A New Definition of Fractional Derivatives With Mittag-Leffler Kernel of Two Parameters





fractional derivatives, non-local kernel, non-singular kernel, Mittag-Leffler function of two parameters, fractional time Fourier's law equation


In this paper, a new fractional derivative with Mittag-Leffler kernel of two parameters is proposed. Several functional properties of this derivative are explained and applied to solve the fractional time Fourier’s law equation.


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A. Atangana, On the new fractional derivative and application to nonlinear Fisher’s reaction-diffusion equation, Applied Mathematics and Computation 273 (2016), 948 – 956, DOI: 10.1016/j.amc.2015.10.021.

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science 20(2) (2016), 763 – 769, DOI: 10.2298/TSCI160111018A.

A. Atangana and J. J. Nieto, Numerical solution for the model of RLC circuit via the fractional derivative without singular kernel, Advances in Mechanical Engineering 7(10) (2015), 1 – 7, DOI: 10.1177/1687814015613758.

D. Baleanu and A. Fernandez, On some new properties of fractional derivatives with Mittag-Leffler kernel, Communications in Nonlinear Science and Numerical Simulation 59 (2018), 444 – 462, DOI: 10.1016/j.cnsns.2017.12.003.

D. A. Benson, S. W. Wheatcraft and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resources Research 36(6) (2000), 1403 – 1412, DOI: 10.1029/2000WR900031.

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications 1(2) (2015), 73 – 85, URL: https://www.naturalspublishing.com/Article.asp?ArtcID=8820.

M. Caputo and M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progress in Fractional Differentiation and Applications 2(2) (2016), 1 – 11, DOI: 10.18576/pfda/020101.

J. F. Gomez-Aguilar, A. Atangana and R. F. Escobar-Jimenez, Numerical solutions of fourier’s law involving fractional derivatives with bi-order, Scientia Iranica 25(4) (2018), 2175 – 2185, DOI: 10.24200/sci.2017.4342.

R. Gorenflo, F. Mainardi and S. Rogosin, Mittag-Leffler function: Properties and applications, in: Basic Theory, Volume 1, edited by A. Kochubei and Y. Luchko, Berlin — Boston, De Gruyter, pp. 269 – 296 2019, DOI: 10.1515/9783110571622-011.

J.-H. He, A new fractal derivation, Thermal Science 15(Suppl. 1) (2011), 145 – 147, DOI: 10.2298/TSCI11S1145H.

J. Hristov, Approximate solutions to time-fractional models by integral-balance approach, in: Fractional Dynamics, edited by C. Cattani, H. M. Srivastava and X.-J. Yang, De Gruyter Open, Poland, Warsaw, Poland, pp. 78 – 109, 2016, DOI: 10.1515/9783110472097-006.

J. Hristov, Double integral-balance method to the fractional subdiffusion equation: Approximate solutions, optimization problems to be resolved and numerical simulations, Journal of Vibration and Control 23(17) (2017), 2795 – 2818, DOI: 10.1177/1077546315622773.

X. Jiang and M. Xu, The time fractional heat conduction equation in the general orthogonal curvilinear coordinate and the cylindrical coordinate systems, Physica A: Statistical Mechanics and its Applications 389(17) (2010), 3368 – 3374, DOI: 10.1016/j.physa.2010.04.023.

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, in: North Holland Mathematics Studies, Volume 204, 1st Edition, pp. 3368 – 3374, Elsevier Science, Publishers BV, Amsterdam (2006), URL: https://www.elsevier.com/books/theoryand-applications-of-fractional-differential-equations/kilbas/978-0-444-51832-3.

F.-J. Liu, Z.-B. Li, S. Zhang and H.-Y. Liu, He’s fractional derivative for heat conduction in a fractal medium arising in silkworm cocoon hierarchy, Thermal Science 19(4) (2015), 1155 – 1159, DOI: 10.2298/TSCI1504155L.

J. Losada and J.J. Nieto, Properties of a new fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications 1(2) (2015), 87 – 92, URL: https://www.naturalspublishing.com/Article.asp?ArtcID=8821.

S. Rogosin, The role of the Mittag-Leffler function in fractional modeling, Mathematics 3(2) (2015), 368 – 381, DOI: 10.3390/math3020368.




How to Cite

Chinchole, S. M., & Bhadane, A. (2022). A New Definition of Fractional Derivatives With Mittag-Leffler Kernel of Two Parameters. Communications in Mathematics and Applications, 13(1), 19–26. https://doi.org/10.26713/cma.v13i1.1689



Research Article