THE LOCAL GENERALIZED DERIVATIVE AND MITTAG-LEFFLER FUNCTION

In this paper, we present a general definition of a generalized derivative of local type using the well known Mittag-Leffler function. Some methodological remarks on the local fractional derivatives are also presented.

Keywords:

Fractional calculus, Fractional, calculus,

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[1] Abdeljawad, T., (2015) On conformable fractional calculus. J. Comput. Appl. Math., 279, 57-66.
[2] Agarwal, R. P., Benchohra, M. and Hamani, S. A., (2010) Survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math. 109, 973–1033.
[3] Bonilla, B., Rivero, M. and Trujillo, J. J., (2007) On systems of linear fractional differential equations with constant coefficients, Appl. Math. Comput. 187, 68–78.
[4] Boyce,W. E. and DiPrima, R. C., (1977) Elementary Diflerential Equations, John Wiley and Sons, New York.
[5] Diethelm, K., (2010) The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type (Springer Science and Business Media).
[6] Deshpande, A. S., Daftardar-Gejji, V. and Vellaisamy, P., (2010) Analysis of intersections of trajectories of systems of linear fractional differential equations, Chaos 29, 013113; doi: 10.1063/1.5052067.
[7] Diethelm, K., and Ford, N., (2012) Volterra integral equations and fractional calculus: Do neighboring solutions intersect?, J. Integral Equ. Appl. 24, 25–37.
[8] Guzmán, P. M., Langton, G., Lugo, L. M., Medina, J. and Nápoles Valdés, J. E., (2018) A new definition of a fractional derivative of local type. J. Math. Anal., 9:2, 88-98.
[9] Guzmán, P. M., Lugo Motta Bittencurt, L. M. and Nápoles Valdés, J. E., (2019) A Note on Stability of Certain Lienard Fractional Equation, International Journal of Mathematics and Computer Science, 14, no. 2, 301–315.
[10] Chandra Har, K. Genesis of Calculus, available in https://es.scribd.com/document/14639948/Genesis-of-Calculus
[11] Hayek, N., Trujillo, J., Rivero, M., Bonilla, B., and Moreno, J., (1998) An extension of Picard-Lindeloff theorem to fractional differential equations, Appl. Anal. 70, 347–361.
[12] Khalil, R., Al Horani, M., Yousef, A., Sababheh, M., (2014) A new definition of fractional derivative. J. Comput. Appl. Math., 264, 65-70.
[13] Kilbas, A., Srivastava, H. and Trujillo, J., (2006) Theory and Applications of Fractional Differential Equations, in Math. Studies, North-Holland, New York.
[14] Miller, K. S., (1993) An Introduction to Fractional Calculus and Fractional Differential Equations, J. Wiley and Sons, New YorK.
[15] Mukhopadhyay, S. N. and Bullen, P. S., (2012) Higher order derivatives, CRC Press Taylor & Francis Group.
[16] Nápoles Valdés, J. E., Guzmán, P. M. and Lugo, L. M., (2018) Some New Results on Nonconformable Fractional Calculus, Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 13, Number 2, pp. 167–175.
[17] Oldham, K. and Spanier, J., (1974) The Fractional Calculus, Theory and Applications of Differentiation and Integration of Arbitrary Order, Academic Press, USA.
[18] Podlubny, I., (1999) Fractional Differential Equations, Academic Press, USA.
[19] Ross, B., (1977) Fractional Calculus, Mathematics Magazine, Vol. 50, No. 3, pp. 115-122.
[20] Tarasov, V. E., (2013) No violation of the Leibniz rule. No fractional derivative, Commun. Nonlinear Sci. Numer. Simulat. 18, 2945-2948.
[21] Tarasov, V. E., (2016) Leibniz rule and fractional derivatives of power functions, J. of Computational and Nonlinear Dynamics, Vol. 11, 031014 (1-4).
[22] Tarasov, V. E., (2018) No nonlocality. No fractional derivative, Communications in Nonlinear Science and Numerical Simulation, 62, 157-163.
[23] Zulfeqarr, F., (2017) Ujlayan, A. and Ahuja, A. A new fractional derivative and its fractional integral with some applications, arXiv: 1705.00962v1 [math.CA] 26.

Sigma Mühendislik ve Fen Bilimleri Dergisi-Cover

ISSN: 1304-7191
Yayın Aralığı: Yılda 4 Sayı
Yayıncı: Yıldız Teknik Üniversitesi

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