New direction in fractional differentiation

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Authors
Abdon Atangana
 Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, 9300, Bloemfontein, South Africa.
Ilknur Koca
 Department of Mathematics, Faculty of Sciences, Mehmet Akif Ersoy University, 15100, Burdur, Turkey.
Abstract
Based upon the MittagLeffler function, new derivatives with fractional order were constructed. With the same line of
idea, improper derivatives based on the Weyl approach are constructed in this work. To further model some complex physical
problems that cannot be modeled with existing derivatives with fractional order, we propose, a new derivative based on the
more generalized MittagLeffler function known as Prabhakar function. Some new results are presented together with some
applications.
Keywords
 AtanganaBaleanu fractional derivatives
 Weyl approach
 Prabhakar MittagLeffler function
 integral transform
References
[1] A. Atangana, On the new fractional derivative and application to nonlinear Fisher’s reactiondiffusion equation, Appl. Math. Comput., 273 (2016), 948956.
[2] A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and nonsingular kernel: theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769.
[3] A. Atangana, I. Koca, Chaos in a simple nonlinear system with AtanganaBaleanu derivatives with fractional order, Chaos Solitons Fractals, 89 (2016), 447–454.
[4] A. Atangana, I. Koca, On the new fractional derivative and application to nonlinear Baggs and Freedman model, J. Nonlinear Sci. Appl., 9 (2016), 2467–2480.
[5] M. Caputo, Linear models of dissipation whose Q is almost frequency independent, II, Reprinted from Geophys. J. R. Astr. Soc., 13 (1967), 529–539, Fract. Calc. Appl. Anal., 11 (2008), 4–14.
[6] M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73–85.
[7] A. V. Chechkin, R. Gorenflo, I. M. Sokolov, Fractional diffusion in inhomogeneous media, J. Phys. A, 38 (2005), L679– L684.
[8] A. H. Cloot, J. P. Botha, A generalised groundwater flow equation using the concept of noninteger order derivatives, Water SA, 32 (2006), 1–7.
[9] M. Davison, C. Essex, Fractional differential equations and initial value problems, Math. Sci., 23 (1998), 108–116.
[10] G. Jumarie, Modified RiemannLiouville derivative and fractional Taylor series of nondifferentiable functions further results, Comput. Math. Appl., 51 (2006), 1367–1376.
[11] A. A. Kilbas, M. Saigo, On MittagLeffler type function, fractional calculus operators and solutions of integral equations, Integral Transform. Spec. Funct., 4 (1996), 355–370.
[12] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, NorthHolland Mathematics Studies, Elsevier Science B.V., Amsterdam, (2006).
[13] I. Koca, A method for solving differential equations of qfractional order, Appl. Math. Comput., 266 (2015), 1–5.
[14] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Dedicated to the 60th anniversary of Prof. Francesco Mainardi, Fract. Calc. Appl. Anal., 5 (2002), 367–386.
[15] T. R. Prabhakar, A singular integral equation with a generalized Mittag Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7–15.