Analog Implementation of Fractional-Order Electric Elements Using Caputo–fabrizio and Atangana–baleanu Definitions

Abstract

This study employs the Caputo–Fabrizio and Atangana–Baleanu fractional derivatives to determine the impedance and admittance model of fractional capacitor and inductor. The analog implementation circuits are proposed aiming at fractional-order electric elements based on these two derivatives, which can be widely used in a variety of electrical systems using new fractional operators. Constant phase capacitor and inductor are approximated by the Oustaloup algorithm and the recursive net-grid-type analog circuit, respectively. Based on that, approximation circuits of fractional electric components under Caputo–Fabrizio and Atangana–Baleanu definitions are given. For the purpose of judging whether the implementation topology of fractional-order capacitor and inductor is accurate, taking fractional RC and RL circuit defined by Caputo, Caputo–Fabrizio and Atangana–Baleanu derivatives as examples, the comparison of numerical and circuit simulations is carried out. The correctness of the analog implementation circuits using the Caputo–Fabrizio and Atangana–Baleanu definitions is verified. Fractional-order RC charging circuit experiments based on Caputo, Caputo–Fabrizio and Atangana–Baleanu derivatives are taken as examples. Several experiments with different fractional-order and circuit parameters are carried out. The validity of the implementation methods is ulteriorly proved with experiment data.

Global Dynamics of Fractional-order Model for Malaria Disease Transmission

“…Fitting the model using real data and parameter estimation in the fractional order models is an integral part in the disease modeling. Therefore, in this study we use both the least squares and Nelder mead algorithm methods [7] to fit and estimate the parameters ( h , θ h , α h ) of the proposed model (2.1). The real data used in this study are weekly reported cases in Zimbabwe as presented in Table 3, and the commutative new infections predicted by the model (2.1) is obtained using the equation (3.13):…”

“…Nevertheless, the complexities in the parasite's life cycle, coupled with the highly complex social and environmental interactions, and movement of people between endemic and non-endemic areas continue to promote morbidity and mortality from malaria. Although combined global efforts are underway to develop a malaria vaccine [7], [8], [9], [10], there is currently no perfect vaccine against the disease. As such, concerted attempts are still being made to understand malaria disease dynamics and effective control measures.…”

The generalized nonlocal fractal calculus: an efficient tool for fractal circuit analysis

“…the nonsingular kernels (Atangana and Alkahtani, 2016, 2015; Das, 2020; Gómez-Aguilar et al , 2016, 2017; Gómez‐Aguilar, 2018; Liao et al , 2021, 2022a, 2022b; Sheikh et al , 2020). Therefore, the nonlocal fractal calculus is inapplicable to those circuits whose memory effects are described by nonsingular kernels (Atangana and Alkahtani, 2015; Gómez‐Aguilar et al , 2016, 2017; Gómez‐Aguilar, 2018; Liao et al , 2021, 2022b, 2022a; Sheikh et al , 2020). In addition, according to its basis fractional derivatives, the nonlocal fractal calculus-based dynamical equation of any fractal electrical circuit with memory effect can be given in either Riemann–Liouville sense or Caputo sense, where the latter has been chosen (Banchuin, 2022c; Ali et al , 2023; Banchuin, 2023) due to its simpler initial condition handling methodology.…”

A novel generalized fractional‐order memristor model with fully explicit memory description

“…Note also that such Atangana-Baleanu fractional derivative has been applied, cited, and studied in many research areas ranged from electrical engineering to epidemiology. 27,[31][32][33][34][35][36][37][38][39][40][41][42][43][44] In particular, we choose the Atangana-Baleanu fractional derivative in Liouville-Caputo sense rather than the derivative in Riemann-Liouville sense. This is because the former takes the initial condition into its Laplace transformation, 30,33 and we solve the fractional-order memristor's state equation of by means of the Laplace transformation-based methodology.…”

“…Between two well‐known nonsingular kernel fractional derivatives, i.e., the Caputo‐Fabrizio fractional derivative 29 and the Atangana‐Baleanu fractional derivative 30 ; we choose the latter as it is a generalization of the former and also nonlocal. Note also that such Atangana‐Baleanu fractional derivative has been applied, cited, and studied in many research areas ranged from electrical engineering to epidemiology 27,31–44 . In particular, we choose the Atangana‐Baleanu fractional derivative in Liouville‐Caputo sense rather than the derivative in Riemann‐Liouville sense.…”