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THE LOCAL GENERALIZED DERIVATIVE AND MITTAG-LEFFLER FUNCTION

Yıl 2020, Cilt: 38 Sayı: 2, 1007 - 1017, 01.06.2021

Öz

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.

Kaynakça

  • [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.
Yıl 2020, Cilt: 38 Sayı: 2, 1007 - 1017, 01.06.2021

Öz

Kaynakça

  • [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.
Toplam 23 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Mühendislik
Bölüm Research Articles
Yazarlar

Juan E. Nápoles Valdes Bu kişi benim 0000-0003-2470-1090

Paulo M. Guzmán Bu kişi benim 0000-0002-7490-5668

Luciano M. Lugo Bu kişi benim 0000-0001-9351-2547

Artion Kashurı Bu kişi benim 0000-0003-0115-3079

Yayımlanma Tarihi 1 Haziran 2021
Gönderilme Tarihi 5 Şubat 2020
Yayımlandığı Sayı Yıl 2020 Cilt: 38 Sayı: 2

Kaynak Göster

Vancouver Valdes JEN, Guzmán PM, Lugo LM, Kashurı A. THE LOCAL GENERALIZED DERIVATIVE AND MITTAG-LEFFLER FUNCTION. SIGMA. 2021;38(2):1007-1.

IMPORTANT NOTE: JOURNAL SUBMISSION LINK https://eds.yildiz.edu.tr/sigma/