Nonlocal fractal calculus based analyses of electrical circuits on fractal set

Abstract

Purpose The purpose of this paper is to present the analyses of electrical circuits with arbitrary source terms defined on middle b cantor set by means of nonlocal fractal calculus and to evaluate the appropriateness of such unconventional calculus. Design/methodology/approach The nonlocal fractal integro-differential equations describing RL, RC, LC and RLC circuits with arbitrary source terms defined on middle b cantor set have been formulated and solved by means of fractal Laplace transformation. Numerical simulations based on the derived solutions have been performed where an LC circuit has been studied by means of Lagrangian and Hamiltonian formalisms. The nonlocal fractal calculus-based Lagrangian and Hamiltonian equations have been derived and the local fractal calculus-based ones have been revisited. Findings The author has found that the LC circuit defined on a middle b cantor set become a physically unsound system due to the unreasonable associated Hamiltonian unless the local fractal calculus has been applied instead. Originality/value For the first time, the nonlocal fractal calculus-based analyses of electrical circuits with arbitrary source terms have been performed where those circuits with order higher than 1 have also been analyzed. For the first time, the nonlocal fractal calculus-based Lagrangian and Hamiltonian equations have been proposed. The revised contradiction free local fractal calculus-based Lagrangian and Hamiltonian equations have been presented. A comparison of local and nonlocal fractal calculus in terms of Lagrangian and Hamiltonian formalisms have been made where a drawback of the nonlocal one has been pointed out.

Noise analysis of electrical circuits on fractal set

“…Therefore, T must be added as follows: which yields [ q ( t )] = sec 1− α Asec α = Asec that is physically measurable as sec does. Thus, by applying the q ( t )− v ( t ) relationship of any capacitor of arbitrary circuit defined on κ b (Banchuin, 2021a), we have …”

“…which shows that [v(t)]=1sec1−αWbsecα=Wb/sec that is now physically measurable as sec does (Banchuin, 2020). By applying the φ ( t )− i ( t ) relationship of any inductor of arbitrary circuit defined on κ b (Banchuin, 2021a), we have …”

“…Besides the local fractional calculus, there exists other fractal set-oriented calculus, namely, the fractal calculus (Parvate and Gangal, 2009; Parvate and Gangal, 2011). As a result, the analysis of electrical circuits on Cantor set via such fractal calculus have been made (Ali et al , 2021; Khalili Golmankhaneh et al , 2021b; Banchuin, 2021a).…”

The Generic Nonlocal Fractal Calculus

“…By let F be a Cantor ternary set thus 𝛼 = 0.6309 and assuming that i(0) = 1 A, R = 1 Ω, L = 1 H, C = 1 F and 𝛽 = 0.6309, i(t) can be approximately simulated based on (27) up to n = k 99 via MATHEMATICA as depicted in Fig. 1 where a significantly different dynamic from that of its Golmankhaneh-Baleanu nonlocal fractal calculus-based counterpart obtained from the previous analysis of fractional RLC circuit on fractal set (see [33]) which has also been depicted in this figure, can be observed. Such disagreement is also caused by the different between the assumed 𝑘(𝑡; 𝛽, 𝑚) and 𝑘 (𝑡; 𝛽, 𝑚).…”

“…Since the nonlocality like that of the fractional calculus is necessary for modelling such memory effect, the nonlocal fractal derivatives have been introduced by Golmankhaneh and Baleanu [31], [32] based on the classical Riemann-Liouville and Caputo fractional derivatives which employ a power law-based kernel. These nonlocal fractal derivatives have been successfully applied to various applications e.g., the mathematical modelling of fractional Brownian motion with fractal support [21] and the analysis of fractional electrical circuits defined on fractal set [23], [33] etc. However, they are inconsistent with the local fractal derivative as will be shown later unlike the fractional derivative that is consistent with the conventional operator.…”