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Functionality of circuit via modern fractional differentiations

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Abstract

The significance of the modern fractional derivatives containing the singular kernel with locality and the non-singular kernel with non-locality have recently diverted the researchers because of the numerical or experimental analyses on the behavior between a system conservative and dissipative and the lack of fractionalized analytic methods. This study investigates the effects of modern fractional differentiation on the RLC electrical circuit via exact analytical approach. The modeling of governing differential equation of RLC electrical circuit has been fractionalized through three types of fractional derivatives namely Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional derivatives based on the range as \(0 \le \alpha \le 1,\,\,0 \le \beta \le 1,\,\,0 \le \gamma \le 1\) respectively. The RLC electrical circuit is observed for exponential, periodic and unit step sources via three classified modern fractional derivatives. The exact analytical solutions have been investigated by invoking mathematical Laplace transforms and presented in terms of convolutions product and special function namely Fox-H function. The Comparative mathematical analysis of RLC electrical circuit is based on Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional derivatives which exhibit the presence of heterogeneities in the electrical components causing irreversible dissipative effects. Finally, the several similarities and differences for the periodic and exponential sources have been rectified on the basis of the Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional derivatives for the current.

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  • 10 January 2019

    The article “Functionality of circuit via modern fractional differentiations”, written by Kashif Ali Abro, Ali Asghar Memon and Anwar Ahmed Memon, was originally published electronically on the publisher’s internet portal (currently SpringerLink) on November 2018 with open access.

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Acknowledgements

The authors are highly thankful and grateful to Mehran university of Engineering and Technology, Jamshoro, Pakistan for generous support and facilities of this research work.

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Authors and Affiliations

  1. Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro, Pakistan

    Kashif Ali Abro

  2. Department of Electrical Engineering, Mehran University of Engineering and Technology, Jamshoro, Pakistan

    Ali Asghar Memon & Anwar Ahmed Memon

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  1. Kashif Ali Abro
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Correspondence to Kashif Ali Abro.

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Appendix

Appendix

$$\frac{1}{1 + x} = \sum\limits_{n = 0}^{\infty } {\left( { - x} \right)^{n} } ,$$
(25)
$$(a + b)^{m} = \sum\limits_{m = 0}^{\infty } {\frac{{\left( { - 1} \right)^{m} \varGamma (n + 1)}}{m!\varGamma (n - m + 1)}a^{n - m} \,b^{m} } ,$$
(26)
$$L^{ - 1} \left\{ {F(s)G(s)} \right\} = \int\limits_{0}^{t} {f(t)g(t - z)dz} ,$$
(27)

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Abro, K.A., Memon, A.A. & Memon, A.A. Functionality of circuit via modern fractional differentiations. Analog Integr Circ Sig Process 99, 11–21 (2019). https://doi.org/10.1007/s10470-018-1371-6

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  • DOI: https://doi.org/10.1007/s10470-018-1371-6

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