×
Abdeljawad, Thabet; Baleanu, Dumitru

Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel. (English) Zbl 1412.47086

J. Nonlinear Sci. Appl. 10, No. 3, 1098-1107 (2017).
Summary: In this manuscript we define the right fractional derivative and its corresponding right fractional integral for the newly suggested nonlocal fractional derivative with Mittag-Leffler kernel. Then, we obtain the related integration by parts formula. We use the Q-operator to confirm our results. The related Euler-Lagrange equations are reported and one illustrative example is discussed.

Cited in 143 Documents

MSC:

26A33 Fractional derivatives and integrals

Keywords:

fractional calculus; Mittag-Leffler function; fractional integration by parts; fractional Euler-Lagrange equations
Cite Review PDF
Full Text: DOI arXiv

References:

[1] Abdeljawad, T., Dual identities in fractional difference calculus within Riemann, Adv. Difference Equ., 2013, 1-16 (2013) · Zbl 1369.39015 · doi:10.1186/1687-1847-2013-36
[2] Abdeljawad, T., On delta and nabla Caputo fractional differences and dual identities, Discrete Dyn. Nat. Soc., 2013, 1-12 (2013) · Zbl 1417.26002 · doi:10.1155/2013/406910
[3] Abdeljawad, T., On conformable fractional calculus, J. Comput. Appl. Math., 279, 57-66 (2015) · Zbl 1304.26004 · doi:10.1016/j.cam.2014.10.016
[4] Abdeljawad, T.; Atici, F. M., On the definitions of nabla fractional operators, Abstr. Appl. Anal., 2012, 1-13 (2012) · Zbl 1253.39004 · doi:10.1155/2012/406757
[5] Abdeljawad, T.; Baleanu, D., Fractional differences and integration by parts, J. Comput. Anal. Appl., 13, 574-582 (2011) · Zbl 1225.39008
[6] Abdeljawad, T.; Baleanu, D., Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Adv. Difference Equ., 2016, 1-18 (2016) · Zbl 1419.34211 · doi:10.1186/s13662-016-0949-5
[7] Agrawal, O. P., Formulation of Euler-Lagrange equations for fractional variational problems, J. Math. Anal. Appl., 272, 368-379 (2002) · Zbl 1070.49013 · doi:10.1016/S0022-247X(02)00180-4
[8] Alkahtani, B. S. T., Chua’s circuit model with Atangana-Baleanu derivative with fractional order, Chaos Solitons Fractals, 89, 547-551 (2016) · Zbl 1360.34160 · doi:10.1016/j.chaos.2016.03.020
[9] Atangana, A.; Baleanu, D., New fractional derivative with non-local and non-singular kernel: theory and application to heat transfer model, Therm. Sci., 20, 763-769 (2016)
[10] Atangana, A.; Koca, I., Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Solitons Fractals, 89, 447-457 (2016) · Zbl 1360.34150 · doi:10.1016/j.chaos.2016.02.012
[11] Baleanu, D.; Trujillo, J. J., On exact solutions of a class of fractional Euler-Lagrange equations, Nonlinear Dynam., 52, 331-335 (2008) · Zbl 1170.70328 · doi:10.1007/s11071-007-9281-7
[12] Caputo, M.; Fabrizio, M., A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1, 73-85 (2015) · doi:10.12785/pfda/010201
[13] Gao, F.; Yang, X.-J., Fractional Maxwell fluid with fractional derivative without singular kernel, Therm. Sci., 20, 871-877 (2016)
[14] R. Hilfer, Applications of fractional calculus in physics, World Sci. Publ., Singapore (2000) · Zbl 0998.26002
[15] Khalil, R.; Horani, M. Al; Yousef, A.; Sababheh, M., A new definition of fractional derivative, J. Comput. Appl. Math., 264, 65-70 (2014) · Zbl 1297.26013 · doi:10.1016/j.cam.2014.01.002
[16] Kilbas, A. A.; Saigo, M.; Saxena, K., Generalized Mittag-Leffler function and generalized fractional calculus operators, Integral Transforms Spec. Funct., 15, 31-49 (2004) · Zbl 1047.33011 · doi:10.1080/10652460310001600717
[17] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam (2006) · Zbl 1092.45003
[18] Lorenzo, C. F.; Hartley, T. T., Variable order and distributed order fractional operators, Fractional order calculus and its applications, Nonlinear Dynam., 29, 57-98 (2002) · Zbl 1018.93007 · doi:10.1023/A:1016586905654
[19] Losada, J.; Nieto, J. J., Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1, 87-92 (2015) · doi:10.12785/pfda/010202
[20] Machado, J. T.; Kiryakova, V.; Mainardi, F., Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul., 16, 1140-1153 (2011) · Zbl 1221.26002 · doi:10.1016/j.cnsns.2010.05.027
[21] Magin, R. L., Fractional calculus in bioengineering, Begell House Publishers Inc., CT (2006)
[22] Maraaba, T.; Baleanu, D.; Jarad, F., Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives, J. Math. Phys., 49, 1-11 (2008) · Zbl 1152.81550 · doi:10.1063/1.2970709
[23] Podlubny, I., 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) · Zbl 0918.34010
[24] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional integrals and derivatives, Theory and applications, Edited and with a foreword by S. M. Nikol’ski˘ı, Translated from the 1987 Russian original, Revised by the authors, Gordon and Breach Science Publishers, Yverdon (1993) · Zbl 0818.26003
[25] Yang, X.-J.; Baleanu, D.; Srivastava, H. M., Local fractional integral transforms and their applications, Elsevier/Academic Press, Amsterdam (2005)
[26] Yang, A.-M.; Han, Y.; Li, J.; Liu, W.-X., On steady heat flow problem involving Yang-Srivastava-Machado fractional derivative without singular kernel, Therm. Sci., 20, 717-723 (2016)
[27] Yang, X.-J.; Srivastava, H. M.; Machado, J. A. Tenreiro, A new fractional derivative without singular kernel: application to the modelling of the steady heat flow, Therm. Sci., 20, 753-756 (2016)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.