Electrical circuits RC, LC, and RL described by Atangana–Baleanu fractional derivatives

Summary In this paper, the analytical solutions for the electrical series circuits RC, LC, and RL using novel fractional derivatives of type Atangana–Baleanu with non-singular and nonlocal kernel in Liouville–Caputo and Riemann–Liouville sense were obtained. The fractional equations in the time domain are considered derivatives in the range α∈(0;1]; analytical solutions are presented considering different source terms introduced in the fractional equation. We solved analytically the fractional equation using the properties of Laplace transform operator together with the convolution theorem. On the basis of the Mittag–Leffler function, new behaviors for the voltage and current were obtained; the classical cases are recovered when α=1. Copyright © 2017 John Wiley & Sons, Ltd.

[1]  J. J. Rosales,et al.  Analysis on the time and frequency domain for the RC electric circuit of fractional order , 2013 .

[2]  Maneesha Gupta,et al.  Digital fractional‐order differentiator and integrator models based on first‐order and higher order operators , 2011, Int. J. Circuit Theory Appl..

[3]  Jinde Cao,et al.  Adaptive synchronization of fractional-order memristor-based neural networks with time delay , 2015, Nonlinear Dynamics.

[4]  Ahmed S. Elwakil,et al.  Fractional-order models of supercapacitors, batteries and fuel cells: a survey , 2015, Materials for Renewable and Sustainable Energy.

[5]  A. Elwakil,et al.  Power and energy analysis of fractional-order electrical energy storage devices , 2016 .

[6]  Miguel Angel Moreles,et al.  Mathematical modelling of fractional order circuit elements and bioimpedance applications , 2017, Commun. Nonlinear Sci. Numer. Simul..

[7]  Ilknur Koca,et al.  Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order , 2016 .

[8]  A. E. Çalık,et al.  Investigation of electrical RC circuit within the framework of fractional calculus , 2015 .

[9]  Ahmed M. Soliman,et al.  Fractional‐order mutual inductance: analysis and design , 2016, Int. J. Circuit Theory Appl..

[10]  Ahmed S Elwakil,et al.  Fractional-order circuits and systems: An emerging interdisciplinary research area , 2010, IEEE Circuits and Systems Magazine.

[11]  Badr Saad T. Alkahtani,et al.  Extension of the resistance, inductance, capacitance electrical circuit to fractional derivative without singular kernel , 2015 .

[12]  Jirí Vlach,et al.  RC models of a constant phase element , 2011, Int. J. Circuit Theory Appl..

[13]  Ahmed Alsaedi,et al.  Fractional electrical circuits , 2015 .

[14]  Debasmita Mondal,et al.  Design and performance study of phase‐locked loop using fractional‐order loop filter , 2015, Int. J. Circuit Theory Appl..

[15]  Abdon Atangana,et al.  Numerical solution for the model of RLC circuit via the fractional derivative without singular kernel , 2015 .

[16]  A. E. Çalık,et al.  Analysis of charge variation in fractional order LC electrical circuit , 2016 .

[17]  José Francisco Gómez-Aguilar,et al.  Irving–Mullineux oscillator via fractional derivatives with Mittag-Leffler kernel , 2017 .

[18]  J. F. Gómez‐Aguilar,et al.  A physical interpretation of fractional calculus in observables terms: analysis of the fractional time constant and the transitory response , 2014 .

[19]  Sunil Kumar,et al.  A new analytical modelling for fractional telegraph equation via Laplace transform , 2014 .

[20]  Carl F. Lorenzo,et al.  Energy storage and loss in fractional-order circuit elements , 2015, IET Circuits Devices Syst..

[21]  R. F. Escobar-Jiménez,et al.  Electrical circuits described by a fractional derivative with regular Kernel , 2016 .

[22]  J. F. Aguilar.  Behavior characteristics of a cap-resistor, memcapacitor, and a memristor from the response obtained of RC and RL electrical circuits described by fractional differential equations , 2016 .

[23]  Ahmed S. Elwakil,et al.  Analysis and realization of a switched fractional‐order‐capacitor integrator , 2016, Int. J. Circuit Theory Appl..

[24]  Sophie Hallstedt,et al.  A model-based approach for sustainability and value assessment in the aerospace value chain , 2015 .

[25]  Dumitru Baleanu,et al.  New Derivatives on the Fractal Subset of Real-Line , 2015, Entropy.

[26]  Maysaa Mohamed Al Qurashi,et al.  Analytical Solutions of the Electrical RLC Circuit via Liouville-Caputo Operators with Local and Non-Local Kernels , 2016, Entropy.

[27]  José Francisco Gómez-Aguilar,et al.  Modeling of a Mass-Spring-Damper System by Fractional Derivatives with and without a Singular Kernel , 2015, Entropy.

[28]  A. Atangana,et al.  New Fractional Derivatives with Nonlocal and Non-Singular Kernel: Theory and Application to Heat Transfer Model , 2016, 1602.03408.

[29]  José Francisco Gómez-Aguilar,et al.  Space–time fractional diffusion equation using a derivative with nonsingular and regular kernel , 2017 .

[30]  Costas Psychalinos,et al.  Ultra‐low voltage fractional‐order circuits using current mirrors , 2016, Int. J. Circuit Theory Appl..

[31]  So Ovie,et al.  The effect of protein and carbohydrate of diets on the body composition of heterobranchus logifilis fingerlings , 2007 .

[32]  Zhigang Zeng,et al.  Global Mittag-Leffler stability and synchronization of memristor-based fractional-order neural networks , 2014, Neural Networks.

[33]  Giuseppe Schettino,et al.  CORRIGENDUM: Biological consequences of nanoscale energy deposition near irradiated heavy atom nanoparticles , 2013, Scientific Reports.