Chain Rule for Fractional Order Derivatives
Science Innovation
Volume 3, Issue 6, December 2015, Pages: 63-67
Received: Aug. 28, 2015; Accepted: Sep. 11, 2015; Published: Sep. 24, 2015
Views 4258      Downloads 250
Author
Ali Karci, Department of Computer Engineering, Faculty of Engineering, İnönü University, Malatya, Turkey
Article Tools
Follow on us
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.
Keywords
Fractional Calculus, Derivative, Fractional Order Derivatives, Chain Rule
To cite this article
Ali Karci, Chain Rule for Fractional Order Derivatives, Science Innovation. Vol. 3, No. 6, 2015, pp. 63-67. doi: 10.11648/j.si.20150306.11
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.
ADDRESS
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
U.S.A.
Tel: (001)347-983-5186