**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**30761

##### Application of a SubIval Numerical Solver for Fractional Circuits

**Authors:**
Marcin Sowa

**Abstract:**

**Keywords:**
Fractional Calculus,
Circuit Analysis,
numerical method,
SubIval,
numerical solver

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1474309

**References:**

[1] K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York (1974).

[2] P.W. Ostalczyk, P. Duch, D.W. Brzezi´nski, D. Sankowski, ”Order Functions Selection in the Variable-, Fractional-Order PID Controller”, Advances in Modelling and Control of Non-integer-Order Systems, Springer, 159–170 (2015).

[3] D. Spałek, ”Synchronous Generator Model with Fractional Order Voltage Regulator PIbDa”, Acta Energetica 2/23, 78–84 (2015).

[4] L. Mescia, P. Bia, 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).

[5] R. Garrappa, 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).

[6] I. Sch¨afer, K. Kr¨uger, ”Modelling of lossy coils using fractional derivatives”, Phys. D: Appl. Phys. 41, 1–8 (2008).

[7] M. Sowa, ”DAQ-based measurements for ferromagnetic coil modeling using fractional derivatives”, International Interdisciplinary PhD Workshop IIPhDW 2018 (in print).

[8] A. Jakubowska, J. Walczak, ”Analysis of the transient state in a circuit with supercapacitor”, Poznan University of Technology Academic Journals. Electrical Engineering 81, 27–34 (2015).

[9] W. Mitkowski, P. Skruch, ”Fractional-order models of the supercapacitors in the form of RC ladder networks”, Bull. Pol. Ac.: Tech. 61 (3), 580–587 (2013).

[10] M.A. Ezzat, A.S. El-Karamany, A.A. El-Bary, ”Thermo-viscoelastic materials with fractional relaxation operators”, Applied Mathematical Modelling 39, 23–24, 7499–7512 (2015).

[11] M. Faraji Oskouie, R. Ansari, ”Linear and nonlinear vibrations of fractional viscoelastic Timoshenko nanobeams considering surface energy effects”, Applied Mathematical Modelling 43, 337–350 (2017).

[12] A. Kawala-Janik, M. Podpora, A. Gardecki, W. Czuczwara, J. Baranowski, W. Bauer: ”Game controller based on biomedical signals”, Methods and Models in Automation and Robotics (MMAR) 2015 20th International Conference, 934–939 (2015).

[13] W. Bauer, A. Kawala-Janik, ”Implementation of Bi-fractional Filtering on the Arduino Uno Hardware Platform”, Theory and Applications of Non-Integer Order Systems, Springer, 419–428 (2017).

[14] U.N. Katugampola, ”Mellin transforms of generalized fractional integrals and derivatives”, Applied Mathematics and Computation, 257, 566–580 (2015).

[15] M. Caputo, ”Linear models of dissipation whose Q is almost frequency independent – II”, Geophysical Journal International 13 (5), 529–539 (1967).

[16] T. Kaczorek, Fractional linear systems and electrical circuits, Springer (2015).

[17] S. Momani, M.A. Noor, ”Numerical methods for fourth order fractional integro-differential equations”, Appl. Math. Comput. 182, 754–760 (2006).

[18] N.S. Khodabakhshi, S.M. Vaezpour, 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).

[19] C. Lubich, ”Fractional linear multistep methods for Abel-Volterra integral equations of the second kind”, Math. Comput. 45, 463–469 (1985).

[20] W.-H. Luo, T.-Z. Huang, G.-C. Wu, X.-M. Gu, ”Quadratic spline collocation method for the time fractional subdiffusion equation”, Applied Mathematics and Computation 276, 252–265 (2016).

[21] R. Garrappa, ”On accurate product integration rules for linear fractional differential equations”, Journal of Computational and Applied Mathematics 235, 1085–1097 (2011).

[22] M. Sowa, ”A subinterval-based method for circuits with fractional order elements”, Bull. Pol. Acad. Sci.: Tech. 62 (3), 449–454 (2014).

[23] M. Sowa, ”Solutions of Circuits with Fractional, Nonlinear Elements by Means of a SubIval Solver”, In Non-Integer Order Calculus and its Applications. RRNR 2017. Lecture Notes in Electrical Engineering, vol. 496, Springer (2018).

[24] M. Sowa, ”The subinterval-based method and its potential improvements”, XXXIX International Conference IC-SPETO 2016, Gliwice-Ustron, 18-21.05.2016 (2016).

[25] M. Sowa, ”SubIval computation time assessment”, Proceedings of International Interdisciplinary PhD Workshop 2017. IIPhDW 2017, September 9-11, 2017, Lodz (2017).

[26] M. Sowa, ”Application of SubIval in solving initial value problems with fractional derivatives”, Applied Mathematics and Computation 319, 86–103 (2018).

[27] M. Sowa, ”Application of SubIval, a method for fractional-order derivative computations in IVPs”, In Theory and applications of non-integer order systems. Lecture Notes in Electrical Engineering, vol. 407, Springer (2017).

[28] M. Sowa, ”A local truncation error estimation for a SubIval solver”, Bull. Pol. Acad. Sci.: Tech. (in print) (2018).

[29] http://msowascience.com.

[30] http://www.mathworks.com/products/matlab.html.

[31] http://www.gnu.org/software/octave/.

[32] M. Sowa, ”“gcdAlpha” – a semi-analytical method for solving fractional state equations”, Computer Applications in Electrical Engineering 96, 231–242 (2018).