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Mathematics 2016, 4(2), 30; doi:10.3390/math4020030

New Approach for Fractional Order Derivatives: Fundamentals and Analytic Properties

Department of Computer Engineering, Faculty of Engineering, İnönü University, 44280 Malatya, Turkey
Academic Editor: Hari M. Srivastava
Received: 25 February 2016 / Revised: 18 April 2016 / Accepted: 21 April 2016 / Published: 4 May 2016
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Abstract

The rate of change of any function versus its independent variables was defined as a derivative. The fundamentals of the derivative concept were constructed by Newton and l’Hôpital. The followers of Newton and l’Hôpital defined fractional order derivative concepts. We express the derivative defined by Newton and l’Hôpital as an ordinary derivative, and there are also fractional order derivatives. So, the derivative concept was handled in this paper, and a new definition for derivative based on indefinite limit and l’Hôpital’s rule was expressed. This new approach illustrated that a derivative operator may be non-linear. Based on this idea, the asymptotic behaviors of functions were analyzed and it was observed that the rates of changes of any function attain maximum value at inflection points in the positive direction and minimum value (negative) at inflection points in the negative direction. This case brought out the fact that the derivative operator does not have to be linear; it may be non-linear. Another important result of this paper is the relationships between complex numbers and derivative concepts, since both concepts have directions and magnitudes. View Full-Text
Keywords: derivatives; fractional calculus; fractional order derivatives derivatives; fractional calculus; fractional order derivatives
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Karcı, A. New Approach for Fractional Order Derivatives: Fundamentals and Analytic Properties. Mathematics 2016, 4, 30.

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