On flow of electric current in RL circuit using Hilfer type composite fractional derivative





Resistance-Inductance circuit, Fractional differential equation, Mittag-Leffler function, Laplace transforms, Hilfer derivatives


This paper deals with an interdisciplinary research work between Mathematical sciences and Electrical engineering to develop fractional model of Resistance-Inductance circuit (RL circuit). Authors obtained the analytical solution of this fractional model in terms of Mittag-Leffler function. Graphical interpretation of solution also discussed in this paper.

Author Biographies

Krunal B. Kachhia, Charotar University of Science and Technology.

Department of Mathematical Sciences,  P. D. Patel Institute of Applied Sciences.

J. C. Prajapati, Sardar Patel University.

Department of Mathematics.

K. S. Pandya, Charotar University of Science and Technology.

Department of Electrical Engineering,
Faculty of Technology and Engineering.

R. Jadeja, Marwadi University.

Department of Electrical Engineering,
Faculty of Technology and Engineering.


A. Kilbas, H. Srivastava, and J. Trujillo, Eds., Theory and applications of fractional differential equations. Amsterdam: Elsevier, 2006. [On line]. Available: http://bit.ly/2lLbn3L

A. Carpinteri and F. Mainardi, Eds., Fractals and fractional calculus in continuum mechanics. Springer: Vienna, 1997, doi:10.1007/978-3-7091-2664-6.

A. Obeidat, M. Gharaibeh, M. Al-Ali, and A. Rousan, “Evolution of a current in a resistor”, Fractional calculus and applied analysis, vol. 14, no. 2, pp. 247–259, Jan. 2011, doi: 10.2478/s13540-011-0015-7.

A. Rousan, N. Ayoub, A. Alzoubi, H. Khateeb, M. Al-Qadi, M. Hasan and B. Albiss, “A fractional LC-RC circuit”, Fractional calculus and applied analysis, vol. 9, no. 1, pp. 33-41, 2006. [On line]. Available: http://bit.ly/2mugndn

A. Shukla and J. Prajapati, “On a generalization of Mittag-Leffler function and its properties”, Journal of mathematical analysis and applications, vol. 336, no. 3, pp. 797-811, Dec. 2007, doi: 10.1016/j.jmaa.2007.03.018.

I. Jeses and J. Machado, “Fractional control of heat diffusion systems”, Nonlinear dynamics, vol. 54, no. 3, pp. 263—282, Dec. 2008, doi: 10.1007/s11071-007-9322-2.

A. Wiman, “Über den fundamentalsatz in der teorie der funktionen Ea(x)“, Acta Mathematica, vol. 29,no. 1, pp. 191-201, 1905, doi: 10.1007/BF02403202.

E. Kreyszig, Advanced engineering mathematics, 8th ed. New York (NY): John Wiley & Sons, Inc, 2007.

G. Tsirimokou and C. Psychalinos, "Ultra-low voltage fractional-order circuits using current mirrors", International journal of circuit theory and applications, vol. 44, no. 1, pp. 109-126, Feb. 2015, doi: 10.1002/cta.2066.

I. Podulbuny, Fractional differential equations, New York (NY): Academic Press, 1999.

J. Prajapati and K. Kachhia, “Fractional modeling of temperature distribution and heat flux in the semi infinite solid”, Journal of fractional calculus and applications, vol. 5, no. 2, pp. 38-43, Jul. 2014. [On line]. Available: http://bit.ly/2miHCb2

J. Prajapati, K. Kachhia, and S. Kosta, “Fractional calculus approach to study temperature distribution within a spinning satellite”, Alexandria engineering journal, vol. 55, no. 3, pp. 2345–2350, Sep. 2016, doi: 10.1016/j.aej.2016.05.004.

J. Dubbeldam, Z. Tomovski and T. Sandev, “Space-Time Fractional Schrödinger equation with composite time fractional derivative”, Fractional calculus and applied analysis, vol. 18, no. 5, pp. 1179-1200, Oct. 2015, doi: 10.1515/fca-2015-0068.

J. Gómez-Aguilar, R. Escobar-Jiménez, V. Olivares-Peregrino, M. Taneco-Hernández, and G. Guerrero-Ramírez, “Electrical circuits RC and RL involving fractional operators with bi-order”, Advances in mechanical engineering, vol. 9, no. 6, pp. 1–10, Jun. 2017, doi: 10.1177/1687814017707132.

K. Kachhia and J. Prajapati, “Solution of fractional partial differential equation aries in study of heat transfer through diathermanous materials”, Journal of interdisciplinary mathematics, vol. 18, no. 1-2, pp. 125–132, Mar. 2015, doi: 10.1080/09720502.2014.996017.

K. Kachhia and J. Prajapati, “On generalized fractional kinetic equations involving generalized Lommel-Wright functions”, Alexandria engineering journal, vol. 55, no. 3, pp. 2953–2957, Sep. 2016, doi: 10.1016/j.aej.2016.04.038.

M. Caputo, Elasticità e dissipazione, Bologna: Zanichelli, 1969.

M. Guía, F. Gómez, and J. Rosales, “Analysis on the time and frequency domain for the RC electric circuit of fractional order”, Open physics, vol. 11, no. 10, Oct. 2013, doi: 10.2478/s11534-013-0236-y.

P. Shah, A. Patel, I. Salehbhai, and A. Shukla, “Analytic solution for the RL electric circuit model in fractional order”, Abstract and applied analysis, vol. 2014, ID 343814, Jun. 2014, doi: 10.1155/2014/343814.

R. Hilfer, Applications of fractional calculus in physics, Singapore: World scientific, 2000.

R. Hilfer, “Experimental evidence for fractional time evolution in glass forming materials”, Chemical physics, vol. 284, no. 1-2, pp. 399–408, Nov. 2002, doi: 10.1016/S0301-0104(02)00670-5.

R. Saxena, A. Mathai, and H. Haubold, “Space-time fractional reaction-diffusion equations associated with a generalized Riemann–Liouville fractional derivative”, Axioms, vol. 3, no. 3, pp. 320–334, Aug. 2014, doi: 10.3390/axioms3030320.

R. Saxena, Z. Tomovski and T. Sandev, “Fractional Helmholtz and fractional wave equations with Riesz-Feller and generalized Riemann-Liouville fractional derivatives”, European journal of pure and applied mathematics, vol. 7, no. 3, pp. 312-334, Aug. 2014. [On line]. Available: http://bit.ly/2nZzigI

S. Purohit, “Solution of fractional partial differential equations of quantum mechanics”, Advances in Applied Mathematics and Mechanics, vol. 5, no. 5, pp. 639-651, Oct. 2011, doi: 10.1017/S2070073300002356.

T. Kaczorek and K. Rogowski, “Positive fractional electrical circuits”, in Fractional Linear Systems and Electrical Circuits, vol. 13, Cham: Springer, 2015, pp. 49–80, doi: 10.1007/978-3-319-11361-6_2.

T. Sandev, R. Metzler, and Ž. Tomovski, “Fractional diffusion equation with a generalized Riemann–Liouville time fractional derivative”, Journal of physics A: mathematical and theoretical, vol. 44, no. 25, p. 255203, May 2011, doi: 10.1088/1751-8113/44/25/255203.

T. Sandev, Z. Tomovski, and B. Crnkovic, “Generalized distributed order diffusion equations with composite time fractional derivative”, Computers & mathematics with applications, vol. 73, no. 6, pp. 1028–1040, Mar. 2017, doi: 10.1016/j.camwa.2016.07.009.

V. Toro, Electrical engineering fundamentals, 2nd ed. Delhi: Prentice-Hall, 2002.

Ž. Tomovski, T. Sandev, R. Metzler, and J. Dubbeldam, “Generalized space–time fractional diffusion equation with composite fractional time derivative”, Physica A: Statistical Mechanics and its Applications, vol. 391, no. 8, pp. 2527–2542, Apr. 2012, doi: 10.1016/j.physa.2011.12.035.



How to Cite

K. B. Kachhia, J. C. Prajapati, K. S. Pandya, and R. Jadeja, “On flow of electric current in RL circuit using Hilfer type composite fractional derivative”, Proyecciones (Antofagasta, On line), vol. 38, no. 4, pp. 625-636, Sep. 2019.




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