Fractal and Fractional

Efficient coordination of directional overcurrent relays (DOCRs) is vital for maintaining the stability and reliability of electrical power systems (EPSs). The task of optimizing DOCR coordination in complex power networks is modeled as an optimization problem. This study aims to enhance the performance of protection systems by minimizing the cumulative operating time of DOCRs. This is achieved by effectively synchronizing primary and backup relays while ensuring that coordination time intervals (CTIs) remain within predefined limits (0.2 to 0.5 s). A novel optimization strategy, the fractional-order derivative war optimizer (FODWO), is proposed to address this challenge. This innovative approach integrates the principles of fractional calculus (FC) into the conventional war optimization (WO) algorithm, significantly improving its optimization properties. The incorporation of fractional-order derivatives (FODs) enhances the algorithm’s ability to navigate complex optimization landscapes, avoiding local minima and achieving globally optimal solutions more efficiently. This leads to the reduced cumulative operating time of DOCRs and improved reliability of the protection system. The FODWO method was rigorously tested on standard EPSs, including IEEE three, eight, and fifteen bus systems, as well as on eleven benchmark optimization functions, encompassing unimodal and multimodal problems. The comparative analysis demonstrates that incorporating fractional-order derivatives (FODs) into the WO enhances its efficiency, enabling it to achieve globally optimal solutions and reduce the cumulative operating time of DOCRs by 3%, 6%, and 3% in the case of a three, eight, and fifteen bus system, respectively, compared to the traditional WO algorithm. To validate the effectiveness of FODWO, comprehensive statistical analyses were conducted, including box plots, quantile–quantile (QQ) plots, the empirical cumulative distribution function (ECDF), and minimal fitness evolution across simulations. These analyses confirm the robustness, reliability, and consistency of the FODWO approach. Comparative evaluations reveal that FODWO outperforms other state-of-the-art nature-inspired algorithms and traditional optimization methods, making it a highly effective tool for DOCR coordination in EPSs. Full article

As a non-associative connective in fuzzy logic, the analysis and research of overlap functions have been extended to many generalized cases, such as interval-valued and intuitionistic fuzzy overlap functions (IFOFs). However, overlap functions face challenges in the Pythagorean fuzzy (PF) environment. This paper first extends overlap functions to the PF domain by proposing PF overlap functions (PFOFs), discussing their representable forms, and providing a general construction method. It then introduces a new PF similarity measure which addresses issues in existing measures (e.g., the inability to measure the similarity of certain PF numbers) and demonstrates its effectiveness through comparisons with other methods, using several examples in fractional form. Based on the proposed PFOFs and their induced residual implication, new generalized PF rough sets (PFRSs) are constructed, which extend the PFRS models. The relevant properties of their approximation operators are explored, and they are generalized to the dual-domain case. Due to the introduction of hesitation in IF and PF sets, the approximate accuracy of classical rough sets is no longer applicable. Therefore, a new PFRS approximate accuracy is developed which generalizes the approximate accuracy of classical rough sets and remains applicable to the classical case. Finally, three multi-criteria decision-making (MCDM) algorithms based on PF information are proposed, and their effectiveness and rationality are validated through examples, making them more flexible for solving MCDM problems in the PF environment. Full article

This paper is dedicated to investigating a highly accurate numerical solution for a class of 2D nonlinear time-dependent partial integro-differential equations with multi-term fractional integral items. These integrals are weakly singular with respect to time, which are handled using the product integration rule on graded meshes to compensate for the influence generated by the initial weak singular nature of the exact solution. The temporal derivative is approximated by a generalized Crank–Nicolson difference scheme, while the nonlinear term is approximated by a linearized method. Furthermore, the stability and convergence of the derived time semi-discretization scheme are strictly proved by revising the finite discrete parameters. Meanwhile, the differential matrices of the spatial high-order derivatives based on barycentric rational interpolation are utilized to obtain the fully discrete scheme. Finally, the effectiveness and reliability of the proposed method are validated by means of several numerical experiments. Full article

This paper presents a generalized fractional Stockwell transform (GFST), extending the classical Stockwell transform and fractional Stockwell transform, which are widely used tools in time–frequency analysis. The GFST on $L^2(\mathbb{R},\mathbb{C})$ is defined as a convolution consistent with the classical Stockwell transform, and the fundamental properties of GFST such as linearity, translation, scaling, etc., are discussed. We focus on establishing an orthogonality relation and derive an inversion formula as a direct application of this relation. Additionally, we characterize the range of the GFST on $L^2(\mathbb{R},\mathbb{C})$. Finally, we prove an uncertainty principle of the Heisenberg type for the proposed GFST. Full article

Granular samples are often used to characterize the pore structure of shale. To systematically analyze the influence of particle size on pore characteristics, case studies were performed on two groups of organic-rich deep shale samples. Multiple methods, including small-angle neutron scattering (SANS), low-pressure nitrogen gas adsorption (LP-N₂GA), low-pressure carbon dioxide gas adsorption (LP-CO₂GA), and XRD analysis, were adopted to investigate how the crushing process would affect pore structure parameters and the fractal features of deep shale samples. The research indicates that with the decrease in particle size, the measurements from nitrogen adsorption and SANS experiments significantly increase, with relative effects reaching 95.09% and 51.27%, respectively. However, the impact on carbon dioxide adsorption measurements is minor, with a maximum of only 8.97%. This suggests that the comminution process primarily alters the macropore structure, with limited influence on the micropores. Since micropores contribute the majority of the specific surface area in deep shale, the effect of particle size variation on the specific surface area is negligible, averaging only 16.52%. Shales exhibit dual-fractal characteristics. The distribution range of the mass fractal dimension of the experimental samples is 2.658–2.961, which increases as the particle size decreases. The distribution range of the surface fractal dimension is 2.777–2.834, which decreases with the decrease in particle size. Full article

In this manuscript, we introduce the concept of fuzzy S-metric spaces and study some of their characteristics. We prove a fixed-point theorem for a self-mapping on a complete fuzzy S-metric space. To illustrate the versatility of our new ideas and related fixed-point theorems, we give examples to illustrate their use in a variety of domains, including fractal formation. These examples illustrate how the fuzzy S-contraction can be applied to iterated function systems, enabling the exploration of fractal forms under diverse contractive conditions. In addition, we solve the satellite web coupling problem by employing this coherent framework. Full article

In the current context of the global energy landscape, China is facing a growing challenge in oil and gas exploration and development. It is difficult to evaluate the log data because of the lithological composition of igneous rocks, which displays an unparalleled degree of complexity and unpredictability. Against this backdrop, this study deploys advanced multifractal detrended fluctuation analysis (MF-DFA) to comprehensively analyze key parameters within igneous rock logging data, including natural gamma-ray logging, resistivity logging, compensated neutron logging, and acoustic logging. The results unequivocally demonstrate that these logging data possess distinct multifractal characteristics. This multifractality serves as a powerful tool to elucidate the inherent complexity, heterogeneity, and structural and property variations in igneous rocks caused by diverse geological processes and environmental changes during their formation and evolution, which is crucial for understanding the subsurface reservoir behavior. Subsequently, through a series of rearrangement sequences and the replacement sequence on the original logging data, we identify that the probability density function and long-range correlation are the fundamental sources of the observed multifractality. These findings contribute to a deeper theoretical understanding of the data-generating mechanisms within igneous rock formations. Finally, multifractal detrended cross-correlation analysis (MF-DCCA) is employed to explore the cross-correlations among different types of igneous rock logging data. We uncover correlations among different igneous rocks’ logging data. These parameters exhibit different properties. There are negative long-range correlations between natural gamma-ray logging and resistivity logging, natural gamma-ray logging and compensated neutron logging in basalt, and resistivity logging and compensated neutron logging in diabase. The logging data on other igneous rocks have long-range correlations. These correlation results are of great significance as they provide solid data support for the formulation of oil and gas exploration and development plans. Full article

This study investigated market efficiency across 20 major commodity assets, including crude oil, utilizing fractal analysis. Additionally, a rolling window approach was employed to capture the time-varying nature of efficiency in these markets. A Granger causality test was applied to assess the influence of crude oil on other commodities. Key findings revealed significant inefficiencies in RBOB(Reformulated Blendstock for Oxygenated Blending) Gasoline, Palladium, and Brent Crude Oil, largely driven by geopolitical risks that exacerbated supply–demand imbalances. By contrast, Copper, Kansas Wheat, and Soybeans exhibited greater efficiency because of their stable market dynamics. The COVID-19 pandemic underscored the time-varying nature of efficiency, with short-term volatility causing price fluctuations. Geopolitical events such as the Russia–Ukraine War exposed some commodities to shocks, while others remained resilient. Brent Crude Oil was a key driver of market inefficiency. Our findings align with Fractal Fractional (FF) concepts. The MF-DFA method revealed self-similarity in market prices, while inefficient markets exhibited long-memory effects, challenging the Efficient Market Hypothesis. Additionally, rolling window analysis captured evolving market efficiency, influenced by external shocks, reinforcing the relevance of fractal fractional models in financial analysis. Furthermore, these findings can help traders, policymakers, and researchers, by highlighting Brent Crude Oil as a key market indicator and emphasizing the need for risk management and regulatory measures. Full article

This paper introduces a novel model-free fractional-order composite control methodology specifically designed for precision positioning in permanent magnet synchronous motor (PMSM) drives. The proposed framework ingeniously combines a composite control architecture, featuring a super twisting double fractional-order differential sliding mode controller (STDFDSMC) synergistically integrated with a complementary extended state observer (CESO). The STDFDSMC incorporates an innovative fractional-order double differential sliding mode surface, engineered to deliver superior robustness, enhanced flexibility, and accelerated convergence rates, while simultaneously addressing potential singularity issues. The CESO is implemented to achieve precise estimation and compensation of both intrinsic and extrinsic disturbances affecting PMSM drive systems. Through rigorous application of Lyapunov stability theory, we provide a comprehensive theoretical validation of the closed-loop system’s convergence stability under the proposed control paradigm. Extensive comparative analyses with conventional control methodologies are conducted to substantiate the efficacy of our approach. The comparative results conclusively demonstrate that the proposed control method represents a significant advancement in PMSM drive performance optimization, offering substantial improvements over existing control strategies. Full article

This work presents a fractional order Proportional Integral and Derivative controller with adaptation characteristics in the control parameters depending on the required output, gain scheduling fractional order PID (GS-FO-PID). The fractional order PID is applied to the voltage control of a DC–DC buck quadratic converter (QBC). The DC–DC buck quadratic converter is designed to operate at 12 V, although in the simulation tests, the output voltage ranges from 5 to 36 V. The performance of the GS-FO-PID is compared with the one from a classic PID. The GS-FO-PID presents better performance when the reference voltage is changed. In the same way, the behavior of the converter with the reference fixed to 12 V output is analyzed with load changes; for this case, the amplitude value of the ripple when the converter is driven by the GS-FO-PID almost has no variation. Full article

This paper investigates the fracture surfaces and fracture performance of recycled aggregate concrete (RAC) using fringe projection technology. This non-contact, point-by-point, and full-field scanning technique allows precise measurement of RAC’s fracture surface characteristics. This research focuses on the effects of recycled aggregate replacement rate, water-to-binder (w/b) ratio, and maximum aggregate size on RAC’s fracture properties. A decrease in the w/b ratio significantly reduces surface roughness (Rs) and fractal dimension (D), due to increased cement mortar bond strength at lower w/b ratios, causing cracks to propagate through aggregates and resulting in smoother fracture surfaces. At higher w/b ratios (0.8 and 0.6), both surface roughness and fractal dimension decrease as the recycled aggregate replacement rate increases. At a w/b ratio of 0.4, these parameters are not significantly affected by the replacement rate, indicating stronger cement mortar. Larger aggregates result in slightly higher surface roughness compared to smaller aggregates, due to more pronounced interface changes. True fracture energy is consistently lower than nominal fracture energy, with the difference increasing with higher recycled aggregate replacement rates and larger aggregate sizes. It increases as the w/b ratio decreases. These findings provide a scientific basis for optimizing RAC mix design, enhancing its fracture performance and supporting its practical engineering applications. Full article

The purpose of this paper is to propose a fractal–fractional-order for computer virus propagation dynamics, in accordance with the Atangana–Baleanu operator. We examine the existence of solutions, as well as the Hyers–Ulam stability, uniqueness, non-negativity, positivity, and boundedness based on the fractal–fractional sense. Hyers–Ulam stability is significant because it ensures that small deviations in the initial conditions of the system do not lead to large deviations in the solution. This implies that the proposed model is robust and reliable for predicting the behavior of virus propagation. By establishing this type of stability, we can confidently apply the model to real-world scenarios where exact initial conditions are often difficult to determine. Based on the equivalent integral of the model, a qualitative analysis is conducted by means of an iterative convergence sequence using fixed-point analysis. We then apply a numerical scheme to a case study that will allow the fractal–fractional model to be numerically described. Both analytical and simulation results appear to be in agreement. The numerical scheme not only validates the theoretical findings, but also provides a practical framework for predicting virus spread in digital networks. This approach enables researchers to assess the impact of different parameters on virus dynamics, offering insights into effective control strategies. Consequently, the model can be adapted to real-world scenarios, helping improve cybersecurity measures and mitigate the risks associated with computer virus outbreaks. Full article

Multi-focus image fusion is an important method for obtaining fully focused information. In this paper, a novel multi-focus image fusion method based on fractal dimension (FD) and parameter adaptive unit-linking dual-channel pulse-coupled neural network (PAUDPCNN) in the curvelet transform (CVT) domain is proposed. The source images are decomposed into low-frequency and high-frequency sub-bands by CVT, respectively. The FD and PAUDPCNN models, along with consistency verification, are employed to fuse the high-frequency sub-bands, the average method is used to fuse the low-frequency sub-band, and the final fused image is generated by inverse CVT. The experimental results demonstrate that the proposed method shows superior performance in multi-focus image fusion on Lytro, MFFW, and MFI-WHU datasets. Full article

As the frequency of disasters increases worldwide, it has become increasingly important to raise awareness of the risks and mitigate their effects through effective disaster management. Anticipating disaster risks and ensuring timely evacuations are crucial. This paper proposes SafeWitness, which dynamically captures the evolving characteristics of disasters by integrating crowdsensing and GIS-based geofencing. It not only enables real-time disaster awareness and evacuation support but also provides spatial context awareness by mapping the disaster area based on GIS road information and temporal context awareness by using crowdsensing to track the progress of the disaster. This approach increases the effectiveness of disaster management by providing explicit, data-driven insights for timely decision making and risk mitigation. The experimental results reveal that the proposed method improved the F1-scores in the hazard and warning zones compared to the domain-based approach. The result increased by 12% in the hazard zone and by 55% in the warning zone compared to the traditional technique. Through user sampling, we enhanced the SafeWitness F1-score in the hazard zone by 6 times and in the warning zone by 2.8 times compared to the method without user sampling. In conclusion, SafeWitness offers a more precise perception of disaster areas than traditional domain-based area definitions, and the experimental results demonstrate the effectiveness of user sampling. Decision-makers and disaster management professionals can use the proposed method in urban disaster scenarios. Full article

In this article, we proposed a compact difference-Galerkin spectral method for the fourth-order equation in multi-dimensional space with the time-fractional derivative order $\alpha \in (1,2)$. The novel compact difference-Galerkin spectral method can effectively address the issue of high-order derivative accuracy and handle complex boundary problems. Simultaneously, the main conclusions of this article, including the stability, convergence, and solvability of the method, are derived. Finally, some computational experiments are illustrated to demonstrate the superiority of the compact difference-Galerkin spectral method. Full article

This manuscript studies the M-fractional Landau–Ginzburg–Higgs (M-fLGH) equation in comprehending superconductivity and drift cyclotron waves in radially inhomogeneous plasmas, especially for coherent ion cyclotron wave propagation, aiming to explore the soliton solutions, the parameter’s effect, and modulation instability. Here, we propose a novel approach, namely a newly improved Kudryashov’s method that integrates the combination of the unified method with the generalized Kudryashov’s method. By employing the modified F-expansion and the newly improved Kudryashov’s method, we investigate the soliton wave solutions for the M-fLGH model. The solutions are in trigonometric, rational, exponential, and hyperbolic forms. We present the effect of system parameters and fractional parameters. For special values of free parameters, we derive some novel phenomena such as kink wave, anti-kink wave, periodic lump wave with soliton, interaction of kink and periodic lump wave, interaction of anti-kink and periodic wave, periodic wave, solitonic wave, multi-lump wave in periodic form, and so on. The modulation instability criterion assesses the conditions that dictate the stability or instability of soliton solutions, highlighting the interplay between fractional order and system parameters. This study advances the theoretical understanding of fractional LGH models and provides valuable insights into practical applications in plasma physics, optical communication, and fluid dynamics. Full article

The accurate and efficient simulation of seismic wave energy dissipation and phase dispersion during propagation in subsurface media due to inelastic attenuation is critical for the hydrocarbon-bearing distinction and improving the quality of seismic imaging in strongly attenuating geological media. The fractional viscoelastic equation, which quantifies frequency-independent anelastic effects, has recently become a focal point in seismic exploration. We have developed a novel hybrid staggered-grid Grünwald–Letnikov (HSGGL) finite difference method for solving the fractional viscoelastic equation in the time domain. The proposed method achieves accurate and computationally efficient solutions by using a staggered grid to discretize the first-order partial derivatives of the velocity–stress equations, combined with Grünwald–Letnikov finite difference discretization for the fractional-order terms. To improve the computational efficiency, we employ a preset accuracy to truncate the difference stencil, resulting in a compact fractional-order difference scheme. A stability analysis using the eigenvalue method reveals that the proposed method confers a relaxed stability condition, providing greater flexibility in the selection of sampling intervals. The numerical experiments indicate that the HSGGL method achieves a maximum relative error of no more than 0.17% compared to the reference solution (on a finely meshed domain) while being significantly faster than the conventional global FD method (GFD). In a 500 × 500 computational domain, the computation times for the proposed methods, which meet the specified accuracy levels used, are only approximately 4.67%, 4.47%, 4.44%, and 4.42% of that of the GFD method. This indicates that the novel HSGGL method has the potential as an effective forward modeling tool for understanding complex subsurface structures by employing a fractional viscoelastic equation. Full article

This paper studies three time-fractional models that arise in plasma physics: the modified Korteweg–deVries–Zakharov–Kuznetsov equation, the stochastic potential Korteweg–deVries equation, and the forced Korteweg–deVries equation. These equations are significant in plasma physics for modeling nonlinear ion acoustic waves and thus helping us to understand wave dynamics in plasmas. We introduce a new approach that relies on a new fractional expansion in the natural transform space and residual power series method to construct analytical solutions to the governing models. We investigate the theoretical analysis of the proposed method for these equations to expose this approach’s applicability, efficiency, and effectiveness in constructing analytical solutions to the governing equations. Moreover, we present a comparative discussion between the solutions derived during the work and those given in the literature to confirm that the proposed approach generates analytical solutions that rapidly converge to exact solutions, which proves the effectiveness of the proposed method. Full article

A continuous adaptive stabilization technique for the unstable period-1 orbit of the fractional difference logistic map is presented in this paper. An impulse-based control technique without short oscillatory transients right after the control impulse is designed for the fractional map with a long memory horizon. However, it appears that the coordinate of the unstable period-1 orbit may drift due to the continuous application of the impulse-based control scheme. An adaptive scheme capable of tracking the drifting coordinate of the unstable period-1 orbit is designed and validated by a number of computational experiments. The proposed control scheme is minimally invasive compared to the continuous feedback control as it preserves the model of the system while requiring only a series of sparse, small, instantaneous control impulses to achieve continuous adaptive stabilization of the unstable period-1 orbit of the fractional difference logistic map. Full article