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Distributed-Order Dynamic Systems

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Abstract

As a branch of mathematics, calculus includes differential calculus and integral calculus. Calculus is the study of change, and has widespread applications in science, economics and engineering, and can solve many real world problems. It is well known that a system’s dynamical properties can be described by an ordinary differential equation (ODE) which contains functions of an independent variable.

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References

  • Adams JL, Hartley TT, Lorenzo CF (2008) Identification of complex order-distributions. J Vib Control 14(9–10):1375–1388

    Article  MathSciNet  MATH  Google Scholar 

  • Ahn HS, Chen YQ (2008) Necessary and sufficient stability condition of fractional-order interval linear systems. Automatica 44(11):2985–2988

    Article  MathSciNet  MATH  Google Scholar 

  • Ahn HS, Chen YQ, Podlubny I (2007) Robust stability test of a class of linear time-invariant interval fractional-order systems using Lyapunov inequality. Appl Math Comput 187(1):27–34

    Article  MathSciNet  MATH  Google Scholar 

  • Astrom KJ, Murray RM (2008) Feedback systems: an introduction for scientists and engineers. Princeton University Press, Princeton

    Google Scholar 

  • Atanackovic TM, Budincevic M, Pilipovic S (2005) On a fractional distributed-order oscillator. J Phys A: Math Gen 38(30):6703–6713

    Article  MathSciNet  MATH  Google Scholar 

  • Atanackovic TM, Oparnica L, Pilipovic S (2007) On a nonlinear distributed order fractional differential equation. J Math Anal Appl 328(1):590–608

    Google Scholar 

  • Atanackovic TM, Pilipovic S, Zorica D (2009a) Existence and calculation of the solution to the time distributed order diffusion equation. Phys Scripta T136:014012 (6pp)

    Google Scholar 

  • Atanackovic TM, Pilipovic S, Zorica D (2009b) Time distributed-order diffusion-wave equation. I. Volterra-type equation. Proc Royal Soc A 465:1869–1891

    Google Scholar 

  • Atanackovic TM, Pilipovic S, Zorica D (2009c) Time distributed-order diffusion-wave equation. II. Applications of Laplace and Fourier transformations. Proc Royal Soc A 465:1893–1917

    Google Scholar 

  • Atanackovic TM, Pilipovic S, Zorica D (2011) Distributed-order fractional wave equation on a finite domain stress relaxation in a rod. Int J Eng Sci 49(2):175–190

    Article  MathSciNet  MATH  Google Scholar 

  • Bagley RL, Torvik PJ (1984) On the appearance of the fractional derivative in the behavior of real materials. ASME J Appl Mech 51(2):294–298

    Article  MATH  Google Scholar 

  • Bagley RL, Torvik PJ (2000) On the existence of the order domain and the solution of distributed order equations (Parts I, II). Int J Appl Mech 2(7):865–882, 965–987

    Google Scholar 

  • Bohannan G (2000) Application of fractional calculus to polarization dynamics in solid dielectric materials. PhD Dissertation, Montana State University, November 2000

    Google Scholar 

  • Bonnet C, Partington JR (2002) Analysis of fractional delay systems of retarded and neutral type. Automatica 38(7):1133–1138

    Article  MathSciNet  MATH  Google Scholar 

  • Bonnet C, Partington JR (2007) Stabilization of some fractional delay systems of neutral type. Automatica 43(12):2047–2053

    Article  MathSciNet  MATH  Google Scholar 

  • Caponetto R, Dongola G, Fortuna L, Petras I (2010) Fractional order systems: modeling and control applications. World Scientific Company, Singapore

    Google Scholar 

  • Caputo M (1969) Elasticità e dissipazione. Zanichelli, Bologna

    Google Scholar 

  • Caputo M (1995) Mean fractional-order-derivatives differential equations and filters. Annali dell’Universita di Ferrara 41(1):73–84

    Google Scholar 

  • Caputo M (2001) Distributed order differential equations modelling dielectric induction and diffusion. Fract Calc Appl Anal 4(4):421–442

    MathSciNet  MATH  Google Scholar 

  • Carlson G, Halijak C (1964) Approximation of fractional capacitors \((1/s)^{(1/n)}\) by a regular Newton process. IEEE Trans Circuit Theory 11(2):210–213

    Google Scholar 

  • Chechkin AV, Gorenflo R, Sokolov IM (2002) Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Phys Rev E 66:046129

    Article  Google Scholar 

  • Chen YQ, Ahn HS, Podlubny I (2006) Robust stability check of fractional order linear time invariant systems with interval uncertainties. Signal Process 86(10):2611–2618

    Article  MATH  Google Scholar 

  • Chen W, Sun HG, Zhang XD, Korosak D (2009) Anomalous diffusion modeling by fractal and fractional derivatives. Comput Math Appl 59(5):1754–1758

    Article  MathSciNet  Google Scholar 

  • Connolly JA (2004) The numerical solution of fractional and distributed order differential equations. Thesis, University of Liverpool, Dec 2004

    Google Scholar 

  • Diethelm K, Ford NJ (2009) Numerical analysis for distributed-order differential equations. J Comput Appl Math 225(1):96–104

    Google Scholar 

  • Hartley TT, Lorenzo CF (2003) Fractional-order system identification based on continuous order-distributions. Signal Process 83(11):2287–2300

    Article  MATH  Google Scholar 

  • Hwang C, Cheng YC (2006) A numerical algorithm for stability testing of fractional delay systems. Automatica 42(5):825–831

    Article  MathSciNet  MATH  Google Scholar 

  • Kochubei AN (2008) Distributed order calculus and equations of ultraslow diffusion. J Math Anal Appl 340(1):252–281

    Article  MathSciNet  MATH  Google Scholar 

  • Li HS, Luo Y, Chen YQ (2009) A fractional order proportional and derivative (fopd) motion controller: tuning rule and experiments. IEEE Trans Control Syst Technol 18(2):1–5

    MATH  Google Scholar 

  • Lorenzo CF, Hartley TT (1998) Initialization, conceptualization, and application in the generalized fractional calculus. NASA technical paper, NASA/TP 1998-208415

    Google Scholar 

  • Lorenzo CF, Hartley TT (2002) Variable order and distributed order fractional operators. Nonlinear Dyn 29(1–4):57–98

    Article  MathSciNet  MATH  Google Scholar 

  • Lu JG, Chen GR (2009) Robust stability and stabilization of fractional-order interval systems: an lmi approach. IEEE Trans Autom Control 54(6):1294–1299

    Article  Google Scholar 

  • Lu JG, Chen YQ (2010) Robust stability and stabilization of fractional order interval systems with the fractional order \(\alpha\): the \(0<\alpha <1\) case. IEEE Trans Autom Control 55(1):152–158

    Google Scholar 

  • Magin RL (2006) Fractional calculus in bioengineering. Begell House, Connecticut

    Google Scholar 

  • Mainardi F, Pagnini G (2007) The role of the fox-wright functions in fractional sub-diffusion of distributed order. J Comput Appl Math 207(2):245–257

    Article  MathSciNet  MATH  Google Scholar 

  • Mainardi F, Mura A, Gorenflo R, Stojanovic M (2007a) The two forms of fractional relaxation of distributed order. J Vib Control 9:1249–1268

    Article  MathSciNet  Google Scholar 

  • Mainardi F, Mura A, Pagnini G, Gorenflo R (2007b) Some aspects of fractional diffusion equations of single and distributed order. Appl Math Comput 187:295–305

    Article  MATH  Google Scholar 

  • Mainardi F, Mura A, Pagnini G, Gorenflo R (2008) Time-fractional diffusion of distributed order. J Vib Control 14(9–10):1267–1290

    Article  MathSciNet  MATH  Google Scholar 

  • Matignon D (1996) Stability results on fractional differential equations with applications to control processing. In: Multiconference on computational engineering in systems and application, pp 963–968

    Google Scholar 

  • Meerschaert MM, Nane E, Vellaisamy P (2011) Distributed-order fractional diffusions on bounded domains. J Math Anal Appl 379:216–228

    Article  MathSciNet  MATH  Google Scholar 

  • Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, New York

    MATH  Google Scholar 

  • Oustaloup A, Mathieu B, Lanusse P (1995) The crone control of resonant plants: application to a flexible transmission. Eur J Control 1(2):113–121

    Google Scholar 

  • Podlubny I (1999) Fractional differential equations. Academic Press, San Diego

    MATH  Google Scholar 

  • Podlubny I (1999) Fractional-order systems and \({P}{I}^{\lambda }{D}^{\mu }\) controllers. IEEE Trans Autom Control 44(1):208–214

    Google Scholar 

  • Sokolov IM, Chechkin AV, Klafter J (2004) Distributed-order fractional kinetics. Acta Phys Polonica B 35(4):1323

    Google Scholar 

  • Srokowski T (2008) Lévy flights in nonhomogeneous media: distributed-order fractional equation approach. Phys Rev E 78(3):031135

    Article  Google Scholar 

  • Sun HG, Chen W, Chen YQ (2009) Variable-order fractional differential operators in anomalous diffusion modeling. Phys A: Stat Mech Appl 388(21):4586–4592

    Article  Google Scholar 

  • Sun HG, Chen W, Sheng H, Chen YQ (2010) On mean square displacement behaviors of anomalous diffusions with variable and random orders. Phys Lett A 374(7):906–910

    Article  MATH  Google Scholar 

  • Tsao YY (1987) Fractal concepts in the analysis of dispersion or relaxation processes. PhD Dissertation, Drexel University, June 1987

    Google Scholar 

  • Umarov S, Steinberg S (2006) Random walk models associated with distributed fractional order differential equations. Inst Math Stat 51:117–127

    MathSciNet  Google Scholar 

  • Xu MY, Tan WC (2006) Intermediate processes and critical phenomena: theory, method and progress of fractional operators and their applications to modern mechanics. Sci China: Ser G Phys Mech Astron 49(3):257–272

    Article  MATH  Google Scholar 

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Jiao, Z., Chen, Y., Podlubny, I. (2012). Introduction. In: Distributed-Order Dynamic Systems. SpringerBriefs in Electrical and Computer Engineering(). Springer, London. https://doi.org/10.1007/978-1-4471-2852-6_1

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  • DOI: https://doi.org/10.1007/978-1-4471-2852-6_1

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  • Publisher Name: Springer, London

  • Print ISBN: 978-1-4471-2851-9

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