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“gcdAlpha” – a semi-analytical method for solving fractional state equations

Autorzy
Treść / Zawartość
Warianty tytułu
Konferencja
Computer Applications in Electrical Engineering (23-24.04.2018 ; Poznań, Polska)
Języki publikacji
PL
Abstrakty
EN
The paper discusses a semi-analytical method for solving systems of fractional state equations with various orders. The method bases on an expansion of the system into one where there is a single derivative order. The formulation of the matrices of the new system is explained in detail. Another characteristic feature of the method is also introduced – a consideration of forms, in which the time functions appear and the terms appearing in the solution as a result. A fractional circuit example is presented in order to test the method. The computation time for the method is also studied.
Rocznik
Tom
Strony
231--242
Opis fizyczny
Bibliogr. 30 poz., rys., tab.
Twórcy
autor
  • Silesian University of Technology
Bibliografia
  • [1] Ross B., The development of fractional calculus 1695-1900, Historia Mathematica, Volume 4, Issue 1, ISSN 0315-0860, 1977, pp. 75-89.
  • [2] Oldham K. B., Spanier J., The Fractional Calculus, Academic Press, New York, 1974.
  • [3] Kuatagampola U. N., Mellin transforms of generalized fractional integrals and derivatives, Applied Mathematics and Computation, Volume 257, 2015, pp. 516-580.
  • [4] Caputo M., Linear models of dissipation whose Q is almost frequency independent – II, Geophysical Journal International, Volume 13, Number 5, 1967, pp. 529-539.
  • [5] Jakubowska A., Walczak J., Analysis of the transient state in a circuit with supercapacitor, Poznan University of Technology Academic Journals. Electrical Engineering, Volume 81, 2015, pp. 27-34.
  • [6] Schäfer I., Krüger K., Modelling of lossy coils using fractional derivatives, Journal of Physics D: Applied Physics, Volume 41, Number 4, 2008, pp. 1-8.
  • [7] Ostalczyk P., Duch P., Brzeziński D., Sankowski D., Order Functions Selection in the Variable-, Fractional-Order PID Controller. In: Advances in Modelling and Control of Non-integer-Order Systems, Springer, 2015, pp. 159-170.
  • [8] Baranowski J., Bauer W., Zagórowska M., Kawala-Janik A., Dziwiński T., Piątek P., Adaptive Non-Integer Controller for Water Tank System. in: Theoretical Developments and Applications of Non-Integer Order Systems, Springer, 2016, pp. 271-280.
  • [9] Spałek D., Analytical Solution of Helmholtz Equation in Anisotropic and Nonhomogeneous Region. Journal of Energy and Power Engineering 8, 2014, pp. 1265-1271.
  • [10] Kawala-Janik A., Podpora M., Gardecki A., Czuczwara W., Baranowski J., Bauer W., Game controller based on biomedical signals. in: Methods and Models in Automation and Robotics (MMAR), IEEE, 2015 20th International Conference on, pp. 934-939.
  • [11] Babiarz A., Łęgowski A., Niezabitowski M., Robot Path Control with Al-Alaoui Rule for Fractional Calculus Discretization, in: Theory and Applications of Non-Integer Order Systems, Springer 2017, pp. 405-148.
  • [12] Faraji Oskouie M., Ansari R., Linear and nonlinear vibrations of fractional viscoelastic Timoshenko nanobeams considering surface energy effects. Applied Mathematical Modelling 43, 2017. 337-350.
  • [13] Brociek R., Słota D., Wituła R., Reconstruction Robin Boundary Condition in the Heat Conduction Inverse Problem of Fractional Order, in: Theory and Applications of Non-Integer Order Systems, Springer 2017, 147–156.
  • [14] Cioć R., Luft M., Selected Issues of Fractional Calculus in Modelling Accelerometers Used in Telematic Equipment, in: Activities of Transport Telematics: 13th International Conference on Transport Systems Telematics, Selected Papers, TST 2013, Katowice-Ustroń, Poland, October 23-26, pp. 234-242.
  • [15] Elwakil A.S., Radwan A.G., Freeborn T.J., Allagui A., Maundy B.J., Fouda M., Low-voltage commercial super-capacitor response to periodic linear-with-time current excitation: a case study, IET Circuits, Devices & Systems, Volume 11, Issue 3, 2017, pp. 189-195.
  • [16] Kaczorek T., Rogowski K., Fractional linear systems and electrical circuits, Springer, 2015.
  • [17] Kaczorek T., Positive Linear Systems Consisting of n Subsystems With Different Fractional Orders, IEEE Transactions on Circuits and Systems – I: Regular Papers, Volume 58, Number 6, 2011, pp. 1203-1210.
  • [18] Sowa M., Application of SubIval in solving initial value problems with fractional derivatives, Applied Mathematics and Computation, Volume 319, 2018, pp. 86-103.
  • [19] Sowa M., Error computation strategies in an adaptive step size solver for time fractional problems, in: Selected problems on experimental mathematics, Wyd. Pol. Śl., 2017, pp. 89-102.
  • [20] Sowa M., Application of SubIval, a method for fractional-order derivative computations in IVPs, in: Theory and applications of non-integer order systems. Springer International Publishing, 2017, pp. 489-499.
  • [21] Sowa M., Numerical computations of the fractional derivative in IVPs, examples in MATLAB and Mathematica, Informat. Autom. Pomiary Gosp. Ochr. Środ., 2017, Volume 7, Issue 3, pp. 19-22.
  • [22] Sowa M., A subinterval-based method for circuits with fractional order elements, Bull. Pol. Acad. Sci., Tech. Sci., 2014, Volume 62, Number 3, pp. 449-454.
  • [23] Spałek D., Analytical Solution of Helmholtz Equation in Anisotropic and Nonhomogeneous Region, Journal of Energy and Power Engineering, 2014, Volume 8, pp. 1265-1271.
  • [24] Morgado M. L., Rebelo M., Ferrs L. L., Ford N. J., Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method. Applied Numerical Mathematics, 2017, Volume 114, pp. 108-123.
  • [25] Jakubowska A., Walczak J., Resonance in series fractional order RLβCα circuit, Przegląd Elektrotechniczny, 2014, Volume 90, Number 4, pp. 210-213.
  • [26] Sowa M., A harmonic balance methodology for circuits with fractional and nonlinear elements, in review for Circuits, Systems and Signal Processing, (2018).
  • [27] Diethelm K., Efficient Solution of Multi-Term Fractional Differential Equations Using P(EC)mE Methods, Computing, 2003, Volume 71, pp. 305-319.
  • [28] Math.NET Numerics: https://numerics.mathdotnet.com.
  • [29] Sowa M., webpage: http://msowascience.com.
  • [30] Cook J., Gamma function C# code: http://www.johndcook.com/Gamma.cs.
Typ dokumentu
Bibliografia
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