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Entropy 2013, 15(12), 5305-5323; doi:10.3390/e15125305

New Results on Fractional Power Series: Theories and Applications

1 Department of Mathematics, Faculty of Science, Al Balqa Applied University, Salt 19117, Jordan 2 Department of Basic Sciences and Humanities, College of Engineering, University of Dammam, Dammam 31451, KSA, Saudi Arabia 3 Department of Mathematics, Faculty of Science, University of Jordan, Amman 11942, Jordan
* Author to whom correspondence should be addressed.
Received: 12 September 2013 / Revised: 9 October 2013 / Accepted: 9 October 2013 / Published: 2 December 2013
(This article belongs to the Special Issue Dynamical Systems)
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In this paper, some theorems of the classical power series are generalized for the fractional power series. Some of these theorems are constructed by using Caputo fractional derivatives. Under some constraints, we proved that the Caputo fractional derivative can be expressed in terms of the ordinary derivative. A new construction of the generalized Taylor’s power series is obtained. Some applications including approximation of fractional derivatives and integrals of functions and solutions of linear and nonlinear fractional differential equations are also given. In the nonlinear case, the new and simple technique is used to find out the recurrence relation that determines the coefficients of the fractional power series.
Keywords: Fractional power series; Caputo fractional derivative; fractional differential equations Fractional power series; Caputo fractional derivative; fractional differential equations
This is an open access article distributed under the Creative Commons Attribution License (CC BY) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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El-Ajou, A.; Arqub, O.A.; Zhour, Z.A.; Momani, S. New Results on Fractional Power Series: Theories and Applications. Entropy 2013, 15, 5305-5323.

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