A subinterval-based method for circuits with fractional order elements

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The paper deals with the solution of problems that concern fractional time derivatives. Specifically the author’s interest lies in solving circuit problems with so called fractional capacitors and fractional inductors. A numerical method is proposed that involves polynomial interpolation and the division of the entire time interval (for which computations are performed) into subintervals. Analytical formulae are derived for the integro-differentiation according to the Caputo fractional derivative. The rules that concern the subinterval dynamics throughout the computation are also presented in the paper. For exemplary linear circuit problems (AC and transient) involving fractional order elements the solutions have been obtained. These solutions are compared with ones obtained by means of traditional methods

[1] J.T. Machado, V. Kiryakova, and F. Mainardi, “Recent history of fractional calculus” Commun. Nonlinear Sci. Numer. Simul. 16, 1140-1153 (2011).

[2] C.P. Li, Y.Q. Chen, and J. Kurths, “Fractional calculus and its applications”, Phil. Trans. R. Soc. A 2013, 371 (2013).

[3] 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).

[4] T.J. Freeborn, B. Maundy, and A.S. Elwakil, “Measurement of supercapacitor fractional-order model parameters from voltage-excited step response”, IEEE J. on Emerging and Selected Topics in Circuits and Systems 3 (3), 367-376 (2013).

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

[6] J. Walczak and A. Jakubowska, “The analysis of resonance phenomena in a series RLC circuit with a supercapacitor”, PAK 59 (10), 1105-1108 (2013), (in Polish).

[7] T. Kaczorek, “Positive linear systems with different fractional orders”, Bull. Pol. Ac.: Tech. 58 (3), 453-458 (2010).

[8] T. Kaczorek “Stability of positive fractional switched continuous-time linear systems”, Bull. Pol. Ac.: Tech. 61 (2), 349-352 (2013).

[9] M. Weilbeer, “Efficient numerical methods for fractional differential equations and their analytical background”, PHD Dissertation pp. 105-159, The Carl-Friedrich-Gauß School Faculty for Mathematics and Computer Science, Braunschweig University of Technology, Braunschweig, 2005.

[10] M. Caputo, “Linear model of dissipation whose Q is almost frequency independent - II”, The Geophysical J. Royal Astronomical Society 13, 529-539 (1967).

[11] http://functions.wolfram.com/GammaBetaErf/Beta3/03/01/ 02/0003/.

[12] M. Ishteva, L. Boyadjiev, and R. Scherer, “On the Caputo operator of fractional calculus and C-Laguerre functions”, Mathematical Sciences Research J. 9, 161-170 (2005).

[13] V. Dolejˇs´ı and P. K°us, “Adaptive backward difference formula- Discontinuous Galerkin finite element method for the solution of conservation laws”, Int. J. Numerical Methods in Engineering 73, 1739-1766 (2008).

[14] E. Hairer, S.P. Nørsett, and G. Wanner, “Solving ordinary differential equations I: Nonstiff problems”, Springer Series in Computational Mathematics 8, 368-374 (1993).

[15] M. Włodarczyk and A. Zawadzki, “RLC circuits in aspect of positive fractional derivatives”, Scientific Works of the Silesian University of Technology Q. Electrical Engineering 1 (217), 75-88, (2011), (in Polish).

Bulletin of the Polish Academy of Sciences Technical Sciences

The Journal of Polish Academy of Sciences

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