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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


A New Local Fractional Derivative of q−uniform Type

Discontinuity, Nonlinearity, and Complexity 8(1) (2019) 101--109 | DOI:10.5890/DNC.2019.03.009

Juan E. Nápoles Valdes, Jorge A. Castillo Medina, Paulo M. Guzmán, Luciano Miguel Lugo

UNNE, FaCENA, Corrientes, Argentina, UTN, FRRE, Resistencia, Chaco, Argentina

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In this work we present a new derivative of q-uniform type, which contains several definitions of known q-derivatives. Some examples and properties similar to those of the ordinary calculus are also presented.


  1. [1]  Victor, K. and Cheung, P. (2002), Quantum Calculus, Springer-Verlag.
  2. [2]  Jackson, F.H. (1908), On q-functions and a certain difference operator, Trans. Roy. Soc. Edin., 46, 253-281.
  3. [3]  Jackson, F.H. (1910), On q-definite integrals, Quart. J. Pure and Appl. Math., 41, 193-203.
  4. [4]  Finkelstein, R.J. (1996), The q-Coulomb problem, J. Math. Phys., 37(6), 2628-2636.
  5. [5]  Finkelstein, R.J. (1967), Symmetry group of the hydrogen atom, J. Math. Phys., 8(3), 443-449.
  6. [6]  Ernst, T. (2001), The history of q?calculus and a new method, thesis, Uppsala, University.
  7. [7]  Neamaty, A. and Tourani, M. (2017), The presentation of a new type of quamtun calculus, Tbilisi Math. J., 10(2), 15-28.
  8. [8]  Aldwoah, K.A., Malinowska, A.B., and Torres, DS.F.M. (2012), The power quantum calculus and variational problems, Dynamic of Continuous, Discrete and Impulsive Systems, Serie B: Applications & Algoriths, 19, 93-116.
  9. [9]  Vanterler Da C., Sousa, J., de Oliveira, E.C. (2017), On the local m-derivative, ArXiv:1704.08186v3.
  10. [10]  Hamza, A.E., Sarhan, A.S.M., Shehata, E.M., and Aldwoah, K.A. (2015), A general quantum difference calculus, Advances in Difference Equations, 182, 19 p.
  11. [11]  Liénard, A. (1928), Étude des oscillations entretenues, Revue Génerale de l' Électricité 23, 901-912, 946-954.
  12. [12]  Guckenheimer, J. and Holmes, P. (2002), "Nonlinear oscillations, dynamical systems, and bifurcations of vector fields", volume 42 of Applied Mathematical Sciences. Springer-Verlag, New York. Revised and corrected reprint of the 1983 original.
  13. [13]  Van der Pol, B. (1922), On oscillation hysteresis in a triode generator with two degrees of freedom, Phil. Mag (6) 43, 700-719.
  14. [14]  Van der Pol, B. (1926), On "relaxation-oscillations", Philosophical Magazine, 2(11), 978-992.
  15. [15]  Li, Y., Chen, Y.Q., and Podlubny, I. (2010), Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag Leffler stability, Comput. Math. Appl., 59, 1810-1821.