Differential Equations, Acta Applied Mathematics, 105:189-198, 2009.
[11] Diethelm, K., Ford, N.J., Freed, A.D., Luchko, Y. Algorithms fort he Fractional Calculus: A Selection of Numerical Methdos, Computer Methods in Applied Mechanics and Engineering, 194:743-773, 2005.
[12] Li, C., Chen, A., Ye, J. Numerical Approaches to Fractional Calculus and Fractional Ordinary Differential Equation, Journal of Computational Physics, 230:3352-3368, 2011.
[13] He, J.-H., Elagan, S. K., Li, Z. B. Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus, Physics Letters A, 376:257-259, 2012.
[14] Karcı, A. Kesirli Türev için Yapılan Tanımlamaların Eksiklikleri ve Yeni Yaklaşım, TOK-2013 Turkish Automatic Control National Meeting and Exhibition, Malatya, Turkey, 2013.
[15] Karcı, A. A New Approach for Fractional Order Derivative and Its Applications, Universal Journal of Engineering Sciences, 1:110-117, 2013.
[16] Karcı, A., Karadoğan, A. Fractional Order Derivative and Relationship between Derivative and Complex Functions, IECMSA-2013:2nd International Eurasian Conference on Mathematical Sciences and Applications, Sarajevo, Bosnia and Herzogovina, 2013.
[17] Karcı, A., Karadoğan, A. Fractional Order Derivative and Relationship between Derivative and Complex Functions, Mathematical Sciences and Applications E-Notes, 2:44-54, 2014.
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    Ali Karci. (2015). Chain Rule for Fractional Order Derivatives. Science Innovation, 3(6), 63-67. https://doi.org/10.11648/j.si.20150306.11

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    Ali Karci. Chain Rule for Fractional Order Derivatives. Sci. Innov. 2015, 3(6), 63-67. doi: 10.11648/j.si.20150306.11

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    Ali Karci. Chain Rule for Fractional Order Derivatives. Sci Innov. 2015;3(6):63-67. doi: 10.11648/j.si.20150306.11

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  • @article{10.11648/j.si.20150306.11,
      author = {Ali Karci},
      title = {Chain Rule for Fractional Order Derivatives},
      journal = {Science Innovation},
      volume = {3},
      number = {6},
      pages = {63-67},
      doi = {10.11648/j.si.20150306.11},
      url = {https://doi.org/10.11648/j.si.20150306.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.si.20150306.11},
      abstract = {The concept of derivative is an old concept and there are numerous studies on this concept. Some of these studies are on fractional order derivative. In this paper, we will emphasize that the methods for fractional order derivative are not valid for chain rule, and all definitions for fractional order derivatives have some deficiencies, since the basic concepts of these definitions are based on the pseudo-continuity and gamma function derived from classical derivation. Due to this case, a new definition for chain rule in fractional order derivative was improved. The validity of definition was verified by theorems and examples.},
     year = {2015}
    }
    

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  • TY  - JOUR
    T1  - Chain Rule for Fractional Order Derivatives
    AU  - Ali Karci
    Y1  - 2015/09/24
    PY  - 2015
    N1  - https://doi.org/10.11648/j.si.20150306.11
    DO  - 10.11648/j.si.20150306.11
    T2  - Science Innovation
    JF  - Science Innovation
    JO  - Science Innovation
    SP  - 63
    EP  - 67
    PB  - Science Publishing Group
    SN  - 2328-787X
    UR  - https://doi.org/10.11648/j.si.20150306.11
    AB  - The concept of derivative is an old concept and there are numerous studies on this concept. Some of these studies are on fractional order derivative. In this paper, we will emphasize that the methods for fractional order derivative are not valid for chain rule, and all definitions for fractional order derivatives have some deficiencies, since the basic concepts of these definitions are based on the pseudo-continuity and gamma function derived from classical derivation. Due to this case, a new definition for chain rule in fractional order derivative was improved. The validity of definition was verified by theorems and examples.
    VL  - 3
    IS  - 6
    ER  - 

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

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