Non integer order, state space model of heat transfer process using Atangana-Baleanu operator

• Autorzy: Oprzędkiewicz K.

Identyfikatory

DOI

10.24425/bpasts.2020.131828

• Języki publikacji: EN

Abstrakty

EN

In the paper a new, state space, non integer order model of an one-dimensional heat transfer process is proposed. The model uses a new operator with Mittag-Leffler kernel, proposed by Atangana and Beleanu. The non integer order spatial derivative is expressed by Riesz operator. Analytical formula of the step response is given, the convergence of the model is discussed too. Theoretical results are verified by experiments.

Słowa kluczowe

EN

non integer order systems; heat transfer equation; infinite dimensional systems; fractional order state equation; Riesz operator; Caputo operator; Atangana-Baleanu operator

• Wydawca: Polska Akademia Nauk, Wydział IV Nauk Technicznych
• Czasopismo: Bulletin of the Polish Academy of Sciences. Technical Sciences
• Rocznik: 2020
• Tom: Vol. 68, nr 1
• Strony: 43--50
• Opis fizyczny: Bibliogr. 28 poz., rys., tab., wykr.

Twórcy

autor

Oprzędkiewicz K.

• AGH University of Science and Technology

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), 1490–1500 (2011).
• [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), 763–769, 2016.
• [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 1 (1), 1–13 (2015).
• [6] A. Dzielinski, D. Sierociuk, and G. Sarwas, “Some applications of fractional order calculus”, Bull. Pol. Ac.: Tech. 58 (4), 583–592 (2010).
• [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), 61–103 (2016).
• [8] L. Torres J.F. Gomez, and R.F. Escobar, Fractional Derivatives with Mittag-Leffler Kernel Trends and Applications in Science and Engineering, Springer, Berlin, 2019.
• [9] T. Kaczorek, Selected Problems of Fractional Systems Theory, Springer, Berlin, 2011.
• [10] T. Kaczorek, “Singular fractional linear systems and electrical circuits”, International Journal of Applied Mathematics and Computer Science 21 (2), 379–384 (2011).
• [11] T. Kaczorek, “Reduced-order fractional descriptor observers for a class of fractional descriptor continuous-time nonlinear systems”, International Journal of Applied Mathematics and Computer Science 26 (2), 277–283 (2016).
• [12] T. Kaczorek and K. Borawski, “Fractional descriptor continuous-time linear systems described by the caputo-fabrizio derivative”, International Journal of Applied Mathematics and Computer Science 26 (3), 533–541 (2016).
• [13] T. Kaczorek and K. Rogowski, Fractional Linear Systems and Electrical Circuits, Bialystok University of Technology, Bialystok, 2014.
• [14] A. Kochubei, Fractional-parabolic systems, preprint, arxiv: 1009.4996 [math.ap], 2011.
• [15] W. Mitkowski, “Approximation of fractional diffusion-wave equation”, Acta Mechanica et Automatica 5 (2), 65–68 (2011).
• [16] K. Oprzedkiewicz, “The interval parabolic system”, Archives of Control Sciences 13 (4), 415–430 (2003).
• [17] K. Oprzedkiewicz, “A controllability problem for a class of uncertain parameters linear dynamic systems”, Archives of Control Sciences 14 (1), 85–100 (2004).
• [18] K. Oprzedkiewicz, “An observability problem for a class of uncertain-parameter linear dynamic systems”, International Journal of Applied Mathematics and Computer Science 15 (3), 331–338 (2005).
• [19] K. Oprzedkiewicz and E. Gawin, “A non-integer order, state space model for one dimensional heat transfer process”, Archives of Control Sciences 26 (2), 261–275 (2016).
• [20] 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), 749–756 (2016).
• [21] K. Oprzedkiewicz and W. Mitkowski. “A memory-efficient non-integer-order discrete-time state-space model of a heat transfer process”, International Journal of Applied Mathematics and Computer Science (AMCS) 28 (4), 649–659 (2018).
• [22] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
• [23] E. Popescu, “On the fractional cauchy problem associated with a feller semigroup”, Mathematical Reports 12 (2), 181–188 (2010).
• [24] L. Sajewski, “Minimum energy control of descriptor fractional discrete-time linear systems with two different fractional orders”, International Journal of Applied Mathematics and Computer Science 27 (1), 33–41 (2017).
• [25] 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), 79–89 (2016).
• [26] N. Sene, “Analytical solutions of hristov diffusion equations with non-singular fractional derivatives”, Chaos 29 (1), DOI: 10.1063/1.5082645 (2019).
• [27] 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), 2–11 (2015).
• [28] Q. Yang, F. Liu, and I. Turner, “Numerical methods for fractional partial differential equations with Riesz space fractional derivatives”, Applied Mathematical Modelling 34 (1), 200–218 (2010).

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).