Classical electrical conduction theories, including Ohm's law and the Drude model, assume charge transport in smooth Euclidean time. However, many disordered and glassy materials exhibit anomalous conduction and relaxation that classical models cannot fully explain. In this work, we present a generalized electrical conduction framework based on fractal calculus, where time evolves on a fractal set with dimension 0 < alpha <= 1. Using local fractal derivatives and the associated staircase function, we derive a fractal equation of motion for charge carriers and obtain explicit formulas for drift velocity, current density, and conductivity. The resulting conductivity follows a power-law dependence on the relaxation time, reflecting temporal fractality. The classical Drude result is recovered in the Euclidean limit alpha = 1. The model also produces stretched-exponential relaxation, giving a unified description of steady and transient transport in complex media.

In this paper, we investigate fractal capacitors, resistors, and inductors by incorporating fractal time into their governing equations and analyzing its effects on the charge and current of each component. The corresponding solutions are derived, and illustrative plots are provided to demonstrate the influence of fractal time on the dynamic behavior of these electrical elements.

In this work, we propose a novel passive circuit theory on fractional High-Pass (HP) Negative Group Delay (NGD) topology by using the fractional order reactive impedance and admittance. The fractional order reactive networks are derived from the generalized exponential kernel fractional derivative function. We analytically show that the proposed fractional order topology is capable to satisfy the existence conditions of the HP-NGD function. The design equations of HP-NGD function were derived with respect to the targeted specifications. The established synthesis methods are compared with those of the state-of-the-art on the HP-NGD circuit specifications. The proof of concept (POC) of generalized exponential Kernel fractional HP-NGD circuit is identified from Oustaloup approximation-based impedance and admittance. Study cases of POCs confirm the feasibility of the established HP-NGD circuit theory by means of simulations with a commercial tool. The calculated model and simulation outcomes confirm the HP-NGD behavior and specifications as expected from the proposed theoretical design. The obtained results present NGD minimal values and HP-NGD cut-off frequency of about (-32.9 ns, 45.3 kHz) and (-650 ns, 2.2 kHz) with low-attenuation less than 1 dB from the POC circuits of the fractional-capacitor and fractional- inductor based NGD circuits, respectively. To the best of our knowledge, this is the first time that the fractional order capacitor and inductor whose impedance and admittance were derived based on the generalized exponential kernel fractional derivative is applied to the NGD circuit.

In this paper, we apply the Homotopy Analysis Method (HAM) to fractal ordinary and partial differential equations defined on fractal curves. After briefly reviewing the essential tools of fractal calculus, we adapt the HAM framework to the fractal setting and show how the convergence–control parameter improves the accuracy of the homotopy series. Several illustrative examples of nonlinear fractal ODEs and PDEs are solved, demonstrating the efficiency and flexibility of the method for analytic approximation on fractal geometries.

PurposeThe implementation and test of electronic circuit able to operate with significant negative delay remain an open challenge for design engineers. This study aims to design an innovative multistage low-pass (LP) type negative group delay (NGD) active circuit. The considered LP-NGD topology is constituted by RC-network. After the proof-of-concept (POC) design and simulation, a printed circuit board (PCB) of LP-NGD active topology prototype was fabricated. The LP-NGD PCB was tested to confirm the possibility to propagate arbitrary waveform signal. The time-advance measurement result enables to verify the simulation one.Design/methodology/approachAfter the NGD value, bandwidth choice and number of cells, this innovative design method of multi-stage LP-NGD circuit is established under the following phases: the first phase is the calculation of resistors and capacitors constituting the considered topology. The second phase is the schematic simulation of the LP-NGD POC in the frequency domain. In the third phase, the obtained LP-NGD POC must be optimized. In the fourth phase, the available R, C and operational amplifier are chosen the PCB prototype. In the last phase, the test signal must be chosen to demonstrate the output in time-advance.FindingsAn innovative design theory, including the analytical and algorithm, to determine the multistage RC-circuit parameters is established in function of targeted LP-NGD specifications. The validity of the multistage RC-circuit design theory is verified by the PCB prototype experimentation showing measurement of -0.3 s signal advance.Originality/valueThis research work originality is the analytical and routine algorithm design methodology of multistage LP-NGD electronic circuit. The transient result obtained by considering arbitrary waveform signal tested with a PCB prototype confirm the relevance of the developed multistage LP-NGD RC-circuit design.

Fractal calculus is a new branch of calculus that has been widely applied in many scientific disciplines, e.g., sub-diffusion, super-diffusion, spatial analysis, and electrical engineering. In the area of electrical engineering, the fractal calculus has been applied to the analysis of many electrical circuits and the modeling of memelement and inverse memelement under the effect of fractal time. However, to the best of our knowledge, there exists no application of the fractal calculus to the active electrical circuit. Therefore, for the first time, we apply the fractal calculus to the analysis of an active circuit under the effect of fractal time in this work. The operational amplifier (OPAMP)-based bandpass (BP) filter has been chosen as our candidate active circuit. For a complete analysis, the nonidealities of the OPAMP have been taken into account. It has been found that the filter exhibits a power law dynamic in the frequency domain due to the effect of fractal time without any usage of the fractional order circuit element. The influences of the OPAMP’s nonidealities on both magnitude and phase of the nondifferentiable (ND) transfer function are comprehensively analyzed.

This paper presents a novel proportional-integral-derivative (PID) control framework for first alpha-order systems evolving in fractal time. The main contribution is the extension of classical control theory to systems exhibiting anomalous temporal scaling by employing local fractal derivatives. In contrast to fractional-order PID (FOPID) approaches, which primarily model memory effects, the proposed fractal PID framework captures time-scaling behavior arising in non-smooth environments, such as viscoelastic friction and irregular contact surfaces. The closed-loop dynamics are formulated as a second alpha-order fractal differential equation, from which a characteristic equation is derived to establish conditions for asymptotic stability. It is shown that, for a constant reference input and positive controller gains, the tracking error converges to zero as t ->infinity. In addition, a quantitative performance analysis demonstrates that the fractal-order alpha governs temporal stretching: smaller values of alpha lead to increased rise and settling times and reduced oscillation frequency. The effectiveness of the proposed approach is illustrated through applications to a thermal system with fractal heat input and robotic actuators operating in irregular environments. These results highlight the potential of fractal-time control as a systematic framework for modeling and controlling dynamical systems with non-integer temporal structure.

This paper studies second α -order dynamical systems within the framework of fractal calculus and examines their behavior under fractal proportional–derivative (FPD) and fractal proportional–integral–derivative (FPID) controllers. A brief overview of fractal calculus is provided, and fractal initial value and final value theorems are formulated. The structure of FPID control for second α -order systems is presented, and the stability conditions are analyzed within the fractal setting. The response of FPD controllers is also investigated through analytical expressions and numerical simulations for selected values of the fractal order α . As an illustrative example, a series RLC circuit is modeled using fractal derivatives to demonstrate how fractal-order dynamics influence the controlled system response. The results emphasize the role of the fractal order α in shaping transient behavior and steady-state characteristics.

This paper presents a novel low-pass filtering framework based on Fractal First α -order and Second α -order designs, formulated within the framework of fractal calculus. By incorporating the structure of fractal time, the proposed filters can effectively process signals with intricate, non-differentiable characteristics. The fractal second α -order low-pass filter is applied to a simulated noisy ECG signal, demonstrating significant noise suppression while preserving the essential morphological features of the waveform. A comparative study with the classical Bessel low-pass filter further illustrates the advantages of the fractal approach in capturing scale-invariant and self-similar properties of biomedical signals. These results highlight the potential of fractal-order filters for advanced biomedical signal processing.

In this work, the capability to generate the negative group delay (NGD) phenomenon at those frequencies higher than a certain value, that is, the NGD high-pass (HP) filtering function, of the bilinear double-order transfer function has been demonstrated. Based on the Type-I bilinear double-order filter circuit, our theory has been verified by strong agreements between the formulae and their proof-of-concept (POC) circuit-based simulation results. Both formula-based and POC circuit-based simulations give the minimum group delay of -5 ms yet the cut-off frequency of 10 and 9 rad/s, respectively. Such slight deviation is caused by the approximation error of the bilinear double-order impedance. By employing two orders, the bilinear double-order transfer function has been found to be the basis transfer function for the NGD filtering function with the highest degree of freedom. Based on our design equation for the NGD filtering function, it can be seen that the distances between zero and pole and the characteristic frequency of the bilinear double-order transfer function are governed by such characteristic frequency itself, the cut-off frequency of the NGD filtering function, and the fractional order of the Laplacian operator. In addition, the effects of the fractional order of the Laplacian operator, the fractional order of the transfer function, and the ratio of the abovementioned distances to the characteristics of the newly found double-order NGD filtering function have been studied in detail. We propose a double order negative group delay filtering (NGD) function by using the bilinear double-order transfer function as the basis. Compared with the previous ones, our newly presented NGD function gives the highest degree of freedom. image

In this work, a memorized dielectric relaxation analogy-based model of inverse memristor has been proposed. In contrast to the previous models, a completely explicit memory description of the inverse memristor can be achieved by ours. The proposed model has been found to be well applicable to the practical inverse memristors of both current and voltage-controlled types where the basic properties of inverse memristor have been preserved and the strong agreements between the model-based results and their circuit-based benchmarks can be observed. Based on our model, a rigorous analysis of inverse memristor has been performed to give a mathematical discussion on the simulation results. The power and energy consumption aspects have also been analyzed. In addition, the effect of parasitic element has been explored and it has been found that the inverse memristor realized based on our model is more robust to such undesired effect than those realized based on the others. This is another virtue of the proposed model. The realization methodology of the inverse memristor circuit based on our model with off-the-shelf components and the extension of the model toward the inverse memcapacitor and inverse meminductor have also been given.

Summary In this work, the analysis of negative group delay circuits in fractional domain has been conducted. The low pass and the high pass negative group delay circuits constructed based on passive network have been chosen as our candidate circuits. For performing the analysis in fractional domain, the novel Caputo–Fabrizio derivative‐based fractional impedance have been introduced to both circuits. The crucial parameters, the necessary conditions, and the existence conditions of both candidate negative group delay circuits have been formulated. The simulations have been conducted based on the formulated results where the comparisons with the conventional prototypes have been made. The verifications by the proof‐of‐concept circuits have also been performed. In addition, the effects of variation in the order of fractional impedances have been investigated. In summary, it has been found that considering these negative group delay circuits in the fractional domain, which effectively taking unavoidable nonidealities that cannot be modeled in conventional domain into account, significantly alters their characteristics especially the low pass circuit. However they can retain their low pass and high pass negative group delay functions.

SummaryIn this work, an improved analytical model of fractal memelement in which the pinched point shifting has been considered and the original analytical model of fractal inverse memelement have been proposed. These fractal circuit elements are the memelement, and inverse memelement operates based on the principle of electromagnetic in fractal time/space, which must be applied whenever the current flows through fractal media. These models are important because these memory elements can be realized based on the porous material, which is a fractal media. In addition, they can employ self‐similarity, which is hard to be simulated by using the traditional models. This is because such self‐similarity can be well explained by the fractal set‐based model, yet those traditional models are based on the set of real values. Therefore, for deriving the proposed models, the fractal calculus, which is oriented to the fractal set, has been adopted as the mathematical basis. From the analytical and numerical analyses based on the derived models, it has been found that both memelement and inverse memelement can retain their unique frequency characteristics despite being operated based on the abovementioned principle. In addition, their input–output relationships are mathematically differentiable albeit the inputs and outputs themselves are not.

Purpose The purpose of this paper is to test the capability to properly analyze the electrical circuits of a novel constitutive relation of capacitor. Design/methodology/approach For ceteris paribus, the constitutive relations of the resistor and inductor have been reformulated by following the novel constitutive relation of capacitor. The responses of RL, RC, LC and RLC circuits defined on the fractal set described by these definitions have been derived by means of the fractal calculus and fractal Laplace transformation. A comparative Hamiltonian formalism-based analysis has been performed where the circuits described by the conventional and the formerly proposed revisited constitutive relations have also been considered. Findings This study has found that the novel constitutive relations give unreasonable results unlike the conventional ones. Like such previous revisited constitutive relations, an odd Hamiltonian has been obtained. On the other hand, the conventional constitutive relations give a reasonable Hamiltonian. Originality/value To the best of the author’s knowledge, for the first time, the analysis of fractal set defined electrical circuits by means of unconventional constitutive relations has been performed where the deficiency of the tested capacitive constitutive relation has been pointed out.

Purpose The purpose of this paper is to originally present the generic analytical models of memelement and inverse memelement with time-dependent memory effect. Design/methodology/approach The variable order forward Grünwald–Letnikov fractional derivative and the memristor and inverse memristor models proposed by Fouda et al. have been adopted as the basis. Both analytical and numerical studies have been conducted. The applications to the candidate practical memristor and inverse memelements have also been presented. Findings The generic analytical models of memelement and inverse memelement with time-dependent memory effect, the simplified ones for DC and AC signal-based analyses and the equations of crucial parameters have been derived. Besides the well-known opposite relationships with frequency, the Lissajous patterns of memelement and inverse memelement also use the opposite relationships with the time. The proposed models can be well applied to the practical elements. Originality/value To the best of the authors’ knowledge, for the first time, the models’ memelement and inverse memelement with time-dependent memory effect have been presented. A new contrast between these elements has been discovered. The resulting models are applicable to the practical elements.

In this work, a novel generalized mathematical model of fractional-order memristor with fully explicit memory description has been proposed. For obtaining such full explicit memory description, the Atangana-Baleanu fractional derivative in Liouville-Caputo sense, which employs a nonsingular kernel, has been adopted as the mathematical basis. The proposed model has been derived without regarding to any specific conventional memristor. A comparison with the singular kernel fractional derivative-based model has been made. The behavioral analysis of the fractional-order memristor based on the proposed model has been performed, where both DC and AC stimuli have been considered. In addition, its application to the practical fractional-order memristor-based circuit and its extension to the fractional-order memreactance have also been shown. Unlike the singular kernel fractional derivative-based model, a fully explicit memory description can be obtained by ours. Many other interesting results that are contradict to the previous singular kernel fractional derivative-based ones, e.g., the fractional-order memristor that can be locally active, have been demonstrated. The abovementioned extension can be conveniently performed. In summary, this is the first time that a nonsingular kernel fractional derivative has been applied to the fractional-order memristor modeling and the resulting model with a fully explicit memory description has been proposed. The proposed model is also highly generic, applicable to the practical circuit, and extendable to the fractional-order memreactance.

Purpose The purpose of this paper is to propose a novel nonlocal fractal calculus scheme dedicated to the analysis of fractal electrical circuit, namely, the generalized nonlocal fractal calculus. Design/methodology/approach For being generalized, an arbitrary kernel function has been adopted. The condition on order has been derived so that it is not related to the γ-dimension of the fractal set. The fractal Laplace transforms of our operators have been derived. Findings Unlike the traditional power law kernel-based nonlocal fractal calculus operators, ours are generalized, consistent with the local fractal derivative and use higher degree of freedom. As intended, the proposed nonlocal fractal calculus is applicable to any kind of fractal electrical circuit. Thus, it has been found to be a more efficient tool for the fractal electrical circuit analysis than any previous fractal set dedicated calculus scheme. Originality/value A fractal calculus scheme that is more efficient for the fractal electrical circuit analysis than any previous ones has been proposed in this work.

In this work, the noise performances of the fractal-fractional electrical circuits have been addressed. The nonlocal fractal calculus has been adopted as our mathematical basis. The fractal time component has also been included for the physical measurability of electrical quantities. The derivations of crucial stochastic parameters of circuit responses, which determine their noise performances, have been performed. Numerical simulations have also been conducted where the influences of Hausdorff dimension of the fractal set, orders of fractal-fractional reactive components, and other parameters on the noise performances have been studied. Regardless to any specific circuit, we have found that the noise performances can be improved by increasing the orders of fractal-fractional reactive components. The optimum Hausdorff dimensions, which the best noise performances can be achieved given the orders of fractal-fractional reactive components, have also been calculated. The results proposed in this work serve as the foundation for understanding noise in fractal-fractional electrical circuits and can be extensively applied to large-scaled circuits, for example, the infinite circuit networks and so forth.

Purpose The purpose of this paper is to compare the suitability of fractional derivatives in the modelling of practical capacitors. Such suitability refers to ability to provide the analytical capacitance function that matches the experimental ones of each fractional derivative. Design/methodology/approach The analytical capacitance functions based on various fractional derivatives of both local and nonlocal types including the author's have been derived. The derived capacitance functions have been simulated and compared with the experimental ones of aluminium electrolytic and electrical double layer capacitors (EDLCs). Findings This paper has found that any local fractional derivative with fractional power law-based relationship with the conventional one is suitable for modelling the aluminium electrolytic capacitor (AEC) by incorporating with the conventional capacitance definition. On the other hand, the author's nonlocal fractional derivatives have been found to be more suitable than the others for modelling the EDLC by incorporating with the revisited definition of capacitance. Originality/value The proposed comparative analysis has been originally presented in this work. The criterion for local fractional derivative, to be suitable for modelling the AEC, has been found. The nonlocal fractional operators which are most suitable for modelling the EDLC have been derived where the unsuitable one has been pointed out.

Purpose The purpose of this paper is to comparatively analyze the electrical circuits defined with the conventional and revisited time domain circuit element definitions in the context of fractional conformable calculus and to promote the combined usage of conventional definitions, fractional conformable derivative and conformable Laplace transform. Design/methodology/approach The RL, RC, LC and RLC circuits described by both conventional and revisited time domain circuit element definitions has been analyzed by means of the fractional conformable derivative based differential equations and conformable Laplace transform. The comparison among the obtained results and those based on the methodologies adopted in the previous works has been made. Findings The author has found that the conventional definitions-based solution gives a physically reasonable result unlike its revisited definitions-based counterpart and the solutions based on those previous methodologies. A strong agreement to the time domain state space concept-based solution can be observed. The author has also shown that the scalar valued solution can be directly obtained by singularity free conformable Laplace transform-based methodology unlike such state space concept based one. Originality/value For the first time, the revisited time domain definitions of resistance and inductance have been proposed and applied together with the revisited definition of capacitance in electrical circuit analyses. The advantage of the combined usage of conventional time definitions, fractional conformable derivative and conformable Laplace transform has been suggested and the impropriety of applying the revisited definitions in circuit analysis has been pointed out.