Fractal Fract., Volume 9, Issue 3 (March 2025) – 65 articles

The primary aim of this manuscript is to establish unique fixed point results for a class of $\Psi$-contraction operators in complete G-metric spaces. By combining and extending various fixed point theorems in the context of $\Psi$-contraction operators, we introduce a novel function, denoted as $\psi$, and explore its properties. Our work presents new theoretical results, supported by examples and applications, that enrich the study of G-metric spaces. These results not only generalize and unify a broad range of existing findings in the literature but also expand their use to boundary value problems, Fredholm-type integral equations, and nonlinear Caputo fractional differential equations. In doing so, we offer a more comprehensive understanding of fixed point theory in the G-metric space framework and broaden its scope in applied mathematics. We also offer a numerical spectral approach for solving fractional initial value problems, utilizing shifted Chebyshev polynomials to construct a semi-analytic solution that inherently satisfies the given homogeneous initial conditions. Full article

The compactness, diversity, and nested structures of the old districts in Chinese cities, in terms of their three-dimensional (3D) morphology, are particularly distinctive. However, existing multifractal measurement methods are insufficient in revealing these 3D structures. This paper introduces a 3D multifractal approach based on generalized dimension and Rényi entropy. In particular, a local indicator τ_q(h) is introduced for the analysis of the mapping of 3D units, with the Nanjing Old City serving as a case study. The results indicate the following: (1) The significant fractal characteristics of the Nanjing Old City, with a capacity dimension value of 2.344, indicating its limited 3D spatial occupancy. (2) The fluctuating generalized dimension spectrum ranges from 2.241 to 2.660, which differs from previous studies, suggesting that the 3D morphology does not exhibit typical multifractal characteristics. (3) The 3D map matrix reveals a fragmented open space system, a heterogeneous distribution of high-rise buildings, and cross-scale variations in morphological heterogeneity. This 3D multifractal method aids urban planners in assessing critical issues such as the fragmentation, crowding, and excessive heterogeneity of urban morphology, providing a spatial coordination and scaling of these issues through the 3D map matrix and enhancing the discussion of the broader mechanisms influencing morphological characteristics. Full article

In this work, we establish the existence of at least one solution for a p-Laplacian Langevin differential equation involving the $\psi$-Hilfer fractional derivative with antiperiodic boundary conditions. More precisely, we transform the studied problem into a Hammerstein integral equation, and after that, we use the Schafer fixed point theorem to prove the existence of at least one solution. Two examples are provided to validate the main result. Full article

In this paper, we have investigated the existence of normalized solutions for a class of fractional Kirchhoff equations involving nonlinearity and critical nonlinearity. The nonlinearity satisfies $L^2$-supercritical conditions. We transform the problem into an extremal problem within the framework of Lagrange multipliers by utilizing the energy functional of the equation in the fractional Sobolev space and applying the mass constraint condition (i.e., for given $m>0,$${\int }_{{\mathbb{R}}^N}{|u|}^{2}dx=m^2$). We introduced a new set and proved that it is a natural constraint. The proof is based on a constrained minimization method and some characterizations of the mountain pass levels are given in order to prove the existence of ground state normalized solutions. Full article

This study presents a shifted Bernstein polynomial-based method for numerically solving the variable fractional order control equation governing a viscoelastic bar. Initially, employing a variable order fractional constitutive relation alongside the equation of motion, the control equation for the viscoelastic bar is derived. Shifted Bernstein polynomials serve as basis functions for approximating the bar’s displacement function, and the variable fractional derivative operator matrix is developed. Subsequently, the displacement control equation of the viscoelastic bar is transformed into the form of a matrix product. Substituting differential operators into the control equations, the control equations are discretized into algebraic equations by the method of matching points, which in turn allows the numerical solution of the displacement of the variable fractional viscoelastic bar control equation to be solved directly in the time domain. In addition, a convergence analysis is performed. Finally, algorithm precision and efficacy are confirmed via computation. Full article

The accurate description of fracture distributions is a crucial prerequisite for fracturing design and the evaluation of fracturing effects in tight reservoirs. We employed a fractal L-system to establish a tree branch model and derived a planar simulation method to characterize the distribution of natural weak discontinuities in rock. Weak discontinuities are classified using fractal similarity principles, enabling the determination of shear and opening criteria for weak discontinuities at various levels in water-based and CO₂ fracturing, as well as the pressure drop gradient within fractures after the initiation of weak discontinuities. Based on a weak discontinuity distribution model and fracture criteria, a simulation calculation method for the distribution of fracturing branch fractures was formulated. The results indicate that the number of branch fractures is closely related to the net pressure within the fractures at the wellbore and the difference between the maximum and minimum principal stresses in the reservoir. Compared with water-based fracturing fluids, CO₂ fracturing can facilitate the opening of branch fractures by reducing the opening conditions required for them to occur. The proposed calculation method can generate planar fracture morphologies and opening conditions with fractal characteristics, providing a basis for studying complex fracture formation mechanisms during CO₂ fracturing in tight reservoirs. Full article

Fractals are geometric patterns that appear self-similar across all length scales and are constructed by repeating a single unit on a regular basis. Entropy, as a core thermodynamic function, is an extension based on information theory (such as Shannon entropy) that is used to describe the topological structural complexity or degree of disorder in networks. Topological indices, as graph invariants, provide quantitative descriptors for characterizing global structural properties. In this paper, we investigate two types of generalized Sierpiński graphs constructed on the basis of different seed graphs, and employ six topological indices—the first Zagreb index, the second Zagreb index, the forgotten index, the augmented Zagreb index, the Sombor index, and the elliptic Sombor index—to analyze the corresponding entropy. We utilize the method of edge partition based on vertex degrees and derive analytical formulations for the first Zagreb entropy, the second Zagreb entropy, the forgotten entropy, the augmented Zagreb entropy, the Sombor entropy, and the elliptic Sombor entropy. This research approach, which integrates entropy with Sierpiński network characteristics, furnishes novel perspectives and instrumental tools for addressing challenges in chemical graph theory, computer networks, and other related fields. Full article

This paper investigates stability switches induced by Hopf bifurcation in a fractional three-neuron network that incorporates both neutral time delay and communication delay, as well as a general structure. Initially, we simplified the characteristic equation by eliminating trigonometric terms associated with purely imaginary roots, enabling us to derive the Hopf bifurcation conditions for communication delay while treating the neutral time delay as a constant. The results reveal that communication delay can drive a stable equilibrium into instability once it exceeds the Hopf bifurcation threshold. Furthermore, we performed a sensitivity analysis to identify the fractional order and neutral delay as the two most sensitive parameters influencing the bifurcation value for the illustrative example. Notably, in contrast to neural networks with only retarded delays, our numerical observations show that the Hopf bifurcation curve is non-monotonic, highlighting that the neural network with a fixed communication delay can exhibit stability switches and eventually stabilize as the neutral delay increases. Full article

This paper extends the fractional calculus of variations to include generalized fractional derivatives with dependence on a given kernel, encompassing a wide range of fractional operators. We focus on variational problems involving the composition of functionals, deriving the Euler–Lagrange equation for this generalized case and providing optimality conditions for extremal curves. We explore problems with integral and holonomic constraints and consider higher-order derivatives, where the fractional orders are free. Full article

Mathematics and physics are deeply interconnected. In fact, physics relies on mathematical tools like calculus and differential equations. The aim of this article is to introduce tempered Riemann–Liouville (RL) fractional operators and their properties with applications in mathematical physics. The tempered RL substantial fractional derivative is also defined, and its properties are discussed. The Laplace transforms (LTs) of the introduced fractional operators are evaluated. The Hyers–Ulam stability and the existence of a novel tempered fractional differential equation are examined. Moreover, a fractional integro-differential kinetic equation is formulated, and the LT is used to find its solution. A growth model and its graphical representation are established, highlighting the role of novel fractional operators in modeling complex dynamical systems. The developed mathematical framework offers valuable insights into solving a range of scenarios in mathematical physics. Full article

In this paper, we explore the outcomes related to the existence of nonlocal functional boundary value problems associated with pantograph equations utilizing $\psi$-Hilfer fractional derivatives. The nonlinear term relies on unknown functions which contain a proportional delay term and their fractional derivatives in a higher order. We discuss various existence results for the different “smoothness” requirements of the unknown function by means of Mawhin’s coincidence theory at resonance. We wrap up by providing a detailed explanation accompanied by an illustration of one of the outcomes. Full article

This paper consists of an exploration of the wave structures of the Benjamin–Ono equation along with a $\beta$-time fractional derivative. The model concerned is utilized to demonstrate internal waves of deep-stratified fluids. Bright, rational, periodic, and many more kinds of solutions for waves are achieved by utilizing the extended sinh-Gordon equation expansion (EShGEE) technique and the improved $G^{\prime }/G$-expansion scheme. An influence of fractional-order derivatives was also explored which gives the non-existing results. The Mathematica tool is utilized to gain and verify the results. The results are represented by 3-D, 2-D, and contour graphs. A stability analysis is utilized to confirm that results are precise as well as exact. Modulation instability (MI) is also performed for the steady-state solutions to the concerned model. Full article

In this paper, we have studied the transient process and the realizability of fractional systems via intermittent control. For any system under intermittent control input, a transient oscillation process is inevitable when the input switches, which is irrelevant to mathematical model. But this process is usually neglected when considering the achievements of fractional intermittent control systems as the initial value is changed by the switching input. The obtained theoretical results cannot agree with the real physical model. The input signal is treated as a piecewise signal by means of convolution operation and unit step function, and the output is drawn by convoluting the control input with a time decay function. We have drawn the conclusions that the initial value of the fractional model can not be updated by any outer input and that a transient process must exist that is related to all historic process and the memory property of a fractional system. If the response function of a system is taken as the time decay function, the results obtained are in good agreement with the actual model and can be used to analyze the transient phenomena in nature. Some examples are presented to verify our theoretical achievements. Full article

Efficient and safe extraction of coalbed methane is essential for reshaping China’s energy composition. This study integrates CO₂ adsorption, N₂ adsorption, and corrected mercury intrusion porosimetry (MIP) data to analyze the full pore size distribution (PSD) of six coal samples from the Qinshui and Tiefa Basins. By applying multifractal theory, we identified key heterogeneity features across different coal ranks, followed by a discussion of the factors influencing these parameters. The results indicate the following: (1) Coal matrix compressibility significantly impacts MIP results when mercury intrusion pressure exceeds 10 MPa, with corrected mesopore and macropore volume reductions ranging from 59.85–96.31% and 3.11–15.53%, respectively. (2) Pore volume distribution varies with coal rank, as macropores dominate in low-rank coal, while micropores contribute most in medium- and high-rank coal, accounting for over 90% of the total specific surface area. Multifractal analysis of CO₂, N₂, and corrected MIP data confirms notable multifractal characteristics across the full pore size range. (3) As the degree of coalification increases, as indicated by the rise in the R_o,max value, there is a notable negative correlation observed among the multifractal parameters D_min-D₀, D₀-D_max, Δα, and H. A positive correlation exists between moisture content and volatile matter content with D_min-D₀, Δα, and H, while a significant negative correlation is shown between the concentration of minerals and D_min-D₀, Δα, and H. There exists a favorable correlation between inertinite concentration and D₀-D_max. This work presents a theoretical foundation and empirical proof for the secure and effective extraction of coalbed methane in the researched region. Full article

In order to study the failure and fractal characteristics of unloaded rocks, with the help of the true triaxial unloading rock test system and the acoustic emission (AE) monitoring system, rock failure tests were conducted under varying intermediate principal stress and the mechanical response features of the rocks were analyzed. An investigation was conducted into the rocks’ AE patterns and multifractal features. The results showed that the rocks’ AE macroscopic and microscopic main failure modes differed slightly under unloading. As the intermediate principal stress ${\sigma }_2$ increased, the fractal dimension of the cracks in the rocks first increased and then decreased. The distribution of rock failure was initially concentrated, then dispersed, and concentrated again at the end. As the ${\sigma }_2$ increased, the number of failure events within a specified area in the rock samples under unloading, as represented by the ring-down count, first increased and then decreased. Meanwhile, the fractal dimension $\Delta \alpha$ first decreased and then increased. These results characterized the process whereby the failure distribution pattern of the rocks changed from being concentrated to dispersed and back to concentrated again. Full article

Missing values in time series data present a significant challenge, often degrading the performance of downstream tasks such as classification and forecasting. Traditional approaches address this issue by first imputing the missing values and then independently solving the predictive tasks. Recent methods have leveraged self-attention models to enhance imputation quality and accelerate inference. These models, however, predict values based on all input observations—including the missing values—thereby potentially compromising the fidelity of the imputed data. In this paper, we propose the Uncertainty-Aware Self-Attention (UASA) model to overcome these limitations. Our approach introduces two novel techniques: (i) A self-attention mechanism with a partially observed diagonal that effectively captures complex non-local dependencies in time series data—a characteristic also observed in fractional-order systems. This approach draws inspiration from fractional calculus, where non-integer-order derivatives better characterize complex dynamical systems with long-memory effects, providing a more comprehensive mathematical framework for handling temporal data. And (ii) uncertainty quantification in data imputation to better inform downstream tasks. The UASA model comprises an upstream component for data imputation and a downstream component for time series prediction, trained jointly in an end-to-end fashion to optimize both imputation accuracy and task-specific objectives simultaneously. For classification tasks, the UASA model demonstrates remarkable performance even under high missing data rates, achieving a ROC-AUC of $99.5\%$, a PR-AUC of $58.5\%$, and an F1-SCORE of $49.3\%$. For forecasting tasks on the AUST-Gait dataset, the UASA model achieves a Mean Squared Error (MSE) of 0.72 under $0\%$ missing data conditions (i.e., complete data input). Under the end-to-end training strategy evaluated across all missing data rates, the model achieves an average MSE of 0.74, showcasing its adaptability and robustness across diverse missing data scenarios. Full article

In this paper, a second-order predefined-time terminal sliding mode (SPTSM) is proposed, which is investigated for the practical applications of the speed regulation system of a permanent magnet synchronous motor (PMSM) by using predefined-time stability theory and Lyapunov stability theory. At first, we propose the SPTSM, which involves the controller’s design by using the novel reaching law with predefined-time terminal sliding mode (PTSM) and the novel sliding mode surface with PTSM. Second, we derive the novel SPTSM controller for the universal second-order nonlinear single-input single-output (SISO) system and the practical applications of the speed regulation system of the PMSM separately. Then, numerical simulation results of the speed regulation system of the PMSM are also included to check the effect of the theoretical results and the corresponding parameters on the convergence rates, so that the results can be guidance for the selection of SPTSM controller parameters. Finally, the dynamic responsiveness and robustness of the system are validated through numerical simulations and experimental results. It has been observed that the robust SPTSM controller, which is designed with the PTSM-PTSM, referring to the sliding mode that involves a reaching law with PTSM and a sliding mode surface with PTSM, exhibits superior performance. Full article

The intent of quantum calculus, briefly q-calculus, is to find q-analogues of mathematical entities so that the original object is achieved when a certain limit is taken. In the case of q-analogue of the ordinary derivative, the limit is $q\to 1^{-}$. Also, the study of integral as well as differential operators has remained a significant field of inquiry from the early developments of function theory. In the present article, a subclass $S_{sc}\left(\mu ,q\right)$ of functions being analytic in $\mathcal{D}=\left\{z\in \mathbb{C}:\left|z\right|<1\right\}$ is introduced. The definition of $S_{sc}\left(\mu ,q\right)$ involves the concepts of subordination, that of q-derivative and q-Ruscheweyh operators. Since coefficient estimates and coefficient functionals provide insights into different geometric properties of analytic functions, for this newly defined subclass, we investigate coefficient estimates up to $a_4$, in which both bounds for $|a_2|$ and $|a_3|$ are sharp, while that of $|a_4|$ is sharp in one case. We also discuss the sharp Fekete–Szegö functional for the said class. In addition, Toeplitz determinant bounds up to $T_3\left(2\right)$ (sharp in some cases) and sufficient condition are obtained. Several consequences derived from our above-mentioned findings are also part of the discussion. Full article

This paper proposes a method for searching for pipeline leaks by analyzing the Hurst exponent of acoustic signals. The investigations conducted on the laboratory setup and the current pipelines of the water supply system. During the experiments, through defects of the round shape-type pipeline with diameters from 1 to 5 mm were modeled. For calculating Hurst exponent, rescaled range analysis (R/S analysis), and detrended fluctuation analysis (DFA) were used. The research results have shown that pipeline leaks are reliably detected by analyzing the Hurst exponent of acoustic signals. The signals of a defect-free pipeline are close to the level of a deterministic signal. When a leak occurs in a pipeline, the Hurst exponent decreases. Pipeline fluctuations are anti-persistent nature. It is shown that a change in the size of the through hole in the pipeline wall does not have a significant effect on the value of the Hurst exponent of acoustic signals. These results are explained by using spectral analysis and CFD modeling (Computational Fluid Dynamics modeling) methods in the Ansys Fluent software (v. 19.2). It has been established that the spectral components that contribute most to the fractal structure of signals are concentrated within the frequency range from 0 to 2 kHz. Full article

This work introduces the Legendre cardinal functions for the first time. Based on Jacobi and Lobatto grids, two approaches are employed to determine these basis functions. These functions are then utilized within the pseudospectral method to solve the fractional Klein–Gordon equation (FKGE). Two numerical schemes based on the pseudospectral method are considered. The first scheme reformulates the given equation into a corresponding integral equation and solves it. The second scheme directly addresses the problem by utilizing the matrix representation of the Caputo fractional derivative operator. We provide a convergence analysis and present numerical experiments to demonstrate the convergence of the schemes. The convergence analysis shows that convergence depends on the smoothness of the unknown function. Notable features of the proposed approaches include a reduction in computations due to the cardinality property of the basis functions, matrices representing fractional derivative and integral operators, and the ease of implementation. Full article

This paper examines the self-similarity (long memory) in prices of crude oil markets, namely Brent and West Texas Instruments (WTI), by means of fractals. Specifically, price series are decomposed by stationary wavelet transform (SWT) to obtain their short and long oscillations. Then, the Hurst exponent is estimated from each resulting oscillation by rescaled analysis (R/S) to represent hidden fractals in the original price series. The analysis is performed during three periods: the calm period (before the COVID-19 pandemic), the COVID-19 pandemic, and the Russia-Ukraine war. In summary, prices of Brent and WTI exhibited significant increases in persistence in long movements during the COVID-19 pandemic and the Russia-Ukraine war. In addition, they showed a significant increase in anti-persistence in short movements during the pandemic and a significant decrease in anti-persistence during the Russia-Ukraine war. It is concluded that both COVID-19 and the Russia-Ukraine war significantly affected long memory in the short and long movements of Brent and WTI prices. Full article

In fractal theory, the fractal dimension has been accepted as a quantitative parameter to measure the complexity of fluctuations and the persistence of wind speeds. Typhoons are extreme wind events that damage structures. In this study, on the basis of wind field measurements, the fractal dimension characteristics of four typhoons in southeastern China are examined. Typhoon wind speeds at different heights and locations are presented. Monofractal dimension analysis was first conducted, and the results revealed that the typhoon wind speeds were persistent, with fractal dimensions smaller than 1.5. For all four typhoons considered, with the onset of landfall, the fractal dimensions approach 1.5, indicating unpredictable trends in the time series. Multifractality is accepted to exist in the typhoon wind speed dataset, and multifractal analysis was also conducted on the basis of the measured typhoon wind speeds. The results show that the fractal parameters calculated by multifractal analysis are generally greater than those calculated via monofractal analysis. This research aims to improve the understanding of the inner dynamic characteristics of typhoon wind speeds. These fractal parameters can provide quantitative references for future typhoon simulations and predictions. Full article

The early warning of disasters such as ground pressure in deep hard rock mines has long constrained the safe and efficient development of mining activities. Based on fractal theory and fractal dimension interpretation, this study constructs a microseismic monitoring system for mining areas, extracting key elements, particularly time and energy elements. Using the box-counting method of fractal theory, the study investigates the fractal dimensions of microseismic time–energy elements, data interpretation, and disaster source early warning. Through parameter analysis, events related to local potential failure are identified and extracted, and disaster characteristics are revealed based on microseismic activity. A time–energy fractal dimension-based analysis method is developed for preliminary fractal analysis and prediction of regional damage. A time–energy-centered early warning model is constructed, narrowing the prediction range to a scale of 10 m. Based on the fractal interpretation of time–energy data, the prediction and early warning of rock mass failure in mining areas are achieved, with the reliability of nested energy warnings ranging between 91.7% and 96.2%. A comprehensive evaluation criterion for fractal dimension values is established, enabling accurate delineation of warning zones and providing scientific decision-making support for mine safety promotion. Full article

This paper considers a numerical method for solving the stochastic semilinear subdiffusion equation which is driven by integrated fractional Gaussian noise and the Hurst parameter $H\in (1/2,1)$. The finite element method is employed for spatial discretization, while the L1 scheme and Lubich’s first-order convolution quadrature formula are used to approximate the Caputo time-fractional derivative of order $\alpha \in (0,1)$ and the Riemann–Liouville time-fractional integral of order $\gamma \in (0,1)$, respectively. Using the semigroup approach, we establish the temporal and spatial regularity of the mild solution to the problem. The fully discrete solution is expressed as a convolution of a piecewise constant function with the inverse Laplace transform of a resolvent-related function. Based on the Laplace transform method and resolvent estimates, we prove that the proposed numerical scheme has the optimal convergence order $O({\tau }^{min\{H+\alpha +\gamma -1-\epsilon ,\alpha \}}),\epsilon >0$. Numerical experiments are presented to validate these theoretical convergence orders and demonstrate the effectiveness of this method. Full article

Scholars from several disciplines have recently expressed interest in the field of fractional q-calculus based on fractional integrals and derivative operators. This article mathematically applies the fractional q-differential and q-integral operators in geometric function theory. The linear q-derivative operator ${\mathbf{S}}_{\mu ,\delta ,q}^{n,m}$ and subordination are used in this study to define and construct new classes of $\alpha$-convex functions associated with the cardioid domain. Additionally, this paper explores acute inequality problems for newly defined classes ${\mathcal{R}}_q^{\alpha }(a,c,m,L,P),$ of $\alpha$-convex functions in the open unit disc ${\mathcal{U}}_s$, such as initial coefficient bounds, coefficient inequalities, Fekete–Szegö problems, the second Hankel determinants, and logarithmic coefficients. The results presented in this paper are simple to comprehend and demonstrate how current research relates to earlier research. We found all of the estimates, and they are sharp. Full article

A fractional-order Lorenz system is optimized to maximize its maximum Lyapunov exponent and Kaplan-York dimension using the Non-dominated Sorting Genetic Algorithm II (NSGA-II) algorithm. The fractional-order Lorenz system is integrated with a recent process called the “modified two-stage Runge-Kutta” (M2sFRK) method, which is very fast and efficient. A Pseudo-Random Number Generator (PRNG) was built using one of the optimized systems that was obtained. The M2sFRK method allows for obtaining a very fast optimization time and also designing a very efficient PRNG with linear complexity, $O\left(n\right)$. The designed PRNG generates 24 random bits at each iteration step, and the random sequences pass all the National Institute of Standards and Technology (NIST) and TestU01 statistical tests, making the PRNG suitable for cryptographic applications. The presented methodology could be extended to any other chaotic system. Full article

Rheological complex models of various elastoviscous and viscoelastic fractional-type substances with polarized piezoelectric properties are of interest due to the widespread use of viscoelastic–plastic bodies under loading. The word “overview” used in the title means and corresponds to the content of the manuscript and aims to emphasize that it presents an overview of a new class of complex rheological models of the fractional type of ideal elastoviscous, as well as viscoelastic, materials with piezoelectric properties. Two new elementary rheological elements were introduced: a rheological basic Newton’s element of ideal fluid fractional type and a basic Faraday element of ideal elastic material with the property of polarization under mechanical loading and piezoelectric properties. By incorporating these newly introduced rheological elements into classical complex rheological models, a new class of complex rheological models of materials with piezoelectric properties described by differential fractional-order constitutive relations was obtained. A set of seven new complex rheological models of materials are presented with appropriate structural formulas. Differential constitutive relations of the fractional order, which contain differential operators of the fractional order, are composed. The seven new complex models describe the properties of ideal new materials, which can be elastoviscous solids or viscoelastic fluids. The purpose of the work is to make a theoretical contribution by introducing, designing, and presenting a new class of rheological complex models with appropriate differential constitutive relations of the fractional order. These theoretical results can be the basis for further scientific and applied research. It is especially important to point out the possibility that these models containing a Faraday element can be used to collect electrical energy for various purposes. Full article

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