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Interpretation of Fractional Derivatives as Reconstruction from Sequence of Integer Derivatives

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Języki publikacji
EN
Abstrakty
EN
In this paper, we propose an “informatic” interpretation of the Riemann-Liouville and Caputo derivatives of non-integer orders as reconstruction from infinite sequence of standard derivatives of integer orders. The reconstruction is considered with respect to orders of derivatives.
Wydawca
Rocznik
Strony
431--442
Opis fizyczny
Bibliogr. 31 poz.
Twórcy
  • Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia
Bibliografia
  • [1] Samko SG, Kilbas AA, Marichev OI. Fractional Integrals and Derivatives Theory and Applications Gordon and Breach, New York, 1993. ISBN-10: 2881248640, 13: 978-2881248641.
  • [2] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, 2006. ISBN: 0444518320.
  • [3] Tarasov VE. No violation of the Leibniz rule. No fractional derivative Communications in Nonlinear Science and Numerical Simulation. 2013: 18 (11): 2945-2948. doi: 10.1016/j.cnsns.2013.04.001.
  • [4] Ortigueira MD, Tenreiro Machado JA. What is a fractional derivative? Journal of Computational Physics. 2015; 293: 4-13. doi: 10.1016/j.jcp.2014.07.019.
  • [5] Tarasov VE. On chain rule for fractional derivatives Communications in Nonlinear Science and Numerical Simulation. 2016; 30 (1-3): 1-4. doi: 10.1016/j.cnsns.2015.06.007.
  • [6] Tarasov VE. Leibniz rule and fractional derivatives of power functions Journal of Computational and Nonlinear Dynamics. 2016: 11 (3): 031014. doi: 10.1115/1.4031364.
  • [7] Kotel’nikov VA. On the transmission capacity of ’ether’ and wire in electric communications Physics-Uspekhi. 2006; 46 (7): 736-744. doi: 10.1070/PU2006v049n07ABEH006160.
  • [8] Shannon CE. Communication in the presence of noise Proceedings of the IEEE. 1998; 86 (2): 447-457. doi: 10.1109/JPROC.1998.659497.
  • [9] Stenger F. Numerical Methods Based on Sinc and Analytic Functions Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-2706-9.
  • [10] Stenger F. Handbook of Sinc Numerical Methods CRC Press, Taylor and Francis, Boca Raton, London, New York, 2011. ISBN: 1439821585, 9781439821589.
  • [11] Whittaker ET. On the functions which are represented by the expansions of the interpolation-theory Proceedings of the Royal Society of Edinburgh. 1915; 35: 181-194. doi: 10.1017/S0370164600017806.
  • [12] McNamee J, Stenger F, Whitney EL. Whittaker’s cardinal function in retrospect Mathematics of Computation. 1971; 25: 141-154. doi: 10.2307/2005140.
  • [13] Stanislavsky AA. Probabilistic interpretation of the integral of fractional-order Theoretical and Mathematical Physics. 2004; 138 (3): 418-431. doi: 10.1023/B:TAMP.0000018457.70786.36.
  • [14] Tenreiro Machado JA. A probabilistic interpretation of the fractional-order differentiation Fractional Calculus and Applied Analysis. 2009; 6 (l): 73-80.
  • [15] Tenreiro Machado JA. Fractional derivatives: Probability interpretation and frequency response of rational approximations Communications in Nonlinear Science and Numerical Simulation. 2009; 14 (9-10): 3492-3497. doi: 10.1016/j.cnsns.2009.02.004.
  • [16] Ben Adda F. Geometric interpretation of the fractional derivative Journal of Fractional Calculus. 1997; 11: 21-51.
  • [17] Ben Adda F. Geometric interpretation of the differentiability and gradient of real order Comptes Rendus de l’Academie des Sciences - Series I - Mathematics. 1997; 326 (8): 931-934. doi: 10.1016/S0764-4442(98)80116-X.
  • [18] Podlubny I, Despotovic V, Skovranek T, McNaughton BH. Shadows on the walls: Geometric interpretation of fractional integration The Journal of Online Mathematics and Its Applications. 2007; 7: 1664.
  • [19] Podlubny I. Geometrical and physical interpretation of fractional integration and fractional differentiation Fractional Calculus and Applied Analysis. 2002; 5 (4): 367-386.
  • [20] Moshrefi-Torbati M, Hammond JK. Physical and geometrical interpretation of fractional operators Journal of the Franklin Institute. 1998; 335 (6): 1077-1086. doi: 10.1016/S0016-0032(97)00048-3.
  • [21] Cioc R. Physical and geometrical interpretation of Grunwald-Letnikov differintegrals: Measurement of path and acceleration Fractional Calculus and Applied Analysis. 2016; 19 (1): 161-172. doi: 10.1515/fca-2016-0009.
  • [22] Nigmatullin RR. A fractional integral and its physical interpretation Theoretical and Mathematical Physics. 1992; 90 (3): 242-251. doi: 10.1007/BF01036529.
  • [23] Rutman RS. On physical interpretations of fractional integration and differentiation Theoretical and Mathematical Physics. 1995; 105 (3): 1509-1519. doi: 10.1007/BF02070871.
  • [24] Heymans N, Podlubny I. Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives Rheologica Acta. 2006; 45 (5): 765-772. doi: 10.1007/s00397-005-0043-5.
  • [25] Gomez-Aguilara JF, Razo-Hernandez R, Granados-Lieberman D. A physical interpretation of fractional calculus in observables terms: analysis of the fractional time constant and the transitory response Revista Mexicana de Fisica. 2014; 60: 32-38.
  • [26] Bandyopadhyay B, Kamal S. Essence of fractional order calculus, physical interpretation and applications in Stabilization and Control of Fractional Order Systems: A Sliding Mode Approach Springer International Publishing, Heidelberg, 2015: 1-54. doi: 10.1007/978-3-319-08621-7.
  • [27] Tarasov VE. Toward lattice fractional vector calculus Journal of Physics A. 2014; 47 (35): 355204. doi: 10.1088/1751-8113/47/35/355204.
  • [28] Tarasov VE. Lattice fractional calculus Applied Mathematics and Computation. 2015; 257: 12-33. doi: 10.1016/j.amc.2014.11.033.
  • [29] Tarasov VE. Exact discrete analogs of derivatives of integer orders: Differences as infinite series Journal of Mathematics. 2015; 2015: 134842. doi: 10.1155/2015/134842.
  • [30] Tarasov VE. Exact discretization by Fourier transforms Communications in Nonlinear Science and Numerical Simulation. 2016; 37: 31-61. doi: 10.1016/j.cnsns.2016.01.006.
  • [31] Tarasov VE. United lattice fractional integro-differentiation Fractional Calculus and Applied Analysis. 2016; 19 (3): 625-664. doi: 10.1515/fca-2016-0034.
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Bibliografia
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