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

Authors

DOI:

https://doi.org/10.26713/cma.v13i1.1689

Keywords:

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

Abstract

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.

Downloads

Download data is not yet available.

References

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.

Downloads

Published

23-05-2022
CITATION

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

Issue

Section

Research Article