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Avtomat. i Telemekh., 2013, Issue 4, Pages 3–42 (Mi at4973)  

This article is cited in 17 scientific papers (total in 17 papers)


Fractional integro-differential calculus and its control-theoretical applications. I. Mathematical fundamentals and the problem of interpretation

A. G. Butkovskiia, S. S. Postnova, E. A. Postnovaab

a Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia
b Moscow State University, Moscow, Russia

Abstract: The review is devoted to using the fractional integro-differential calculus for description of the dynamics of various systems and control processes. Consideration was given to the basic notions of the fractional integro-differential calculus and the problem of interpretation of the fractional operators. Presented were examples of physical systems described in terms of the apparatus under consideration.

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English version:
Automation and Remote Control, 2013, 74:4, 543–574

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Presented by the member of Editorial Board: И. В. Рублев

Received: 05.03.2012

Citation: A. G. Butkovskii, S. S. Postnov, E. A. Postnova, “Fractional integro-differential calculus and its control-theoretical applications. I. Mathematical fundamentals and the problem of interpretation”, Avtomat. i Telemekh., 2013, no. 4, 3–42; Autom. Remote Control, 74:4 (2013), 543–574

Citation in format AMSBIB
\by A.~G.~Butkovskii, S.~S.~Postnov, E.~A.~Postnova
\paper Fractional integro-differential calculus and its control-theoretical applications.~I. Mathematical fundamentals and the problem of interpretation
\jour Avtomat. i Telemekh.
\yr 2013
\issue 4
\pages 3--42
\jour Autom. Remote Control
\yr 2013
\vol 74
\issue 4
\pages 543--574

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    1. A. G. Butkovskii, S. S. Postnov, E. A. Postnova, “Fractional integro-differential calculus and its control-theoretical applications. II. Fractional dynamic systems: Modeling and hardware implementation”, Autom. Remote Control, 74:5 (2013), 725–749  mathnet  crossref  mathscinet  zmath  isi
    2. A. H. Bhrawy, M. A. Alghamdi, “The operational matrix of caputo fractional derivatives of modified generalized laguerre polynomials and its applications”, Adv. Differ. Equ., 2013, 307  crossref  mathscinet  zmath  isi  elib  scopus
    3. A. H. Bhrawy, M. A. Alghamdi, “A new legendre spectral galerkin and pseudo-spectral approximations for fractional initial value problems”, Abstract Appl. Anal., 2013, 306746  crossref  mathscinet  zmath  isi  scopus
    4. D. A. Zenyuk, Yu. N. Orlov, “O primenenii drobnogo ischisleniya Rimana–Liuvillya dlya opisaniya raspredelenii veroyatnostei”, Preprinty IPM im. M. V. Keldysha, 2014, 018, 21 pp.  mathnet
    5. V. A. Kubyshkin, S. S. Postnov, “Investigation of two kinds of optimal control problem for fractional order pendulum using the method of moments”, Autom. Remote Control, 76:7 (2015), 1302–1314  mathnet  crossref  isi
    6. U. N. Katugampola, “Mellin transforms of generalized fractional integrals and derivatives”, Appl. Math. Comput., 257 (2015), 566–580  crossref  mathscinet  zmath  isi  elib  scopus
    7. Ch. Thaiprayoon, S. K. Ntouyas, J. Tariboon, “On the nonlocal katugampola fractional integral conditions for fractional langevin equation”, Adv. Differ. Equ., 2015, 374  crossref  mathscinet  isi  scopus
    8. L. Zhang, Ch. Yu, T. Liu, “Control of finite-time anti-synchronization for variable-order fractional chaotic systems with unknown parameters”, Nonlinear Dyn., 86:3 (2016), 1967–1980  crossref  zmath  isi  elib  scopus
    9. V. E. Tarasov, “Geometric interpretation of fractional-order derivative”, Fract. Calc. Appl. Anal., 19:5 (2016), 1200–1221  crossref  mathscinet  zmath  isi  elib  scopus
    10. R. Cioc, “Physical and geometrical interpretation of grunwald-letnikov differintegrals: measurement of path and acceleration”, Fract. Calc. Appl. Anal., 19:1 (2016), 161–172  crossref  mathscinet  zmath  isi  elib  scopus
    11. D. A. Zenyuk, L. V. Klochkova, Yu. N. Orlov, “Modelirovanie nestatsionarnykh sluchainykh protsessov kineticheskimi uravneniyami s drobnymi proizvodnymi”, Zhurnal SVMO, 18:2 (2016), 125–133  mathnet  elib
    12. R. M. Evans, U. N. Katugampola, D. A. Edwards, “Applications of fractional calculus in solving Abel-type integral equations: surface-volume reaction problem”, Comput. Math. Appl., 73:6 (2017), 1346–1362  crossref  mathscinet  isi  scopus
    13. B. Ahmad, S. K. Ntouyas, J. Tariboon, A. Alsaedi, “A study of nonlinear fractional-order boundary value problem with nonlocal Erdelyi-Kober and generalized Riemann-Liouville type integral boundary conditions”, Math. Model. Anal., 22:2 (2017), 121–139  crossref  mathscinet  isi  scopus
    14. H. Chen, U. N. Katugampola, “Hermite-Hadamard and Hermite-Hadamard-Fejer type inequalities for generalized fractional integrals”, J. Math. Anal. Appl., 446:2 (2017), 1274–1291  crossref  mathscinet  zmath  isi  elib  scopus
    15. S. Serhiienko, I. Serhiienko, “Performance enhancement of the relay automatic control system with a fractional-order controller”, 2017 International Conference on Modern Electrical and Energy Systems, MEES 2017, IEEE, 2017, 76–79  crossref  isi
    16. Ch. Promsakon, N. Phuangthong, S. K. Ntouyas, J. Tariboon, “Nonlinear sequential Riemann-Liouville and Caputo fractional differential equations with generalized fractional integral conditions”, Adv. Differ. Equ., 2018, 385  crossref  mathscinet  isi  scopus
    17. D. A. Zenyuk, G. G. Malinetskii, “Odnomernyi bryusselyator s drobnymi proizvodnymi po vremeni”, Preprinty IPM im. M. V. Keldysha, 2019, 098, 32 pp.  mathnet  crossref
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