Application of SubIval, a Method for Fractional-Order Derivative Computations in IVPs

  • Marcin SowaEmail author
Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 407)


The paper concerns a numerical method for the computations of the fractional derivative in initial value problems. The method bases on a partition of the integrodifferentiation interval into subintervals. It has been referred to previously as the subinterval-based method and is now called SubIval (for simpler reference). The subintervals are modified during a time stepping process – this is determined by an original subinterval dynamics algorithm. The method is mainly built in order to aid in solving problems of circuit theory. Hence, two examples have been introduced to ascertain the method, where both use a fractional order capacitor and a fractional order coil.


Fractional calculus Numerical analysis Circuit analysis Integrodifferential equations 


  1. 1.
    Caputo, M.: Linear models of dissipation whose Q is almost frequency independent - II. Geophys. J. Int. 13(5), 529–539 (1967)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Munkhammar, J.D.: Riemann-Liouville fractional derivatives and the Taylor-Riemann series. UUDM Proj. Rep. 7, 1–18 (2004)Google Scholar
  3. 3.
    Katugampola, U.N.: A new approach to generalized fractional derivatives. Bull. Math. Anal. Appl. 6(4), 1–15 (2014)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Spałek, D.: Synchronous generator model with fractional order voltage regulator PI\(^{b}\)D\(^{a}\). Acta Energetica 2(23), 78–84 (2015)Google Scholar
  5. 5.
    Baranowski, J., Bauer, W., Zagórowska, M., Kawala-Janik, A., Dziwiński, T., Pia̧tek, P.: Adaptive non-integer controller for water tank system. Theor. Develop. Appl. Non-Integer Syst. 271–280 (2016)Google Scholar
  6. 6.
    Schäfer, I., Krüger, K.: Modelling of lossy coils using fractional derivatives. Phys. D: Appl. Phys. 41, 1–8 (2008)Google Scholar
  7. 7.
    Jakubowska, A., Walczak, J.: Analysis of the transient state in a series circuit of the class RL\(_\beta \)C\(_\alpha \). Circuits Syst. Signal Process. 35, 1831–1853 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Mescia, L., Bia, P., Caratelli, D.: Fractional derivative based FDTD modeling of transient wave propagation in Havriliak–Negami media. IEEE Trans. Microw. Theory Tech. 62(9), (2014)Google Scholar
  9. 9.
    Brociek, R., Słota, D., Wituła R.: Reconstruction of the thermal conductivity coefficient in the time fractional diffusion equation. In: Advances in Modelling and Control of Non-integer-Order Systems, pp. 239–247 (2015)Google Scholar
  10. 10.
    Kawala-Janik, A., Podpora, M., Gardecki, A., Czuczwara, W., Baranowski, J., Bauer, W.: Game controller based on biomedical signals. In: 2015 20th International Conference on Methods and Models in Automation and Robotics (MMAR), pp. 934–939 (2015)Google Scholar
  11. 11.
    Wang, H., Du, N.: Fast solution methods for space-fractional diffusion equations. J. Comput. Appl. Math. 255, 376–383 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kaczorek, T.: Minimum energy control of fractional positive electrical circuits with bounded inputs. Circuits Syst. Signal Process. 1–15 (2015)Google Scholar
  13. 13.
    Zhou, K., Chen, D., Zhang, X., Zhou, R., Iu, H.H.-C.: Fractional-order three-dimensional circuit network. IEEE Trans. Circuits Syst. I Regul. Pap. 62(10), 2401–2410 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Mitkowski, W., Skruch, P.: Fractional-order models of the supercapacitors in the form of RC ladder networks. Bull. Pol. Acad.: Tech. 61(3), 580–587 (2013)Google Scholar
  15. 15.
    Momani, S., Noor, M.A.: Numerical methods for fourth order fractional integro-differential equations. Appl. Math. Comput. 182, 754–760 (2006)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Arikoglu, A., Ozkol, I.: Solution of fractional integro-differential equations by using fractional differential transform method. Chaos Solitons Fractals 40(2), 521–529 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Saeedi, H., Mohseni Moghadam, M.: Numerical solution of nonlinear Volterra integro-differential equations of arbitrary order by CAS wavelets. Commun. Nonlinear Sci. Numer. Simul. 16, 1216–1226 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Huang, L., Xian-Fang, L., Zhao, Y., Duan, X.-Y.: Approximate solution of fractional integro-differential equations by Taylor expansion method. Comput. Math. Appl. 62, 1127–1134 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Rawashdeh, E.A.: Numerical solution of fractional integro-differential equations by collocation method. Appl. Math. Comput. 176, 1–6 (2006)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Lubich, C.: Fractional linear multistep methods for Abel-Volterra integral equations of the second kind. Math. Comput. 45, 463–469 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Cui, M.: Compact finite difference method for the fractional diffusion equation. J. Comput. Phys. 228, 7792–7804 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Klamka, J., Czornik, A., Niezabitowski, M., Babiarz, A.: Controllability and minimum energy control of linear fractional discrete-time infinite-dimensional systems. In: 11th IEEE International Conference on Control & Automation (ICCA), pp. 1210–1214 (2014)Google Scholar
  23. 23.
  24. 24.
    Abdeljawad, T.: On Riemann and Caputo fractional differences. Comput. Math. Appl. 62, 1602–1611 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Włodarczyk, M., Zawadzki, A.: RLC circuits in aspect of positive fractional derivatives. Sci. Works Sil. Univ. Techn.: Electr. Eng. 1, 75–88 (2011)Google Scholar
  26. 26.
    Podlubny, I., Kacenak, M.: The Matlab mlf code. MATLAB Central File Exchange (2001–2009). File ID 8738 (2001)Google Scholar
  27. 27.
    Sowa, M.: A subinterval-based method for circuits with fractional order elements. Bull. Pol. Acad.: Tech. 62(3), 449–454 (2014)MathSciNetGoogle Scholar
  28. 28.
    Sowa, M.: The subinterval-based (“SubIval”) method and its potential improvements. In: Proceedings of the 39th Conference on Fundamentals of Electrotechnics and Circuit Theory IC-SPETO (2016)Google Scholar
  29. 29.
    Majka, Ł.: Measurement verification of the nonlinear coil models. In: Proceedings of the 39th Conference on Fundamentals of Electrotechnics and Circuit Theory IC-SPETO (2016)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute of Electric Engineering and Computer ScienceSilesian University of TechnologyGliwicePoland

Personalised recommendations