Abstract
A function f(t) of the independent variable t changing with every increment dt can be formulated as a functional sequence. If g(f(t)) is a derivative or an integral of f(t) and the value of dt is interpreted subject to an error ΔT, then g(f(t)) is Grünwald-Letnikov differintegral of that sequence with an order closely related to dt and ΔT. This paper illustrates this relationship and proposes a geometrical and physical interpretation of a fractional order Grünwald-Letnikov differintegrals using the example of path and acceleration measurements of a point in motion.
References
[1] A.G. Butkovskii, S.S. Postnov, E.A. Postnova, Fractional integro-differential calculus and its control-theoretical application. I. Mathematical fundamentals and the problem of interpretation. Automation and Remote Control74, No 4 (2013), 543-574.10.1134/S0005117913040012Search in Google Scholar
[2] S. Das, Functional Fractional Calculus for System Identification and Controls. Springer-Verlag, Berlin Heidelberg (2008).Search in Google Scholar
[3] J.F. Gómez-Aguilar, R. Razo-Hernández, D. Granados-Lieberman, A physical interpretation of fractional calculus in observables terms: Analysis of the fractional time constant and the transitory response. Revista Mexicana de Física60 (2014), 32-38.Search in Google Scholar
[4] R. Herrmann, Towards a geometric interpretation of generalized fractional integrals – Erdélyi-Kober type integrals on RN as an example. Fract. Calc. Appl. Anal. 17, No 2 (2014), 361–370; DOI: 10.2478/sl3540-Ol4-Ol74-4; http://www.degruyter.com/view/j/fca.2014.17.issue-2/issue-files/fca.2014.17.issue-2.xml.10.2478/sl3540-014-0l74-4Search in Google Scholar
[5] R. Hilfer, Fractional Time Evolution. In: Applications of Fractional Calculus in Physics, R. Hilfer (Ed.), World Scientific, Singapore (2000), 89-130.10.1142/3779Search in Google Scholar
[6] N. Heymans, I. Podlubny, Physical interpretation of initial condition for fractional differential equations with Riemann-Liouville fractional derivatives. Rheologica Acta45, No 5 (2006), 765-771.10.1007/s00397-005-0043-5Search in Google Scholar
[7] A. Kilbas, H. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies 204, Elsevier, Amsterdam (2006).10.1016/S0304-0208(06)80001-0Search in Google Scholar
[8] J.A. Tenreiro Machado, And I say to myself: “What a fractional world!” Fract. Calc. Appl. Anal. 14, No 4 (2011), 635–654; DOI:10.2478/s13540-011-0037-1; http://www.degruyter.com/view/j/fca.2011.14.issue-4/issue-files/fca.2011.14.issue-4.xml.10.2478/s13540-011-0037-1Search in Google Scholar
[9] P. Ostalczyk, Zarys Rachunku Różniczkowo-Całkowego Ułamkowych Rzędów. Teoria i Zastosowanie w Praktyce. Wydawnictwo Politechniki Łódzkiej,Łódź (2008).Search in Google Scholar
[10] I. Podlubny, Fractional Differential Equations. Mathematics in Science and Engineering, Vol. 198, Academic Press, San Diego (1999).Search in Google Scholar
[11] I. Podlubny, Geometric and physical nterpretation of fractional integration and fractional differentiation. Fract. Calc. Appl. Anal. 5, No 4 (2002), 357-366; http://www.math.bas.bg/∼fcaa.Search in Google Scholar
[12] R.S. Rutman, On physical interpretation of fractional integration and differentiation. Theoretical and Mathematical Physics105, No 3 (1995), 1509-1519.10.1007/BF02070871Search in Google Scholar
[13] J. Sabatier, O.P. Agrawal, J.A. Tenreiro Machado (Eds.), Advances in Fractional Calculus. Theoretical Developments and Applications in Physics and Engineering. Springer, Netherlands (2007).10.1007/978-1-4020-6042-7Search in Google Scholar
[14] S. Samko, A. Kilbas, O. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, London (1993).Search in Google Scholar
Please cite to this paper as published in:
Fract. Calc. Appl. Anal., Vol. 19, No 1 (2016), pp. 161–172, DOI: 10.1515/fca-2016-0009
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