On the generalized fractional derivatives and their Caputo modification
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Authors
F. Jarad
- Mathematics Department, Faculty of Arts and Sciences, ÇankayaUniversity, 06790, Etimesgut, Ankara, Turkey.
T. Abdeljawad
- Department of Mathematics and Physical Sciences, Prince Sultan University, P. O. Box 66833, 11586 Riyadh, Saudi Arabia.
D. Baleanu
- Mathematics Department, Faculty of Arts and Sciences, ÇankayaUniversity, 06790, Etimesgut, Ankara, Turkey.
- Institute of Space Sciences, Magurele-Bucharest, Romania.
Abstract
In this manuscript, we define the generalized fractional derivative on \(AC^n_\gamma
[a, b]\), the space of functions defined on [a, b]
such that
\(\gamma^{n-1}f\in AC[a, b]\), where
\(\gamma=x^{1-p}\frac{d}{dx}\). We present some of the properties of generalized fractional derivatives of these
functions and then we define their Caputo version.
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ISRP Style
F. Jarad, T. Abdeljawad, D. Baleanu, On the generalized fractional derivatives and their Caputo modification, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 5, 2607--2619
AMA Style
Jarad F., Abdeljawad T., Baleanu D., On the generalized fractional derivatives and their Caputo modification. J. Nonlinear Sci. Appl. (2017); 10(5):2607--2619
Chicago/Turabian Style
Jarad, F., Abdeljawad, T., Baleanu, D.. "On the generalized fractional derivatives and their Caputo modification." Journal of Nonlinear Sciences and Applications, 10, no. 5 (2017): 2607--2619
Keywords
- Riemann-Liouville fractional derivatives
- Caputo fractional derivatives
- Hadamard fractional derivatives
- Caputo-Hadamard fractional derivatives
- generalized fractional integral
- generalized Caputo fractional derivatives.
MSC
References
-
[1]
T. Abdeljawad, Dual identities in fractional difference calculus within Riemann, Adv. Difference Equ., 2013 (2013), 16 pages.
-
[2]
T. Abdeljawad, On delta and nabla Caputo fractional differences and dual identities, Discrete Dyn. Nat. Soc., 2013 (2013), 12 pages.
-
[3]
T. Abdeljawad, D. Baleanu, Fractional differences and integration by parts, J. Comput. Anal. Appl., 13 (2011), 574–582.
-
[4]
T. Abdeljawad, D. Baleanu, Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Adv. Difference Equ., 2016 (2016), 18 pages.
-
[5]
R. Almeida, A. B. Malinowska, T. Odzijewicz, Fractional differential equations with dependence on the Caputo- Katugampola derivative, J. Comput. Nonlinear Dyn., 11 (2016), 11 pages.
-
[6]
F. M. Atıcı, S. Şengül, Modeling with fractional difference equations, J. Math. Anal. Appl., 369 (2010), 1–9.
-
[7]
M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73–85.
-
[8]
Y. Y. Gambo F. Jarad, D. Baleanu, T. Abdeljawad, On Caputo modification of the Hadamard fractional derivatives, Adv. Difference Equ., 2014 (2014), 12 pages.
-
[9]
F. Gao, X.-J. Yang, Fractional Maxwell fluid with fractional derivative without singular kernel, Therm. Sci., 20 (2016), S871–S877.
-
[10]
C. Goodrich, A. C. Peterson, Discrete fractional calculus, Springer, Cham (2015)
-
[11]
R. Hilfer (Ed.), Applications of fractional calculus in physics, World Scientific Publishing Co., Inc., River Edge, NJ (2000)
-
[12]
F. Jarad, T. Abdeljawad, D. Baleanu, Caputo-type modification of the Hadamard fractional derivatives, Adv. Difference Equ., 2012 (2012), 8 pages.
-
[13]
U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput., 218 (2011), 860–865.
-
[14]
U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6 (2014), 1–15.
-
[15]
A. A. Kilbas, Hadamard-type fractional calculus, J. Korean Math. Soc., 38 (2001), 1191–1204.
-
[16]
A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam (2006)
-
[17]
J. Losada, J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 87–92.
-
[18]
J. T. Machado, V. Kiryakova, F. Mainardi, Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1140–1153.
-
[19]
R. L. Magin, Fractional calculus in bioengineering, Begell House Publishers, CT (2006)
-
[20]
I. Podlubny, Fractional differential equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA (1999)
-
[21]
S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives, Theory and applications, Edited and with a foreword by S. M. Nikolski˘ı, Translated from the 1987 Russian original, Revised by the authors, Gordon and Breach Science Publishers, Yverdon (1993)
-
[22]
X.-J. Yang, D. Baleanu, H. M. Srivastava, Local fractional integral transforms and their applications, Elsevier/Academic Press, Amsterdam (2016)
-
[23]
X.-J. Yang, F. Gao, J. A. Tenreiro Machado, D. Baleanu, A new fractional derivative involving the normalized sinc function without singular kernel, ArXiv, 2017 (2017), 11 pages.
