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Functionality of circuit via modern fractional differentiations

  • Kashif Ali Abro
  • Ali Asghar Memon
  • Anwar Ahmed Memon
Open Access
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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.

Keywords

RLC electrical circuit Modern fractional operators Special functions Laplace transform Graphical illustrations 

1 Introduction

There is no denying fact that the modern fractional operators are utilized for the vivid behavior of electrical circuits; for instance, the governing ordinary differential equations of circuits design analog and digital filters of fractional-order, systems with memristors, meminductors or memcapacitors and fractional-order description of magnetically-coupled coils or the behavior of circuits. Owing to such effects of the magnetic and electric fields, these elements involve irreversible dissipative effects (non-linearities, thermal memory or ohmic friction) and have a non-conservative feature [1, 2, 3, 4, 5, 6, 7]. These models have been extended to the scope of modern fractional derivatives; for instance, Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional derivatives and many other fractional operators. These fractional derivatives (Caputo, Caputo-Fabrizio and Atangana-Baleanu) have become the burning topics in the research due to two reasons/weaknesses for instance, (1) problem of the singular kernel with locality and (2) problem of the non-singular kernel with non-locality. The Caputo fractional derivative operator is the most suitable time fractional operator [8]. But this operator contains two serious pitfalls and drawbacks [9, 10, 11]. Frist is, the analytical solutions via Caputo fractional derivative operator are always expressed in terms of generalized special functions (Fox-H function [12], Mittage-Leffler function [13], Robotnov function [13], Lorenzo-Hartley function [13], Wright’s generalized hyper-geometric function [14] and few others). Second is, the singularity issue in the order of fractional derivative does not provide the full description of a memory. In order to avoid the problem of the singular kernel, Michele Caputo and Mauro Fabrizio proposed a fractional derivative by employing exponential function [15]. Indeed, the claim of singular kernel for fractional derivative operator is not based on their observations; even they suggested their fractional derivative operator was appropriate for various physical problems as well. On the other hand, there are some discrepancies regarding Caputo and Capotu-Fabrizio fractional derivatives which are recently addressed by Atangana [16]. Atangana-Baleanu suggested two general definitions of fractional order derivatives in Riemann–Liouville and Caputo sense. They claimed that their fractional derivative has fractional integral as anti-derivative of their operators. Atangana-Baleanu fractional order derivative has non-locality as well as non-singularity of the kernel based on the generalized Mittag–Leffler function [16]. The transient heat diffusion equation expressed through the Caputo-Fabrizio time fractional derivative with relaxation term was considered by Hristov [17] in which the Cattaneo constitutive relation with a Jeffrey’s kernel was also modified with non-singular fading memories. Atanganaa and Kocab [18] presented stability of the iterative method for the nonlinear Baggs and Freedman model via modern fractional derivative. They investigated special solutions and some properties of the inner product and the Hilbert space. In another study, the same authors Atanganaa and Kocab [19] implemented Atangana-Baleanu fractional derivatives to a simple nonlinear system. They generated a chaotic behavior which was not obtained by local derivative. Qasem et al. [20] observed the combined analysis of heat and mass transfer with and without magnetic field and porosity via Caputo-Fabrizio fractional derivatives approach. Alkahtani and Atangana [21] presented the numerical simulations to control the movement of waves on the area of shallow water through Caputo-Fabrizio fractional operator.

In this continuation, this manuscript emphasizes the comparative analysis of different newly presented fractional derivatives, we discuss here few recent comparisons of modern fractional derivatives as well. Nadeem et al. [22] studied the free convection flow of a generalized Casson fluid due to the combined gradients of temperature and concentration based on exponential kernel (Caputo and Fabrizio) and generalized Mittag–Leffler function (Atangana-Balaenu). Arshad et al. [23] observed the effects of heat and mass transfer of second grade fluids through the comparative analysis of Atangana-Baleanu and Caputo Fabrizio fractional derivatives. Kashif et al. [24] analyzed the RL and RC electrical circuits by the mathematical approaches of Caputo- Fabrizio and Atangana-Balaenu fractional derivatives. They the presented comparison of both fractional derivatives through three two types of sources namely periodic and exponential sources. Shaikh et al. [25] investigated a comparative analysis of Atangana-Baleanu and Caputo-Fabrizio fractional models to the nanofluids for enhancement of the performance of solar collectors. The studies on modern fractional derivatives can be continued but we end here categorically as diffusion models [26, 27], fluids [28, 29, 30], heat transfer [31], magnetohydrodynamics [32, 33], Nanofluids [34], non-autonomous system [35, 36], dynamics and control theory [37, 38] and many others. Inspiring from these modern approaches to fractional derivatives, the present study adopted comparative analysis based on Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional derivatives approach for the RLC electrical circuit via exact analytical approach. The modelling of governing differential equation of RLC electrical circuits have 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. Three types of sources are implemented namely exponential, periodic and unit step and 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. In the ending, the several similarities and differences for three types of sources (periodic, exponential and unit step) have been rectified on the basis of the Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional derivatives for the current.

2 Mathematical modeling of RLC electrical circuit

By invoking the Kirchhoff laws for expressing the non-homogeneous integer-order differential equation corresponding to RLC electrical circuit as depicted in Fig. 1
$$\frac{{d^{2} I(t)}}{{dt^{2} }} + \frac{R}{L}\frac{dI(t)}{dt} + \frac{I(t)}{LC} = \frac{E(t)}{L},$$
(1)
$$E(t) = Ue^{ - at} \,{\text{and}}\,E(t) = U\sin (\omega \,t),\,\,\,\,\,I(0) = 0,$$
(2)
Here \(I\) symbolizes for current, \(t\) specifies for time, \(R\) represents the resistance, \(L\) indicates the inductance, \(C\) denotes the capacitance and \(E(t)\) signifies source of voltage. Subject to the initial and boundary conditions are taken into consideration for Eq. (1) fulfuling the exact analytical solution for the validations of results. Meanwhile, Eq. (2) represents the source terms which are categorized as an exponential, the periodic and unit step sources of the current. In order to develop ordinary differential equations of current say Eq. (1) for fractionalized differential equations, we introduce newly defined Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional operators as
$${}^{C}\left( {\frac{{d^{\alpha } I(t)}}{{dt^{\alpha } }}} \right) = \left\{ {\begin{array}{*{20}l} {\frac{1}{{\varGamma \left( {1 - \alpha } \right)}}\int\limits_{0}^{t} {\frac{1}{{(z - t)^{\alpha } }}\,I(t)dt,} } \hfill & {0 < \alpha < 1} \hfill \\ {\frac{dI(t)}{dt},} \hfill & {\alpha = 1.} \hfill \\ \end{array} } \right.$$
(3)
$${}^{CF}\left( {\frac{{d^{\beta } I(t)}}{{dt^{\beta } }}} \right) = \left\{ {\begin{array}{*{20}l} {\frac{1}{1 - \beta }\int\limits_{0}^{t} {Exp\left[ {\frac{{ - \beta (z - t)^{\beta } }}{(1 - \beta )}} \right]} \,I(t)dt,} \hfill & {0 < \beta < 1} \hfill \\ {\frac{dI(t)}{dt},} \hfill & {\beta = 1.} \hfill \\ \end{array} } \right.$$
(4)
$${}^{AB}\left( {\frac{{d^{\gamma } I(t)}}{{dt^{\gamma } }}} \right) = \left\{ {\begin{array}{*{20}l} {\frac{1}{1 - \gamma }\int\limits_{0}^{t} {\,{\mathbf{E}}_{\gamma } \left( {\frac{{ - \gamma (z - t)^{\gamma } }}{1 - \gamma }} \right)} \,I(t)dt,} \hfill & {0 < \gamma < 1} \hfill \\ {\frac{dI(t)}{dt},} \hfill & {\gamma = 1.} \hfill \\ \end{array} } \right.$$
(5)
where, \({}^{C}\left( {\frac{{d^{\alpha } I(t)}}{{dt^{\alpha } }}} \right)\) is Caputo fractional operator of order \(0 < \alpha < 1\) is defined as in [39, 40, 41], \({}^{CF}\left( {\frac{{d^{\beta } I(t)}}{{dt^{\beta } }}} \right)\) is Caputo-Fabrizio fractional operators of order \(0 < \beta < 1\) proposed by [42] and \({}^{AB}\left( {\frac{{d^{\gamma } I(t)}}{{dt^{\gamma } }}} \right)\) Atangana-Baleanu fractional operator of order \(0 < \gamma < 1\) is defined as in [43]. Implementing Eqs. (35) on governing differential equation of RLC electrical circuit (1), we investigated the following fractional order governing differential equation of RLC electrical circuit in terms of Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional operators respectively
$${}^{C}\left( {\frac{{d^{2\alpha } I(t)}}{{dt^{2\alpha } }}} \right) + \frac{R}{L}{}^{C}\left( {\frac{{d^{\alpha } I(t)}}{{dt^{\alpha } }}} \right) + \frac{I(t)}{LC} = \frac{E(t)}{L},$$
(6)
$${}^{CF}\left( {\frac{{d^{2\beta } I(t)}}{{dt^{2\beta } }}} \right) + \frac{R}{L}{}^{CF}\left( {\frac{{d^{\beta } I(t)}}{{dt^{\beta } }}} \right) + \frac{I(t)}{LC} = \frac{E(t)}{L},$$
(7)
$${}^{AB}\left( {\frac{{d^{2\gamma } I(t)}}{{dt^{2\gamma } }}} \right) + \frac{R}{L}{}^{AB}\left( {\frac{{d^{\gamma } I(t)}}{{dt^{\gamma } }}} \right) + \frac{I(t)}{LC} = \frac{E(t)}{L}.$$
(8)
The system of Caputo fractional, Caputo-Fabrizio fractional and Atangana-Baleanu fractional differential equations for RLC electrical circuit (68) with adequate imposed initial and boundary conditions say Eq. (2) can be elucidated in principle by numerous approaches, but their effectiveness usually depends upon the domain of definition. Keeping effectiveness in mind for the exact analytical solutions of fractional differential equations of non-integer order governed by RLC electrical circuit, we utilize Laplace transforms as a systematic, powerful and efficient tool for eliminating the time variable.
Fig. 1

Geometrical configuration of RLC electrical circuit

3 Mathematical analysis of the RLC electrical circuit

3.1 Exact analytical solution via Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional operator

Case I: periodic source

\(E(t) = U\,Sin(\omega \,t),\,I(0) = 0\)

Invoking Laplace transform on governing fractional differential Eqs. (68) of RLC electrical circuit and keeping in mind an exponential source, say Eqs. (2)2 and (2)4, we get
$$\frac{{Rq^{\alpha } }}{L}\bar{I}(q) + \frac{{\bar{I}(q)}}{LC} + q^{2\alpha } \bar{I}(q) = \frac{U\omega }{{(q^{2} + \omega^{2} )L}},$$
(9)
$$\frac{{q^{2} \bar{I}(q)}}{{(1 - \beta )^{2} \left[ {q + \frac{\beta }{1 - \beta }} \right]^{2} }} + \frac{{Rq\bar{I}(q)}}{{L(1 - \beta )\left[ {q + \frac{\beta }{1 - \beta }} \right]}} + \frac{{\bar{I}(q)}}{LC} = \frac{U\omega }{{(q^{2} + \omega^{2} )L}},$$
(10)
$$\frac{{q^{2\gamma } \bar{I}(q)}}{{(1 - \gamma )^{2} \left[ {q^{\gamma } + \frac{\gamma }{1 - \gamma }} \right]^{2} }} + \frac{{Rq^{\gamma } \bar{I}(q)}}{{L(1 - \gamma )\left[ {q^{\gamma } + \frac{\gamma }{1 - \gamma }} \right]}} + \frac{{\bar{I}(q)}}{LC} = \frac{U\omega }{{(q^{2} + \omega^{2} )L}}.$$
(11)
rewriting Eqs. (911) into equivalent form, we arrive
$$\bar{I}(q) = \frac{U\omega \,C}{{L(q^{2} + \omega^{2} )\left[ {q^{ 2\alpha } + \frac{R}{L}q^{\alpha } + \frac{1}{LC}} \right]}},$$
(12)
$$\bar{I}(q) = \frac{{U\omega \,C\left( {q^{2} + q\beta \eta_{0}^{{}} + \beta^{2} \eta_{0}^{2} } \right)}}{{L(q^{2} + \omega^{2} )\left[ {\varUpsilon_{0}^{{}} q^{2} + \varUpsilon_{1}^{{}} q + \varUpsilon_{3}^{{}} } \right]}},$$
(13)
$$\bar{I}(q) = \frac{{U\omega \,C\left( {q^{2\gamma } + q^{\gamma } \gamma \eta_{1}^{{}} + \gamma^{2} \eta_{1}^{2} } \right)}}{{L(q^{2} + \omega^{2} )\left[ {\varUpsilon_{3}^{{}} q^{2\gamma } + \varUpsilon_{4}^{{}} q^{\gamma } + \varUpsilon_{5}^{{}} } \right]}},$$
(14)
where, the letting parameters are \(\eta_{0}^{{}} = \frac{1}{1 - \beta }\), \(\eta_{1}^{{}} = \frac{1}{1 - \gamma }\), \(\varUpsilon_{0}^{{}} = \left( {1 + \eta_{0}^{{}} LC} \right)\), \(\varUpsilon_{1}^{{}} = \left( {LC + 2\beta \eta_{0}^{{}} } \right)\), \(\varUpsilon_{2}^{{}} = \beta \eta_{0}^{{}} \left( {LC + \beta \eta_{0}^{{}} } \right)\), \(\varUpsilon_{3}^{{}} = \left( {1 + \eta_{1}^{{}} LC} \right)\), \(\varUpsilon_{4}^{{}} = \left( {LC + 2\gamma \eta_{1}^{{}} } \right)\), \(\varUpsilon_{5}^{{}} = \gamma \eta_{1}^{{}} \left( {LC + \gamma \eta_{1}^{{}} } \right)\). Expanding Eqs. (1214) in terms of series form by using “Appendix” (25) and (26), we have
$$\bar{I}(q) = \frac{U\omega C}{{(q^{2} - \omega^{2} )}}\sum\limits_{{\delta_{0}^{{}} = 0}}^{\infty } {\left( { - LC} \right)^{{\delta_{0}^{{}} }} } \sum\limits_{{\delta_{1}^{{}} = 0}}^{\infty } {\frac{{\left( { - R} \right)^{{\delta_{1}^{{}} }} \varGamma \left( {\delta_{1}^{{}} + 1} \right)}}{{\delta_{1}^{{}} !\varGamma \left( {\delta_{0}^{{}} - \delta_{1}^{{}} + 1} \right)}}\frac{1}{{q^{{\delta_{1}^{{}} \alpha - 2\alpha \delta_{0}^{{}} }} }}} ,$$
(15)
$$\bar{I}(q) = \frac{U\omega C}{{\varUpsilon_{2}^{{}} (q^{2} - \omega^{2} )}}\sum\limits_{{\delta_{0}^{{}} = 0}}^{\infty } {\left( {\frac{{\varUpsilon_{0}^{{}} }}{{\varUpsilon_{2}^{{}} }}} \right)^{{\delta_{0}^{{}} }} } \sum\limits_{{\delta_{1}^{{}} = 0}}^{\infty } {\frac{{\left( { - \frac{{\varUpsilon_{1}^{{}} }}{{\varUpsilon_{0}^{{}} }}} \right)^{{\delta_{1}^{{}} }} \varGamma \left( {\delta_{1}^{{}} + 1} \right)}}{{\delta_{1}^{{}} !\varGamma \left( {\delta_{0}^{{}} - \delta_{1}^{{}} + 1} \right)}}\frac{1}{{q^{{2\delta_{0}^{{}} - \delta_{1}^{{}} + 2}} }}} + \frac{{2U\omega \beta \eta_{0}^{{}} C}}{{\varUpsilon_{2}^{{}} (q^{2} - \omega^{2} )}}\sum\limits_{{\delta_{0}^{{}} = 0}}^{\infty } {\left( {\frac{{\varUpsilon_{0}^{{}} }}{{\varUpsilon_{2}^{{}} }}} \right)^{{\delta_{0}^{{}} }} } \times \sum\limits_{{\delta_{1}^{{}} = 0}}^{\infty } {\frac{{\left( { - \frac{{\varUpsilon_{1}^{{}} }}{{\varUpsilon_{0}^{{}} }}} \right)^{{\delta_{1}^{{}} }} \varGamma \left( {\delta_{1}^{{}} + 1} \right)}}{{\delta_{1}^{{}} !\varGamma \left( {\delta_{0}^{{}} - \delta_{1}^{{}} + 1} \right)}}} \frac{1}{{q^{{2\delta_{0}^{{}} - \delta_{1}^{{}} + 1}} }} + \frac{{U\omega \beta^{2} \eta_{0}^{2} C}}{{\varUpsilon_{2}^{{}} (q^{2} - \omega^{2} )}}\sum\limits_{{\delta_{0}^{{}} = 0}}^{\infty } {\left( { - \frac{{\varUpsilon_{0}^{{}} }}{{\varUpsilon_{2}^{{}} }}} \right)^{{\delta_{0}^{{}} }} } \sum\limits_{{\delta_{1}^{{}} = 0}}^{\infty } {\frac{{\left( {\frac{{\varUpsilon_{1}^{{}} }}{{\varUpsilon_{0}^{{}} }}} \right)^{{\delta_{1}^{{}} }} \varGamma \left( {\delta_{1}^{{}} + 1} \right)}}{{\delta_{1}^{{}} !\varGamma \left( {\delta_{0}^{{}} - \delta_{1}^{{}} + 1} \right)}}} \frac{1}{{q^{{2\delta_{0}^{{}} - \delta_{1}^{{}} }} }},$$
(16)
$$\bar{I}(q) = \frac{U\omega C}{{\varUpsilon_{5}^{{}} (q^{2} - \omega^{2} )}}\sum\limits_{{\delta_{0}^{{}} = 0}}^{\infty } {\left( {\frac{{\varUpsilon_{3}^{{}} }}{{\varUpsilon_{5}^{{}} }}} \right)^{{\delta_{0}^{{}} }} } \sum\limits_{{\delta_{1}^{{}} = 0}}^{\infty } {\frac{{\left( { - \frac{{\varUpsilon_{4}^{{}} }}{{\varUpsilon_{3}^{{}} }}} \right)^{{\delta_{1}^{{}} }} \varGamma \left( {\delta_{1}^{{}} + 1} \right)}}{{\delta_{1}^{{}} !\varGamma \left( {\delta_{0}^{{}} - \delta_{1}^{{}} + 1} \right)}}\frac{1}{{q^{{2\gamma \delta_{0}^{{}} - \gamma \delta_{1}^{{}} + 2\gamma }} }}} + \frac{{2U\omega \gamma \eta_{1}^{{}} C}}{{\varUpsilon_{5}^{{}} (q^{2} - \omega^{2} )}}\sum\limits_{{\delta_{0}^{{}} = 0}}^{\infty } {\left( {\frac{{\varUpsilon_{3}^{{}} }}{{\varUpsilon_{5}^{{}} }}} \right)^{{\delta_{0}^{{}} }} } \times \sum\limits_{{\delta_{1}^{{}} = 0}}^{\infty } {\frac{{\left( { - \frac{{\varUpsilon_{4}^{{}} }}{{\varUpsilon_{3}^{{}} }}} \right)^{{\delta_{1}^{{}} }} \varGamma \left( {\delta_{1}^{{}} + 1} \right)}}{{\delta_{1}^{{}} !\varGamma \left( {\delta_{0}^{{}} - \delta_{1}^{{}} + 1} \right)}}} \frac{1}{{q^{{2\gamma \delta_{0}^{{}} - \gamma \delta_{1}^{{}} + 1}} }} + \frac{{U\omega \gamma^{2} \eta_{1}^{2} C}}{{\varUpsilon_{5}^{{}} (q^{2} - \omega^{2} )}}\sum\limits_{{\delta_{0}^{{}} = 0}}^{\infty } {\left( { - \frac{{\varUpsilon_{3}^{{}} }}{{\varUpsilon_{5}^{{}} }}} \right)^{{\delta_{0}^{{}} }} } \sum\limits_{{\delta_{1}^{{}} = 0}}^{\infty } {\frac{{\left( {\frac{{\varUpsilon_{4}^{{}} }}{{\varUpsilon_{3}^{{}} }}} \right)^{{\delta_{1}^{{}} }} \varGamma \left( {\delta_{1}^{{}} + 1} \right)}}{{\delta_{1}^{{}} !\varGamma \left( {\delta_{0}^{{}} - \delta_{1}^{{}} + 1} \right)}}} \frac{1}{{q^{{2\gamma \delta_{0}^{{}} - \gamma \delta_{1}^{{}} }} }},$$
(17)
Inverting Eqs. (1517) by Laplace transform having the property of convolution defined in “Appendix” (27), we find final expression of current flowing in RLC electrical in terms of special function namely Fox-H function as
$${}^{C}\left( {I(t)} \right) = \frac{UC}{t}\sum\limits_{{\delta_{0}^{{}} = 0}}^{\infty } {\left( {\frac{ - LC}{{t^{2\alpha } }}} \right)^{{\delta_{0}^{{}} }} } \int\limits_{0}^{t} {\sin \omega (t - z)\,} {\mathbf{H}}_{1,3}^{1,1} \left[ {\left( { - Rt^{\alpha } } \right)\left| {\begin{array}{*{20}c} {\left( { - \delta_{0}^{{}} ,0} \right)} \\ {\left( {0,1} \right),\left( { - \delta_{0}^{{}} , - \delta_{1}^{{}} } \right),\left( {1 + 2\alpha \delta_{0}^{{}} ,\alpha } \right)} \\ \end{array} } \right.} \right]\,\,dz,$$
(18)
$${}^{CF}\left( {I(t)} \right) = \frac{UC}{{\varUpsilon_{2}^{{}} t^{2\beta + 1} }}\sum\limits_{{\delta_{0}^{{}} = 0}}^{\infty } {\left( {\frac{{\varUpsilon_{0}^{{}} }}{{\varUpsilon_{2}^{{}} t^{2\beta } }}} \right)^{{\delta_{0}^{{}} }} \int\limits_{0}^{t} {\sin \omega (t - z)\,} } {\mathbf{H}}_{1,3}^{1,1} \left[ {\left( { - \frac{{\varUpsilon_{1}^{{}} t^{\beta } }}{{\varUpsilon_{0}^{{}} }}} \right)\left| {\begin{array}{*{20}c} {\left( { - \delta_{0}^{{}} ,0} \right)} \\ {\left( {0,1} \right),\left( { - \delta_{0}^{{}} , - \delta_{1}^{{}} } \right),\left( {1 + 2\beta \delta_{0}^{{}} + 2\beta ,\beta } \right)} \\ \end{array} } \right.} \right]\,\,dz + \frac{{2U\beta \eta_{0}^{{}} C}}{{\varUpsilon_{2}^{{}} t^{\beta + 1} }}\sum\limits_{{\delta_{0}^{{}} = 0}}^{\infty } {\left( {\frac{{\varUpsilon_{0}^{{}} }}{{\varUpsilon_{2}^{{}} t^{2\beta } }}} \right)^{{\delta_{0}^{{}} }} } \int\limits_{0}^{t} {\sin \omega (t - z)\,} {\mathbf{H}}_{1,3}^{1,1} \left[ {\left( { - \frac{{\varUpsilon_{1}^{{}} t^{\beta } }}{{\varUpsilon_{0}^{{}} }}} \right)\left| {\begin{array}{*{20}c} {\left( { - \delta_{0}^{{}} ,0} \right)} \\ {\left( {0,1} \right),\left( { - \delta_{0}^{{}} , - \delta_{1}^{{}} } \right),\left( {1 + 2\beta \delta_{0}^{{}} + \beta ,\beta } \right)} \\ \end{array} } \right.} \right]\,\,dz + \frac{{U\beta^{2} \eta_{0}^{2} C}}{{\varUpsilon_{2}^{{}} t}}\sum\limits_{{\delta_{0}^{{}} = 0}}^{\infty } {\left( { - \frac{{\varUpsilon_{0}^{{}} }}{{\varUpsilon_{2}^{{}} t^{2\beta } }}} \right)^{{\delta_{0}^{{}} }} } \int\limits_{0}^{t} {\sin \omega (t - z)\,} {\mathbf{H}}_{1,3}^{1,1} \left[ {\left( { - \frac{{\varUpsilon_{1}^{{}} t^{\beta } }}{{\varUpsilon_{0}^{{}} }}} \right)\left| {\begin{array}{*{20}c} {\left( { - \delta_{0}^{{}} ,0} \right)} \\ {\left( {0,1} \right),\left( { - \delta_{0}^{{}} , - \delta_{1}^{{}} } \right),\left( {1 + 2\beta \delta_{0}^{{}} ,\beta } \right)} \\ \end{array} } \right.} \right]\,\,dz,$$
(19)
$${}^{AB}\left( {I(t)} \right) = \frac{UC}{{\varUpsilon_{5}^{{}} t^{2\gamma + 1} }}\sum\limits_{{\delta_{0}^{{}} = 0}}^{\infty } {\left( {\frac{{\varUpsilon_{3}^{{}} }}{{\varUpsilon_{5}^{{}} t^{2\gamma } }}} \right)^{{\delta_{0}^{{}} }} \int\limits_{0}^{t} {\sin \omega (t - z)\,} } {\mathbf{H}}_{1,3}^{1,1} \left[ {\left( { - \frac{{\varUpsilon_{4}^{{}} t^{\gamma } }}{{\varUpsilon_{3}^{{}} }}} \right)\left| {\begin{array}{*{20}c} {\left( { - \delta_{0}^{{}} ,0} \right)} \\ {\left( {0,1} \right),\left( { - \delta_{0}^{{}} , - \delta_{1}^{{}} } \right),\left( {1 + 2\gamma \delta_{0}^{{}} + 2\gamma ,\gamma } \right)} \\ \end{array} } \right.} \right]\,\,dz + \frac{{2U\gamma \eta_{1}^{{}} C}}{{\varUpsilon_{5}^{{}} t^{\gamma + 1} }}\sum\limits_{{\delta_{0}^{{}} = 0}}^{\infty } {\left( {\frac{{\varUpsilon_{3}^{{}} }}{{\varUpsilon_{5}^{{}} t^{2\gamma } }}} \right)^{{\delta_{0}^{{}} }} } \int\limits_{0}^{t} {\sin \omega (t - z)\,} {\mathbf{H}}_{1,3}^{1,1} \left[ {\left( { - \frac{{\varUpsilon_{4}^{{}} t^{\gamma } }}{{\varUpsilon_{3}^{{}} }}} \right)\left| {\begin{array}{*{20}c} {\left( { - \delta_{0}^{{}} ,0} \right)} \\ {\left( {0,1} \right),\left( { - \delta_{0}^{{}} , - \delta_{1}^{{}} } \right),\left( {1 + 2\gamma \delta_{0}^{{}} + \gamma ,\gamma } \right)} \\ \end{array} } \right.} \right]\,\,dz + \frac{{U\gamma^{2} \eta_{1}^{2} C}}{{\varUpsilon_{5}^{{}} t}}\sum\limits_{{\delta_{0}^{{}} = 0}}^{\infty } {\left( { - \frac{{\varUpsilon_{3}^{{}} }}{{\varUpsilon_{5}^{{}} t^{2\gamma } }}} \right)^{{\delta_{0}^{{}} }} } \int\limits_{0}^{t} {\sin \omega (t - z)\,} {\mathbf{H}}_{1,3}^{1,1} \left[ {\left( { - \frac{{\varUpsilon_{4}^{{}} t^{\gamma } }}{{\varUpsilon_{3}^{{}} }}} \right)\left| {\begin{array}{*{20}c} {\left( { - \delta_{0}^{{}} ,0} \right)} \\ {\left( {0,1} \right),\left( { - \delta_{0}^{{}} , - \delta_{1}^{{}} } \right),\left( {1 + 2\gamma \delta_{0}^{{}} ,\gamma } \right)} \\ \end{array} } \right.} \right]\,\,dz.$$
(20)
where, the Fox-H function is defined as [44]
$$\sum\limits_{\lambda }^{\infty } {\frac{{\left( { - I} \right)^{\lambda } \,\prod_{h = 1}^{f} \,\varGamma \left( {u_{h}^{{}} + U_{h}^{{}} \lambda } \right)}}{{\lambda \,!\,\,\prod_{h = 1}^{g} \,\varGamma \left( {v_{h}^{{}} + V_{h}^{{}} \lambda } \right)}}} = {\mathbf{H}}_{f,g + 1}^{1,f} \left. {\left[ {I\left| {\begin{array}{*{20}c} {(1 - u_{1}^{{}} ,U_{1}^{{}} ),(1 - u_{2}^{{}} ,U_{2}^{{}} ), \ldots ,(1 - u_{f}^{{}} ,U_{f}^{{}} )} \\ {(0,1),(1 - v_{1}^{{}} ,V_{1}^{{}} ),(1 - v_{2}^{{}} ,V_{2}^{{}} ), \ldots ,(1 - v_{f}^{{}} ,V_{f}^{{}} )} \\ \end{array} } \right.} \right.} \right].$$
(21)
Equations (1820) are the exact analytical solutions for the governing fractional differential equations of RLC electrical circuits via Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional derivatives respectively. These exact analytical solutions validate the imposed initial and boundary conditions as well.

Case-II: Exponential source

\(E(t) = U\,e^{at} ,\,I(0) = 0\)

Employing similar algorithm, we have investigated the exact analytical solutions for second case of exponential source \(E(t) = U\,e^{at} ,\,I(0) = 0\) as
$${}^{C}\left( {I(t)} \right) = \frac{UC}{t}\sum\limits_{{\delta_{0}^{{}} = 0}}^{\infty } {\left( {\frac{ - LC}{{t^{2\alpha } }}} \right)^{{\delta_{0}^{{}} }} } \int\limits_{0}^{t} {e^{a(t - z)} \,} {\mathbf{H}}_{1,3}^{1,1} \left[ {\left( { - Rt^{\alpha } } \right)\left| {\begin{array}{*{20}c} {\left( { - \delta_{0}^{{}} ,0} \right)} \\ {\left( {0,1} \right),\left( { - \delta_{0}^{{}} , - \delta_{1}^{{}} } \right),\left( {1 + 2\alpha \delta_{0}^{{}} ,\alpha } \right)} \\ \end{array} } \right.} \right]\,\,dz,$$
(22)
$${}^{CF}\left( {I(t)} \right) = \frac{UC}{{\varUpsilon_{2}^{{}} t^{2\beta + 1} }}\sum\limits_{{\delta_{0}^{{}} = 0}}^{\infty } {\left( {\frac{{\varUpsilon_{0}^{{}} }}{{\varUpsilon_{2}^{{}} t^{2\beta } }}} \right)^{{\delta_{0}^{{}} }} \int\limits_{0}^{t} {e^{a(t - z)} \,} } {\mathbf{H}}_{1,3}^{1,1} \left[ {\left( { - \frac{{\varUpsilon_{1}^{{}} t^{\beta } }}{{\varUpsilon_{0}^{{}} }}} \right)\left| {\begin{array}{*{20}c} {\left( { - \delta_{0}^{{}} ,0} \right)} \\ {\left( {0,1} \right),\left( { - \delta_{0}^{{}} , - \delta_{1}^{{}} } \right),\left( {1 + 2\beta \delta_{0}^{{}} + 2\beta ,\beta } \right)} \\ \end{array} } \right.} \right]\,\,dz + \frac{{2U\beta \eta_{0}^{{}} C}}{{\varUpsilon_{2}^{{}} t^{\beta + 1} }}\sum\limits_{{\delta_{0}^{{}} = 0}}^{\infty } {\left( {\frac{{\varUpsilon_{0}^{{}} }}{{\varUpsilon_{2}^{{}} t^{2\beta } }}} \right)^{{\delta_{0}^{{}} }} } \int\limits_{0}^{t} {e^{a(t - z)} \,} {\mathbf{H}}_{1,3}^{1,1} \left[ {\left( { - \frac{{\varUpsilon_{1}^{{}} t^{\beta } }}{{\varUpsilon_{0}^{{}} }}} \right)\left| {\begin{array}{*{20}c} {\left( { - \delta_{0}^{{}} ,0} \right)} \\ {\left( {0,1} \right),\left( { - \delta_{0}^{{}} , - \delta_{1}^{{}} } \right),\left( {1 + 2\beta \delta_{0}^{{}} + \beta ,\beta } \right)} \\ \end{array} } \right.} \right]\,\,dz + \frac{{U\beta^{2} \eta_{0}^{2} C}}{{\varUpsilon_{2}^{{}} t}}\sum\limits_{{\delta_{0}^{{}} = 0}}^{\infty } {\left( { - \frac{{\varUpsilon_{0}^{{}} }}{{\varUpsilon_{2}^{{}} t^{2\beta } }}} \right)^{{\delta_{0}^{{}} }} } \int\limits_{0}^{t} {e^{a(t - z)} \,} {\mathbf{H}}_{1,3}^{1,1} \left[ {\left( { - \frac{{\varUpsilon_{1}^{{}} t^{\beta } }}{{\varUpsilon_{0}^{{}} }}} \right)\left| {\begin{array}{*{20}c} {\left( { - \delta_{0}^{{}} ,0} \right)} \\ {\left( {0,1} \right),\left( { - \delta_{0}^{{}} , - \delta_{1}^{{}} } \right),\left( {1 + 2\beta \delta_{0}^{{}} ,\beta } \right)} \\ \end{array} } \right.} \right]\,\,dz,$$
(23)
$${}^{AB}\left( {I(t)} \right) = \frac{UC}{{\varUpsilon_{5}^{{}} t^{2\gamma + 1} }}\sum\limits_{{\delta_{0}^{{}} = 0}}^{\infty } {\left( {\frac{{\varUpsilon_{3}^{{}} }}{{\varUpsilon_{5}^{{}} t^{2\gamma } }}} \right)^{{\delta_{0}^{{}} }} \int\limits_{0}^{t} {e^{a(t - z)} \,} } {\mathbf{H}}_{1,3}^{1,1} \left[ {\left( { - \frac{{\varUpsilon_{4}^{{}} t^{\gamma } }}{{\varUpsilon_{3}^{{}} }}} \right)\left| {\begin{array}{*{20}c} {\left( { - \delta_{0}^{{}} ,0} \right)} \\ {\left( {0,1} \right),\left( { - \delta_{0}^{{}} , - \delta_{1}^{{}} } \right),\left( {1 + 2\gamma \delta_{0}^{{}} + 2\gamma ,\gamma } \right)} \\ \end{array} } \right.} \right]\,\,dz + \frac{{2U\gamma \eta_{1}^{{}} C}}{{\varUpsilon_{5}^{{}} t^{\gamma + 1} }}\sum\limits_{{\delta_{0}^{{}} = 0}}^{\infty } {\left( {\frac{{\varUpsilon_{3}^{{}} }}{{\varUpsilon_{5}^{{}} t^{2\gamma } }}} \right)^{{\delta_{0}^{{}} }} } \int\limits_{0}^{t} {e^{a(t - z)} \,} {\mathbf{H}}_{1,3}^{1,1} \left[ {\left( { - \frac{{\varUpsilon_{4}^{{}} t^{\gamma } }}{{\varUpsilon_{3}^{{}} }}} \right)\left| {\begin{array}{*{20}c} {\left( { - \delta_{0}^{{}} ,0} \right)} \\ {\left( {0,1} \right),\left( { - \delta_{0}^{{}} , - \delta_{1}^{{}} } \right),\left( {1 + 2\gamma \delta_{0}^{{}} + \gamma ,\gamma } \right)} \\ \end{array} } \right.} \right]\,\,dz + \frac{{U\gamma^{2} \eta_{1}^{2} C}}{{\varUpsilon_{5}^{{}} t}}\sum\limits_{{\delta_{0}^{{}} = 0}}^{\infty } {\left( { - \frac{{\varUpsilon_{3}^{{}} }}{{\varUpsilon_{5}^{{}} t^{2\gamma } }}} \right)^{{\delta_{0}^{{}} }} } \int\limits_{0}^{t} {e^{a(t - z)} \,} {\mathbf{H}}_{1,3}^{1,1} \left[ {\left( { - \frac{{\varUpsilon_{4}^{{}} t^{\gamma } }}{{\varUpsilon_{3}^{{}} }}} \right)\left| {\begin{array}{*{20}c} {\left( { - \delta_{0}^{{}} ,0} \right)} \\ {\left( {0,1} \right),\left( { - \delta_{0}^{{}} , - \delta_{1}^{{}} } \right),\left( {1 + 2\gamma \delta_{0}^{{}} ,\gamma } \right)} \\ \end{array} } \right.} \right]\,\,dz.$$
(24)
It is worth pointing out that the exact analytical solutions of fractional differential equations for RLC electrical circuits have been traced out by three varieties of fractional operators namely, Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional operators. Such investigated solutions can be retrieved for ordinary differential equations by making fractional parameters equal to one. Meanwhile, the behavior of current for RLC electrical circuit is also described for ordinary differential operator and fractional differential operators via graphical depictions via embedded parameters.

4 Parametric discussions

This section is organized for two types of sources for RLC circuit namely periodic and exponential sources for exploring the outcomes of different passive and pertinent parameter embedded like resistance \(R\), inductance \(L\), capacitance \(C\) and fractional parameters \(\alpha ,\beta ,\gamma\) through ordinary, Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional derivatives. In short, the following outcomes are discussed below:
  • Effects of resistance via Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional operators

Periodic source

The profile of current of RLC electrical circuit is depicted for the analysis of periodic source through Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional derivatives in Fig. 2. The profile of current is tested by increasing the numerical values of resistance which results decreasing behavior of current through above four approaches. It is noted that profile of current with respect to delay via Caputo fractional derivative and Caputo-Fabrizio fractional derivative is more in comparison with Atangana-Baleanu fractional derivative. From physical point of view, this delay results in system equilibrium poor.
Fig. 2

Plot of numerical simulation for an RLC electrical circuit through three differential fractional approaches corresponding to periodic source with different values of resistance R; for all figures I(t) is measured at Amperes and t is measured at seconds

Exponential source

In order to have comparative analysis of RLC electrical circuit for profile of current, Fig. 3 is prepared for the exponential source. It is pointed out from Fig. 3 that the increase in the resistance does not have scattering behavior of the current via Caputo fractional derivative in contrast with Caputo-Fabrizio and Atangana-Baleanu fractional derivative. This may be due to the fact of non-singular and non-local kernel among fractional derivatives. Physically, enhancing the resistance causes energies wastes and power losses in the RLC electrical circuits, such phenomenon is controlled via modern fractional derivatives namely Caputo-Fabrizio and Atangana-Baleanu fractional derivative.
Fig. 3

Plot of numerical simulation for an RLC electrical circuit through three different fractional approaches corresponding to exponential source with different values of resistance R; for all figures I(t) is measured at Amperes and t is measured at seconds

  • Effects of inductance via Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional operators

Periodic source

Figure 4 is drawn to test the ability of Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional derivatives for inductance on the periodic source. Here the damping is produced that refers to dissipate the energy stored in the oscillations. It is also observed that as inductance increases then number of oscillation are reduced by Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional derivatives respectively. Due to this fact the current obtained via fractional derivatives is increasing as the inductance increases, ultimately the number of oscillations is decreased. From physical and Practical point of view, the efficiency of the RLC electrical circuit can be minimized/decreased when the inductance increases. Hence, this fact suggests that the Atangana-Baleanu fractional operator is more effective than the Caputo, Caputo-Fabrizio fractional operators due to its significance of the non-locality and non-singularity involved in their kernels.
Fig. 4

Plot of numerical simulation for an RLC electrical circuit through three different fractional approaches corresponding to periodic source with different values of inductance L; for all figures I(t) is measured at Amperes and t is measured at seconds

Exponential source

The same observation is perceived while testing the ability of Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional derivatives for inductance on the exponential source as depicted in Fig. 5.
Fig. 5

Plot of numerical simulation for an RLC electrical circuit through three different fractional approaches corresponding to exponential source with different values of inductance L; for all figures I(t) is measured at Amperes and t is measured at seconds

  • Effects of fractional parameters via Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional operators

Periodic source

Fig. 6 is prepared to analyze the issue of singularity at the end point of the interval that is faced by Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional derivatives. Hence, the rate of change in current with respect to time can significantly be analyzed by their kernel of non-local and non-singular type. Here, the profile of current is depicted via three approaches separately in which it is noted that decreasing the values of Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional parameters \(0 \le \alpha \le 1,\,\,0 \le \beta \le 1,\,\,0 \le \gamma \le 1\) respectively, an attenuation of the amplitudes is achieved. Viewing this fact of fractional parameters say \(0 \le \alpha \le 1,\,\,0 \le \beta \le 1,\,\,0 \le \gamma \le 1\), effects of irreversible dissipative causes like ohmic friction or in other words the current changes due to the order derivative. In short, one can conclude that the system increases its “damping capacity” because fractional operators provide a non-local effect of dissipation of energy (internal friction).
Fig. 6

Plot of numerical simulation for an RLC electrical circuit through three different fractional approaches corresponding to periodic source with different values of fractional parameters \(\alpha ,\beta ,\gamma\) for all figures I(t) is measured at Amperes and t is measured at seconds

Exponential source

Figure 7 elaborates the behavior of the current for the exponential source with decreasing the values of Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional parameters \(0 \le \alpha \le 1,\,\,0 \le \beta \le 1,\,\,0 \le \gamma \le 1\) respectively. It is noted from fractional parameters \(\alpha \,,\beta ,\gamma\) that the non-conservative behavior appears in system when \(\alpha \, < 1,\beta < 1,\gamma < 1\) while the system displays Markovian nature when \(\alpha \, = \beta = \gamma = 1\). This may be due to the fact that the fractional derivatives are analyzed in this work involves the non-locality and non-singularity among the kernels. From the physical aspects, the investigated results of this work can also be analyzed to the experimental results as well.
Fig. 7

Plot of numerical simulation for an RLC electrical circuit through three different fractional approaches corresponding to exponential source with different values of fractional parameters \(\alpha ,\beta ,\gamma\) for all figures I(t) is measured at Amperes and t is measured at seconds

5 Concluding remarks

Analytical study of RLC electrical circuit is analysis through Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional derivatives in which the modeling of governing differential equation of RLC electrical circuits have been fractionalized based on the ranges of fractional operators 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. However, the major outcomes are:
  • The profile of current is tested by increasing the numerical values of resistance which results decreasing behavior of current and the current with respect to delay is noted. From physical point of view, this delay results in system equilibrium poor. Physically, enhancing the resistance causes energies wastes and power losses in the RLC electrical circuits.

  • The effects of the inductance suggested that the damping is produced that refers to dissipate the energy stored in the oscillations. The efficiency of the RLC electrical circuit can be minimized/decreased when the inductance increases. Hence, this fact suggests that the Atangana-Baleanu fractional operator is more effective than the Caputo, Caputo-Fabrizio fractional operators due to its significance of the non-locality and non-singularity involved in their kernels.

  • An attenuation of the amplitudes is achieved when Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional parameters \(0 \le \alpha \le 1,\,\,0 \le \beta \le 1,\,\,0 \le \gamma \le 1\) are decreased respectively while effects of irreversible dissipative causes like ohmic friction. The system increases its “damping capacity” because fractional operators provide a non-local effect of dissipation of energy (internal friction).

  • The non-conservative behavior appears in system when \(\alpha \, < 1,\beta < 1,\gamma < 1\) while the system displays Markovian nature when \(\alpha \, = \beta = \gamma = 1\).

Notes

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.

Compliance with ethical standards

Conflict of interest

The authors declare no conflict of interest.

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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Basic Sciences and Related StudiesMehran University of Engineering and TechnologyJamshoroPakistan
  2. 2.Department of Electrical EngineeringMehran University of Engineering and TechnologyJamshoroPakistan

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