A local truncation error estimation for a SubIval solver

• Autorzy: Sowa M.

Identyfikatory

DOI

10.24425/124264

• Języki publikacji: EN

Abstrakty

EN

The paper concerns an analysis for SubIval (the subinterval-based method for fractional derivative computations in initial value problems). A time step size adaptive solver is discussed, for which the formula of a local truncation error is derived. A general form for a system of linear equations is given for the considered class of problems (for which the analysis is performed in the paper). Two circuit examples are introduced to display the usefulness of the SubIval solver. For the examples that have been chosen it is possible to obtain referential solutions through completely different methods. The results obtained through the numerical solver are compared with evaluations of the referential solutions. The error estimation results obtained for the time steps of the SubIval solver are compared with the actual errors, being the differences between the numerical solutions and the referential solutions. The paper also contains a comparison of the accuracy of results obtained through the SubIval solver with the accuracies of other solvers.

Słowa kluczowe

EN

fractional calculus; numerical method; adaptive step size; local truncation error; circuit analysis

PL

rachunek różniczkowy; metoda numeryczna; analiza obwodów

• Wydawca: Polska Akademia Nauk, Wydział IV Nauk Technicznych
• Czasopismo: Bulletin of the Polish Academy of Sciences. Technical Sciences
• Rocznik: 2018
• Tom: Vol. 66, nr 4
• Strony: 475--484
• Opis fizyczny: Bibliogr. 57 poz., wykr., tab.

Twórcy

autor

Sowa M.

• Silesian University of Technology, 2a Akademicka St., 44-100 Gliwice, Poland

Bibliografia

• [1] K.B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York (1974).
• [2] T. Kaczorek and K. Borawski, “Positive stable realization problem for linear continuous-time fractional-order systems with symmetric system Metzler matrix”, PAK 60(10), 822–825 (2014).
• [3] R. Brociek, D. Słota, and R. Wituła, “Reconstruction Robin Boundary Condition in the Heat Conduction Inverse Problem of Fractional Order”, Theory and Applications of Non-Integer Order Systems, Springer, 147–156 (2017).
• [4] 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, 2–11 (2015).
• [5] S. Kukla and U. Siedlecka, “An analytical solution to the problem of time-fractional heat conduction in a composite sphere”, Bull. Pol. Ac.: Tech. 65(2), 179–186 (2017).
• [6] W. Bauer and A. Kawala-Janik, “Implementation of Bifractional Filtering on the Arduino Uno Hardware Platform”, Theory and Applications of Non-Integer Order Systems, Springer, 419–428 (2017).
• [7] A. Kawala-Janik, M. Podpora, A. Gardecki, W. Czuczwara, J. Baranowski, and W. Bauer, “Game controller based on biomedical signals”, Methods and Models in Automation and Robotics (MMAR) 2015 20th International Conference, 934–939 (2015).
• [8] P.W. Ostalczyk, P. Duch, D.W. Brzeziński, and D. Sankowski, “Order Functions Selection in the Variable-, Fractional-Order PID Controller”, Advances in Modelling and Control of Noninteger-Order Systems, Springer, 159–170 (2015).
• [9] J. Baranowski, W. Bauer, M. Zagórowska, A. Kawala-Janik, T. Dziwiński, and P. Piątek, “Adaptive Non-Integer Controller for Water Tank System”, Theoretical Developments and Applications of Non-Integer Order Systems, Springer, 271–280 (2016).
• [10] D. Spałek, “Synchronous generator model with fractional order voltage regulator PIbDa”, Acta Energetica 2/23, 78–84 (2015).
• [11] P. Mercorelli, “A Discrete-Time Fractional Order PI Controller for a Three Phase Synchronous Motor Using an Optimal Loop Shaping Approach”, Theory and Applications of Non-Integer Order Systems, Springer, 477–487 (2017).
• [12] R. Garrappa and G. Maione, “Fractional Prabhakar derivative and applications in anomalous dielectrics: a numerical approach”, Theory and Applications of Non-Integer Order Systems, Springer, 429–439 (2017).
• [13] L.Mescia, P. Bia, and D. Caratelli, “Fractional derivative based FDTD modeling of transient wave propagation in Havriliak-Negami media”, IEEE Transactions on Microwave Theory and Techniques 62(9), 1920–1929 (2014).
• [14] G. Litak, B. Ducharne, G. Sebald, and D. Guyomar, “Dynamics of magnetic field penetration into soft ferromagnets”, Journal of Applied Physics 117(24), 243907 (2015).
• [15] A. Babiarz, A. Łęgowski, and M. Niezabitowski, “Robot path control with Al-Alaoui rule for fractional calculus discretization”, Theory and Applications of Non-Integer Order Systems, Springer, 405–418 (2017).
• [16] W. Sumelka, “Fractional calculus for continuum mechanics – anisotropic non-locality”, Bull. Pol. Ac.: Tech. 64(2), 361–372 (2017).
• [17] M. Faraji Oskouie and R. Ansari, “Linear and nonlinear vibrations of fractional viscoelastic Timoshenko nanobeams considering surface energy effects”, Applied Mathematical Modelling 43, 337–350 (2017).
• [18] M.A. Ezzat, A.S. El-Karamany, and A.A. El-Bary, “Thermoviscoelastic materials with fractional relaxation operators”, Applied Mathematical Modelling 39(23–24), 7499–7512 (2015).
• [19] W. Mitkowski and P. Skruch, “Fractional-order models of the supercapacitors in the form of RC ladder networks”, Bull. Pol. Ac.: Tech. 61(3), 580–587 (2013).
• [20] J. Walczak and A. Jakubowska, "Analysis of parallel resonance RLC circuit with supercapacitor", Computer Applications in Electrical Engineering 12, 30–37 (2014).
• [21] A. Jakubowska-Ciszek and J. Walczak, “Analysis of the transient state in a parallel circuit of the class RLC”, Applied Mathematics and Computation 319, 287–300 (2018).
• [22] K.J. Latawiec, R. Stanisławski, M. Łukaniszyn, W. Czuczwara, and M. Rydel, “Fractional-order modeling of electric circuits: modern empiricism vs. classical science”, Progress in Applied Engineering (PAEE) 2017, 1–4 (2017).
• [23] I. Sch¨afer and K. Kr¨uger, “Modelling of lossy coils using fractional derivatives”, Phys. D: Appl. Phys. 41, 1–8 (2008).
• [24] A. King and F.T. Agerkvist, “State-space modeling of loudspeakers using fractional derivatives”, Audio Engineering Society Convention 139, Audio Engineering Society (2015).
• [25] J.F. Gómez Aguilar and M.M. Hern´andez: “Space-time fractional diffusion-advection equation with Caputo derivative”, Abstract and Applied Analysis, 8 p. (2014).
• [26] M. Caputo, “Linear models of dissipation whose Q is almost frequency independent – II”, Geophysical Journal International 13(5), 529–539 (1967).
• [27] J.D. Munkhammar, “Riemann-Liouville fractional derivatives and the Taylor-Riemann series”, UUDM Project Report 7, 1–18 (2004).
• [28] T. Abdeljawad, “On Riemann and Caputo fractional differences”, Computers and Mathematics with Applications 62, 1602–1611 (2011).
• [29] T. Kaczorek, “Positivity of a class of fractional descriptor continuous-time nonlinear systems”, Bull. Pol. Ac.: Tech. 65(2), 561–565 (2015).
• [30] A. Jakubowska and J. Walczak, “Analysis of the transient state in a circuit with supercapacitor”, Poznan University of Technology Academic Journals. Electrical Engineering 81, 27–34 (2015).
• [31] N.S. Khodabakhshi, S.M. Vaezpour, and D. Baleanu, “Numerical solutions of the initial value problem for fractional differential equations by modification of the Adomian decomposition method”, Fractional Calculus and Applied Analysis 17(2), 382–400 (2014).
• [32] S. Momani and M.A. Noor, “Numerical methods for fourth order fractional integro-differential equations”, Appl. Math. Comput. 182, 754–760 (2006).
• [33] A. Arikoglu and I. Ozkol, “Solution of fractional differential equations by using differential transform method”, Chaos, Solitons and Fractals 34, 1473–1481 (2007).
• [34] C. Lubich, “Fractional linear multistep methods for Abel-Volterra integral equations of the second kind”, Math. Comput. 45, 463–469 (1985).
• [35] R. Garrappa, “On accurate product integration rules for linear fractional differential equations”, Journal of Computational and Applied Mathematics 235, 1085–1097 (2011).
• [36] W.-H. Luo, T.-Z. Huang, G.-C. Wu, and X.-M. Gu, “Quadratic spline collocation method for the time fractional subdiffusion equation”, Applied Mathematics and Computation 276, 252–265 (2016).
• [37] A. Pirkhedri and H.H.S. Javadi, “Solving the time-fractional diffusion equation via Sinc–Haar collocation method”, Applied Mathematics and Computation 257, 317–326 (2015).
• [38] D.W. Brzeziński and P. Ostalczyk, “High-accuracy numerical integration methods for fractional order derivatives and integrals computations”, Bull. Pol. Ac.: Tech. 62(4), 723–733 (2014).
• [39] M. Sowa, “A subinterval-based method for circuits with fractional order elements”, Bull. Pol. Ac.: Tech. 62(3), 449–454 (2014).
• [40] M. Sowa, “Application of SubIval in solving initial value problems with fractional derivatives”, Applied Mathematics and Computation 319, 86–103 (2018).
• [41] V. Dolejˇs´ı and P. K˚us, “Adaptive backward difference formula– Discontinuous Galerkin finite element method for the solution of conservation laws“ Int. J. Numer. Meth. Engng 73, 1739–1766 (2008).
• [42] M. Sowa, “Application of SubIval, a Method for Fractional-Order Derivative Computations in IVPs”, Theory and Applications of Non-Integer Order Systems, Springer, 405–418 (2017).
• [43] http://functions.wolfram.com/GammaBetaErf/Beta3/03/01/02/0003/
• [44] http://www.sympy.org
• [45] M. Włodarczyk and A. Zawadzki: “Obwody RLC w aspekcie pochodnych niecałkowitych rzędów”, Prace Naukowe Politechniki Śląskiej 1, 75–88 (2011).
• [46] http://msowascience.com
• [47] J.D. Cook, “C# code for gamma and log gamma”, http://www.johndcook.com/Gamma.cs
• [48] http://numerics.mathdotnet.com
• [49] https://www.mathworks.com/products/matlab.html
• [50] https://www.gnu.org/software/octave/
• [51] R. Garrappa, “Numerical Solutions of Fractional Differential Equations: A Survey and a Software Tutorial”, Mathematics 6(2), 16 p. (2018).
• [52] R. Garrappa, “Trapezoidal methods for fractional differential equations: Theoretical and computational aspects”, Mathematics and Computers in Simulation 110, 96–112 (2015).
• [53] R. Garrappa, “Software for Fractional Differential Equations and related problems”, https://www.dm.uniba.it/Members/garrappa/Software.
• [54] R. Scherer, S.L. Kalla, Y. Tang, and J. Huang, “The Gr¨unwald-Letnikov method for fractional differential equations”, Computers and Mathematics with Applications 62, 902–917 (2011).
• [55] I. Podlubny, Fractional differential equations, San Diego, Academic Press (1999).
• [56] R. Garrappa, “Numerical evaluation of two and three parameter Mittag-Leffler functions”, SIAM J. Numer. Anal. 53(3), 1350–1369 (2015).
• [57] http://www.mathworks.com/matlabcentral/fileexchange/48154-the-mittag-leffler-function.

Uwagi

PL

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).