Application of a SubIval Numerical Solver for Fractional Circuits
The paper discusses the subinterval-based numerical
method for fractional derivative computations. It is now referred
to by its acronym – SubIval. The basis of the method is briefly
recalled. The ability of the method to be applied in time stepping
solvers is discussed. The possibility of implementing a time step size
adaptive solver is also mentioned. The solver is tested on a transient
circuit example. In order to display the accuracy of the solver –
the results have been compared with those obtained by means of a
semi-analytical method called gcdAlpha. The time step size adaptive
solver applying SubIval has been proven to be very accurate as
the results are very close to the referential solution. The solver is
currently able to solve FDE (fractional differential equations) with
various derivative orders for each equation and any type of source
time functions.
[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).
[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).
@article{"International Journal of Electrical, Electronic and Communication Sciences:77797", author = "Marcin Sowa", title = "Application of a SubIval Numerical Solver for Fractional Circuits", abstract = "The paper discusses the subinterval-based numerical
method for fractional derivative computations. It is now referred
to by its acronym – SubIval. The basis of the method is briefly
recalled. The ability of the method to be applied in time stepping
solvers is discussed. The possibility of implementing a time step size
adaptive solver is also mentioned. The solver is tested on a transient
circuit example. In order to display the accuracy of the solver –
the results have been compared with those obtained by means of a
semi-analytical method called gcdAlpha. The time step size adaptive
solver applying SubIval has been proven to be very accurate as
the results are very close to the referential solution. The solver is
currently able to solve FDE (fractional differential equations) with
various derivative orders for each equation and any type of source
time functions.", keywords = "Numerical method, SubIval, fractional calculus,
numerical solver, circuit analysis.", volume = "12", number = "8", pages = "546-5", }
