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Fractional discrete-continuous model of heat transfer process

Treść / Zawartość
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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The paper proposes a new, state space, finite dimensional, fractional order model of a heat transfer in one dimensional body. The time derivative is described by Caputo operator. The second order central difference describes the derivative along the length. The analytical formulae of the model responses are proved. The stability, convergence, and positivity of the model are also discussed. Theoretical results are verified by experiments.
Rocznik
Strony
287--306
Opis fizyczny
Bibliogr. 34 poz., rys., tab., wzory
Twórcy
  • AGH University of Science and Technology in Krakow, Faculty of Electrical Engineering, Automatics, Computer Science and Robotics, Department of Automatics and Biomedical Engineering, Kraków, Poland
  • AGH University of Science and Technology in Krakow, Faculty of Electrical Engineering, Automatics, Computer Science and Robotics, Department of Automatics and Biomedical Engineering, Kraków, Poland
Bibliografia
  • [1] R. Almeida and D.F.M. Torres: Necessary and sufficient conditions for the fractional calculus of variations with caputo derivatives. Communications in Nonlinear Science and Numerical Simulation, 16(3), (2011), 1490-1500, DOI: 10.1016/j.cnsns.2010.07.016.
  • [2] A. Atangana and D. Baleanu: New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer. Thermal Sciences, 20(2), (2016), 763-769, DOI: 10.2298/TSCI160111018A.
  • [3] R. Caponetto, G. Dongola, L. Fortuna, and I. Petras: Fractional order systems: Modeling and Control Applications. In: Leon O. Chua, editor, World Scientific Series on Nonlinear Science, pages 1-178. University of California, Berkeley, 2010.
  • [4] S. Das: Functional Fractional Calculus for System Identyfication and Control. Springer, Berlin, 2010.
  • [5] M. Dlugosz and P. Skruch: The application of fractional-order models for thermal process modelling inside buildings. Journal of Building Physics, 39(5), (2016), 440-451, DOI: 10.1177/1744259115591251.
  • [6] A. Dzielinski, D. Sierociuk, and G. Sarwas: Some applications of fractional order calculus. Bulletin of the Polish Academy of Sciences, Technical Sciences, 58(4), (2010), 583-592, DOI: 10.2478/v10175-010-0059-6.
  • [7] C.G. Gal and M. Warma Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions. Evolution Equations and Control Theory, 5(1), (2016), 61-103, DOI: 10.3934/eect.2016.5.61.
  • [8] T. Kaczorek Fractional positive contiuous-time linear systems and their reachability. International Journal of Applied Mathematics and Computer Science, 18(2), (2008), 223-228, DOI: 10.2478/v10006-008-0020-0.
  • [9] T. Kaczorek: Singular fractional linear systems and electrical circuits. International Journal of Applied Mathematics and Computer Science, 21(2), (2011), 379-384, DOI: 10.2478/v10006-011-0028-8.
  • [10] T. Kaczorek and K. Rogowski: Fractional Linear Systems and Electrical Circuits. Bialystok University of Technology, Bialystok, 2014.
  • [11] A. Kochubei: Fractional-parabolic systems, preprint, arxiv:1009.4996 [math.ap], 2011.
  • [12] W. Mitkowski: Approximation of fractional diffusion-wave equation. Acta Mechanica et Automatica, 5(2), (2011), 65-68.
  • [13] W. Mitkowski: Finite-dimensional approximations of distributed rc networks. Bulletin of the Polish Academy of Sciences. Technical Sciences, 62(2), (2014), 263-269, DOI: 10.2478/bpasts-2014-0026.
  • [14] W. Mitkowski,W. Bauer, and M. Zagorowska: Rc-ladder networks with supercapacitors. Archiver of Electrical Engineering, 67(2), (2018), 377-389, DOI: 10.24425/119647.
  • [15] K. Oprzedkiewicz: The discrete-continuous model of heat plant. Automatyka, 2(1), (1998), 35-45 (in Polish).
  • [16] K. Oprzedkiewicz: The interval parabolic system. Archives of Control Sciences, 13(4), (2003), 415-430.
  • [17] K. Oprzedkiewicz: Acontrollability problem for a class of uncertain parameters linear dynamic systems. Archives of Control Sciences, 14(1), (2004), 85-100.
  • [18] K. Oprzedkiewicz: An observability problem for a class of uncertainparameter linear dynamic systems. International Journal of Applied Mathematics and Computer Science, 15(3), (2005), 331-338.
  • [19] K. Oprzedkiewicz: Non integer order, state space model of heat transfer proces using Caputo-Fabrizio operator. Bulletin of the Polish Academy of Sciences. Technical Sciences, 66(3), (2018), 249-255, DOI: 10.24425/122105.
  • [20] K. Oprzedkiewicz: Non integer order, state space model of heat transfer process using Atangana-Baleanu operator. Bulletin of the Polish Academy of Sciences. Technical Sciences, 68(1), (2020), 43-50, DOI: 10.24425/bpasts.2020.131828.
  • [21] K. Oprzedkiewicz: Positivity problem for the one dimensional heat transfer process. ISA Transactions, 112, (2021), 281-291
  • [22] K. Oprzedkiewicz: Fractional order, discrete model of heat transfer process using time and spatial Grünwald-Letnikov operator. Bulletin of the Polish Academy of Sciences. Technical Sciences, 69(1), (2021), 1-10, DOI: 10.24425/bpasts.2021.135843.
  • [23] K. Oprzedkiewicz, K. Dziedzic, and Ł. Więckowski: Non integer order, discrete, state space model of heat transfer process using Grünwald-Letnikov operator. Bulletin of the Polish Academy of Sciences. Technical Sciences, 67(5), (2019), 905-914, DOI: 10.24425/bpasts.2019.130873.
  • [24] K. Oprzedkiewicz and E. Gawin: A non-integer order, state space model for one dimensional heat transfer process. Archives of Control Sciences, 26(2), (2016), 261-275, DOI: 10.1515/acsc-2016-0015.
  • [25] K. Oprzedkiewicz, E. Gawin, and W. Mitkowski: Modeling heat distribution with the use of a non-integer order, state space model. International Journal of Applied Mathematics and Computer Science, 26(4), (2016), 749756, DOI: 10.1515/amcs-2016-0052.
  • [26] K. Oprzedkiewicz and W. Mitkowski: A memory-efficient noninteger-order discrete-time state-space model of a heat transfer process. International Journal of Applied Mathematics and Computer Science, 28(4), (2018), 649-659, DOI: 10.2478/amcs-2018-0050.
  • [27] K. Oprzedkiewicz,W. Mitkowski, E.Gawin, and K. Dziedzic: The Caputo vs. Caputo-Fabrizio operators in modeling of heat transfer process. Bulletin of the Polish Academy of Sciences. Technical Sciences, 66(4), (2018), 501-507, DOI: 10.24425/124267.
  • [28] K. Oprzędkiewicz, E. Gawin, and W. Mitkowski: Parameter identification for non integer order, state space models of heat plant. In MMAR 2016: 21th international conference on Methods and Models in Automation and Robotics: 29 August-01 September 2016, Międzyzdroje, Poland, pages 184-188, 2016.
  • [29] P. Ostalczyk: Discrete Fractional Calculus. Applications in Control and Image Processing. Worlsd Scientific Publishing, Singapore, 2016.
  • [30] I. Podlubny: Fractional Differential Equations. Academic Press, San Diego, 1999.
  • [31] G. Recktenwald: Finite-difference approximations to the heat equation. 2011.
  • [32] M. Rózanski: Determinants of two kinds of matrices whose elements involve sine functions. Open Mathematics, 17(1), (2019), 1332-1339, DOI: 10.1515/math-2019-0096.
  • [33] N. Al Salti, E. Karimov, and S. Kerbal: Boundary-value problems for fractional heat equation involving caputo-fabrizio derivative. New Trends in Mathematical Sciences, 4(4), (2016), 79-89, arXiv:1603.09471.
  • [34] D. Sierociuk, T. Skovranek, M. Macias, I. Podlubny, I. Petras, A. Dzielinski, and P. Ziubinski: Diffusion process modeling by using fractional-order models. Applied Mathematics and Computation, 257(1), (2015), 2-11, DOI: 10.1016/j.amc.2014.11.028.
Uwagi
1. This paper was sponsored by AGH UST project no 16.16.120.773.
2. Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-8ca62a8a-2d8d-4c0c-aaa7-2229a5c11cde
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