Fractal Fract., Volume 9, Issue 4 (April 2025) – 70 articles

The successful exploration and development of shale oil in the clay-rich Gulong shale have sparked increased research into the influence of clay minerals on shale reservoirs. However, compared to chlorite in sandstones, limited studies have focused on the occurrence of chlorite in continental shales and its effects on shale reservoir properties. This study offers a comprehensive analysis of chlorite in Gulong shale samples from three wells at different diagenetic stages. Four primary chlorite occurrences are identified in the Gulong shale: Type I, which is chlorite filling dissolved pores in carbonate; Type II, which is isolated chlorite; Type III, which is chlorite filling organic matter; and Type IV, which is chlorite filling authigenic microquartz. Types I and III chlorites exhibit higher porosity, offering more storage space for shale reservoirs. Chlorites of Types I, III, and IV, filled with other substances, display higher fractal dimensions, indicating more complex pore structures. These complex pores are favorable for oil adsorption but hinder oil seepage. The processes of organic matter expulsion and dissolution, which intensify with increasing diagenesis, promote the development of Types I and III chlorites, thereby positively influencing the shale reservoir porosity of Gulong shale. This study underscores the influence of chlorite occurrences on shale reservoir properties, providing valuable insights for the future exploration and development of shale oil and gas. Full article

Pore structure plays a critical role in evaluating shale “sweet spots”. Compared to marine shale, lacustrine shale has more diverse lithofacies types and greater heterogeneity in pore structure due to frequently changing environmental conditions. Using methods such as mercury intrusion porosimetry (MIP), field emission scanning electron microscopy (FE-SEM), nuclear magnetic resonance (NMR), and X-ray diffraction (XRD), this study investigates the micropore structures and heterogeneity of different lithofacies in the Jurassic Dongyuemiao Member lacustrine shale. Image processing and multifractal theory were employed to identify the controlling factors of pore structure heterogeneity. The key findings are as follows. (1) Based on mineral content and laminae types, the lithofacies types of Dongyuemiao lacustrine shale are classified into four types: shell–laminae mixed shale (SLMS), silty–laminae clay shale (SLCS), clast–laminae clay shale (CLCS), and clay shale (CS). (2) Based on genesis, shale reservoirs’ pore and permeability space are categorized into inorganic pores, organic pores, and micro-fractures. Inorganic pores consist of inter-particle pores and intra-particle pores. Pore size distribution curves for all four lithofacies exhibit two main peaks, with pore sizes concentrated in the ranges of 2–10 nm and 50–80 nm. Mesopores and macropores dominate, accounting for over 80% of the total pore volume. Mesopores are most developed in CLCS, representing 56.3%. (3) Quartz content is positively correlated with the multifractal dimension, while clay content shows a negative correlation. Higher quartz content, coupled with lower clay content, weakens pore structure heterogeneity. A negative correlation exists between total organic carbon (TOC) and the multifractal dimension, indicating that higher organic matter content enhances organic pore development and increases microscopic heterogeneity. (4) Porosity heterogeneity in SLMS is effectively characterized by D₀-D_max, while in the other three lithofacies, it is characterized by D_min-D₀. Permeability across all lithofacies correlates with D₀-D_max. In CS, SLMS, and SLCS, permeability is positively correlated with D₀-D_max, with higher values indicating greater permeability heterogeneity. In CLCS, permeability is negatively correlated with D₀-D_max, such that lower values reflect stronger heterogeneity. Full article

Fractional variable order systems with unusual dynamics in the order are a little-studied topic. In this study, we present three examples of very simple fractional systems with unusual dynamics in the derivative order. These cases involve different approaches to define the variable-order dynamics: (1) an integer-order differential equation that includes the state variable, (2) a differential equation that incorporates the state variable and features both integer- and fractional-order derivatives, and (3) fractional variable-order differential equations nested in the derivative orders. We prove a result that shows how the extended recursion of the last case is generalized. These examples illustrate the richness that simple dynamical systems can reveal through the order of their derivatives. Full article

The pore structure of reservoir rocks was a crucial factor affecting hydrocarbon production. Accurately characterized the micropore structure of different types of rock reservoirs was of great significance for unconventional natural gas exploration. In this study, multiple observation methods (field emission scanning electron microscope (FESEM) and low-field nuclear magnetic resonance (LFNMR)) and physical tests (mercury injection capillary pressure (MICP)) were employed, and double logarithmic plots for fractal fitting were illustrated. The fractal dimension of 15 samples was calculated using fractal theory to systematically investigate the pore–fracture structure and fractal characteristics of hydrocarbon source rock (limestone, mudstone, and sandstone) samples from the Late Carboniferous Taiyuan Formation in the Huainan coalfield. MICP experiments revealed that sandstone reservoirs had larger and more uniformly distributed pore throats compared to mudstone and limestone, exhibiting superior connectivity and permeability. The T₂ spectrum characteristic maps obtained using LFNMR were also consistent with the pore distribution patterns derived from MICP experiments, particularly showed that sandstone types exhibited excellent signal intensity across different relaxation time periods and had a broader T₂ spectrum width, which fully indicated that sandstone types possess superior pore structures and higher connectivity. FESEM experiments demonstrated that sandstone pores were highly developed and uniform, with sandstone fractures dominated by large fractures above the micrometer scale. Meanwhile, the FESEM fractal dimension results indicated that sandstone exhibits good fractal characteristics, validating its certain oil storage capacity. Furthermore, the FESEM fractal dimension exhibited a good correlation with the porosity and permeability of the hydrocarbon source rock reservoirs, suggesting that the FESEM fractal dimension can serve as an important parameter for evaluating the physical properties of hydrocarbon source rock reservoirs. This study enriched the basic geological theories for unconventional natural gas exploration in deep coal-bearing strata in the Huainan coalfield. Full article

High-tech industries are of strategic importance to the national economy, and Beijing has been designated as a science and technology innovation center by the State Council. Accurate analysis of its added value is crucial for technological development. While recent data enhance prediction accuracy, its limited volume poses challenges. The cumulative grey Lotka–Volterra model and grey differential dynamic multivariate model address this by leveraging short-term data effectively. This study applies these two models to analyze influencing factors and predict Beijing’s high-tech industry growth. Results show a competitive relationship with four systems, lacking synergy. In the next five years, a mutually beneficial trend is expected. The Mean Absolute Percentage Error (MAPE) remains within 10%, confirming the model’s reliability. Full article

In this article, we study a fractional lower-order differential equation, $-D_{0+}^{\alpha }{\rm Y}\left(\xi \right)+\mathrm{a}\left(\xi \right){\rm Y}\left(\xi \right)=\mathrm{y}\left(\xi \right),\xi \in (0,1),\alpha \in (1,2),$ with a Dirichlet-type boundary condition, where $\mathrm{a}\left(\xi \right)\in L^1[0,1]$ permits singularity. When the coefficient of perturbation term $\mathrm{a}\left(\xi \right)$ is continuous on $[0,1]$, Graef et al. derived the associated Green’s function under certain conditions on $\mathrm{a}$, but failed to obtain its positivity. We obtain some positive properties of the Green’s function of this problem under certain conditions. The results will fill the gap in the above literature. As for application of the theoretical results, a singular fractional differential equation with a perturbation term is considered. The unique positive solution is obtained by using the fixed point theorem, and an iterative sequence is established to approximate it. In addition, a semipositone fractional differential equation, $-D_{0+}^{\alpha }{\rm Y}\left(\xi \right)=\mu \mathcal{F}(\xi ,{\rm Y}\left(\xi \right)),\xi \in (0,1),\alpha \in (1,2),$ is also considered. The existence of positive solutions is determined under a more general condition, $\mathcal{F}(\xi ,\mathrm{x})\ge -\mathrm{b}\left(\xi \right)\mathrm{x}-\mathrm{e}\left(\xi \right)$, where $\mathrm{b}\left(\xi \right),\mathrm{e}\left(\xi \right)\in L^1[0,1]$ are non-negative functions. Relevant examples are listed to manifest the theoretical results. Full article

Bayoud disease, caused by Fusarium oxysporum f. sp. albedinis, is a major threat to date palm trees. It leads to lower crop yields, financial losses, and decreased biodiversity. The complexity of the disease presents challenges to effective disease management. This study introduces a mathematical model comprising six compartments for palm trees: susceptible trees, resistant varieties, exposed trees, infected trees, isolated trees under treatment, and recovered trees, along with a contaminant water compartment. The model emphasizes the role of resistant varieties, contamination of irrigation water, and the treatment of infected trees in disease control. Theoretical analyses guarantee positivity, boundedness, and the existence of a unique solution. The existence of equilibrium points (disease-free and endemic) and the reproduction number (${\mathcal{R}}_0$) of the model are calculated analytically and validated through numerical simulations. Stability analysis at disease-free and endemic equilibrium points is conducted in terms of ${\mathcal{R}}_0$. Sensitivity analysis identifies key parameters influencing disease dynamics and is helpful to identify the potential control parameters. An optimal control problem is formulated to minimize infection spread and associated costs via preventive isolation and treatments, irrigation water treatment, and the promotion of resistant varieties. Numerical simulations demonstrate the efficacy of these strategies, highlighting the potential of resistant varieties and treatment measures in reducing infection rates and enhancing tree health. This research offers valuable insights into sustainable Bayoud disease management, underscoring the importance of mathematical modeling in addressing agricultural challenges. Full article

In this research article, we investigate a three-dimensional dynamical system governed by fractal-fractional-order evolution differential equations subject to terminal boundary conditions. We derive existence and uniqueness results using Schaefer’s and Banach’s fixed-point theorems, respectively. Additionally, the Hyers–Ulam stability approach is employed to analyze the system’s stability. We employ vector terminology for the proposed problem to make the analysis simple. To illustrate the practical relevance of our findings, we apply the derived results to a numerical example and graphically illustrate the solution for different fractal-fractional orders, emphasizing the effect of the derivative’s order on system behavior. Full article

This paper presents an advanced simulation-based investigation of tumor growth and chemical diffusion in biological tissues, using $\vartheta$-fractional stochastic integral equations. Based on the theoretical framework developed in [Fractal Fract. 2025, 9(1), 7], we develop an innovative computational model to explore the practical applications of these equations in the biological field. The model focuses on providing new insights into the dynamic interaction between stochastic effects of a fractional nature and complex biological tissue environments, contributing to a deeper understanding of the mechanisms of chemical diffusion within tissues and tumor growth under different conditions. The paper details the numerical techniques used to solve the $\vartheta$-fractional stochastic integral equations, focusing on the stability and accuracy of the solutions, while demonstrating their ability to accurately and effectively capture key biological phenomena. Through extensive computational experiments, the model demonstrates its ability to replicate realistic tumor growth patterns and complex chemical transport dynamics, providing a powerful and flexible tool for understanding tumor behavior and interaction with potential therapies. These results represent an important step toward improving biological models and enhancing biomedical applications, particularly in the areas of targeted drug design and analysis of tumor dynamics under chemotherapeutic influence. Full article

Fractal analysis was used to characterize the organic matter nanopore structure in tectonically deformed shales, providing insights into the heterogeneity and complexity of the pore network. Shale samples from different tectonic deformation styles (undeformed, brittle deformed, and ductile deformed) in the Lower Cambrian Niutitang Formation in western Hunan, South China, were collected. By comprehensively applying techniques such as low-temperature gaseous (CO₂ and N₂) adsorption (LTGA), scanning electron microscopy (SEM), and ImageJ analysis, we accurately obtained key parameters of the pore structure. The results show ductile deformation reduces fractal dimension (D_M) by ~0.2 compared to brittle deformed shale, reflecting the homogenization of organic nanopore structures. Brittle deformation leads to a more complex pore network, while ductile deformation reduces the complexity of the organic nanopore structure. The fractal dimensions are affected by various factors, with micropore development being crucial for undeformed shale, clay and pore length–width ratio dominating in brittle deformed shale, and all-scale pores being key for ductile deformed shale. This study provides the first comparative analysis of fractal dimensions across undeformed, brittle deformed, and ductile deformed shales, revealing distinct pore structure modifications linked to deformation styles. These findings not only enhance our understanding of the influence mechanism of tectonic deformation on shale pore structure and fractal characteristics but also provide a theoretical basis for optimizing shale gas exploration and production strategies. These findings offer a framework for predicting gas storage and flow dynamics in tectonically complex shale reservoirs. For instance, in areas with different tectonic deformation styles, we can better evaluate the gas storage capacity and production potential of shale reservoirs according to the obtained fractal characteristics, which is of great significance for efficient shale gas development. Full article

On the one hand, convex functions are important to derive rigorous convergence rates, and on the other, synchronous functions are significant to solve statistical problems using Chebyshev inequalities. Therefore, fractional integral inequalities involving such functions play a crucial role in creating new models and methods. Although a large class of fractional operators have been used to establish inequalities, nevertheless, these operators having the Fox-H and the Meijer-G functions in their kernel have been applied to establish fractional integral inequalities for such important classes of functions. Taking motivation from these facts, the primary objective of this work is to develop fractional inequalities involving the Fox-H function for convex and synchronous functions. Since the Fox-H function generalizes several important special functions of fractional calculus, our results are significant to innovate the existing literature. The inventive features of these functions compel researchers to formulate deeper results involving them. Therefore, compared with the ongoing research in this field, our results are general enough to yield novel and inventive fractional inequalities. For instance, new inequalities involving the Meijer-G function are obtained as the special cases of these outcomes, and certain generalizations of Chebyshev inequality are also included in this article. Full article

A total of 108 maximum Kolmogorov entropy (S_K) values, calculated by means of chaos theory, are obtained from 108 time series (TSs) (each consisting of 28,463 hourly data points). The total TSs are divided into 54 urban meteorological (temperature (T), relative humidity (RH) and wind speed magnitude (WS)) and 54 pollutants (PM₁₀, PM_2.5 and CO). The measurement locations (6) are located at different heights and the data recording was carried out in three periods, 2010–2013, 2017–2020 and 2019–2022, which determines a total of 3,074,004 data points. For each location, the sum of the maximum entropies of urban meteorology and the sum of maximum entropies of pollutants, S_{K, MV} and S_{K, P}, are calculated and plotted against h, generating six different curves for each of the three data-recording periods. The tangent of each figure is determined and multiplied by the average temperature value of each location according to the period, obtaining, in a first approximation, the magnitude of the entropic forces associated with urban meteorology (F_{K, MV}) and pollutants (F_{K, P}), respectively. It is verified that all the time series have a fractal dimension, and that the fractal dimension of the pollutants shows growth towards the most recent period. The entropic dynamics of pollutants is more dominant with respect to the dynamics of urban meteorology. It is found that this greater influence favors subdiffusion processes (α < 1), which is consistent with a geographic basin with lower atmospheric resilience. By applying a heavy-tailed probability density analysis, it is shown that atmospheric pollution states are more likely, generating an extreme environment that favors the growth of respiratory diseases and low relative humidity, makes heat islands more stable over time, and strengthens heat waves. Full article

Concrete materials exposed to sulfate-rich geological environments are prone to structural durability degradation due to chemical erosion. Silane-based protective materials can enhance the durability of concrete structures under harsh environmental conditions. This study investigates the evolution of acoustic emission (AE) precursor characteristics in silane-protected, sulfate-eroded concrete specimens during uniaxial compression failure. Unlike existing research focused primarily on protective material properties, this work establishes a novel framework linking “silane treatment–AE parameters–failure precursor identification”, thereby bridging the research gap in damage evolution analysis of sulfate-eroded concrete under silane protection. Uniaxial compressive strength tests and AE monitoring were conducted on both silane-protected and unprotected sulfate-eroded concrete specimens. A diagnostic system integrating dynamic analysis of the acoustic emission b-value, mutation detection of energy concentration index ρ, and multifractal detrended fluctuation analysis (MF-DFA) was developed. The results demonstrate that silane-protected specimens exhibited a distinct b-value escalation followed by an abrupt decline prior to peak load, whereas unprotected specimens showed disordered fluctuations. The mutation point of energy concentration ρ for silane-protected specimens occurred at 0.83 σc, representing a 9.2% threshold elevation compared to 0.76 σc for unprotected specimens, confirming delayed damage accumulation in protected specimens. MF-DFA revealed narrowing spectrum width ($∆\alpha$) in unprotected specimens, indicating reduced heterogeneity in AE signals, while protected specimens maintained significant multifractal divergence. $∆f\left(\alpha \right)$ peak localization revealed that weak AE signals dominated during early loading stages in both groups, with crack evolution primarily involving sliding and friction. During the mid-late elastic phase, crack propagation became the predominant failure mode. Experimental evidence confirms the engineering significance of silane protection in extending service life of concrete structures in sulfate environments. The proposed multi-parameter AE diagnostic methodology provides quantitative criteria for the safety monitoring of protected concrete structures in sulfate-rich conditions. Full article

This paper proposes an innovative method that combines the homotopy analysis method with the Jafari transform, applying it for the first time to solve systems of fractional-order linear and nonlinear differential equations. The method constructs approximate solutions in the form of a series and validates its feasibility through comparison with known exact solutions. The proposed approach introduces a convergence parameter ℏ, which plays a crucial role in adjusting the convergence range of the series solution. By appropriately selecting initial terms, the convergence speed and computational accuracy can be significantly improved. The Jafari transform can be regarded as a generalization of classical transforms such as the Laplace and Elzaki transforms, enhancing the flexibility of the method. Numerical results demonstrate that the proposed technique is computationally efficient and easy to implement. Additionally, when the convergence parameter $\hslash =-1$, both the homotopy perturbation method and the Adomian decomposition method emerge as special cases of the proposed method. The knowledge gained in this study will be important for model solving in the fields of mathematical economics, analysis of biological population dynamics, engineering optimization, and signal processing. Full article

In this paper, a counter-example based on a realistic initial condition invalidates the usual approach related to the so-called physical initial condition of the Caputo derivative used to solve fractional-order Cauchy problems. Due to Infinite State representation, we prove that the initial condition of the Caputo derivative has to take into account the distributed states of an associated fractional integrator. Then, we prove that the free response of the counter-example requires the knowledge of the associated fractional integrator free response, and a realistic solution is proposed for the convolution problem based on the Mittag–Leffler function. Moreover, a simple and efficient technique based on Infinite State representation is proposed to solve the previous free response problem. Finally, numerical simulations demonstrate that the usual Caputo technique is based on an unrealistic initial condition without any physical meaning. Full article

This paper introduces a novel fractional Susceptible-Infected-Recovered ($\mathbb{SIR}$) model that incorporates a power Caputo fractional derivative (PCFD) and a density-dependent recovery rate. This enhances the model’s ability to capture memory effects and represent realistic healthcare system dynamics in epidemic modeling. The model’s utility and flexibility are demonstrated through an application using parameters representative of the COVID-19 pandemic. Unlike existing fractional $\mathbb{SIR}$ models often limited in representing diverse memory effects adequately, the proposed PCFD framework encompasses and extends well-known cases, such as those using Caputo–Fabrizio and Atangana–Baleanu derivatives. We prove that our model yields bounded and positive solutions, ensuring biological plausibility. A rigorous analysis is conducted to determine the model’s local stability, including the derivation of the basic reproduction number (${\mathbf{R}}_0$) and sensitivity analysis quantifying the impact of parameters on ${\mathbf{R}}_0$. The uniqueness and existence of solutions are guaranteed via a recursive sequence approach and the Banach fixed-point theorem. Numerical simulations, facilitated by a novel numerical scheme and applied to the COVID-19 parameter set, demonstrate that varying the fractional order significantly alters predicted epidemic peak timing and severity. Comparisons across different fractional approaches highlight the crucial role of memory effects and healthcare capacity in shaping epidemic trajectories. These findings underscore the potential of the generalized PCFD approach to provide more nuanced and potentially accurate predictions for disease outbreaks like COVID-19, thereby informing more effective public health interventions. Full article

The pore–fracture structure of ultra-deep coal is critical for evaluating resource potential and guiding the exploration and development of deep coalbed methane (CBM). In this study, a coal sample was obtained from the Gaogu-4 well at a depth of 4369.4 m in the Shengli Oilfield of Shandong, China. A comprehensive suite of characterization techniques, including Field Emission Scanning Electron Microscopy (FE-SEM), X-ray diffraction (XRD), Mercury Intrusion Porosimetry (MIP), Low-temperature Nitrogen Adsorption (LT-N₂GA), and Low-pressure CO₂ Adsorption (LP-CO₂GA), were employed to investigate the surface morphology, mineral composition, and multi-scale pore–fracture characteristics of the ultra-deep coal. Based on fractal geometry theory, four fractal dimension models were established, and the pore structure parameters were then used to calculate the fractal dimensions of the coal sample. The results show that the ultra-deep coal surface is relatively rough, with prominent fractures and limited pore presence as observed under FE-SEM. Energy Dispersive Spectrometer (EDS) analysis identified the elements such as C, O, Al, Si, S, and Fe, thus suggesting that the coal sample contains silicate and iron sulfide minerals. XRD analysis shows that the coal sample contains kaolinite, marcasite, and clinochlore. The multi-scale pore–fracture structure characteristics indicate that the ultra-deep coal is predominantly composed of micropores, followed by mesopores. Macropores are the least abundant, yet they contribute the most to pore volume (PV), accounting for 70.9%. The specific surface area (SSA) of micropores occupies an absolute advantage, accounting for up to 97.46%. Based on the fractal model, the fractal dimension of the coal surface is 1.4372, while the fractal dimensions of the micropores, mesopores, and macropores are 2.5424, 2.5917, and 2.5038, respectively. These results indicate that the surface morphology and pore–fracture distribution of the ultra-deep coal are non-uniform and exhibit statistical fractal characteristics. The pore–fracture structure dominated by micropores in ultra-deep coal seams provides numerous adsorption sites for CBM, thereby controlling the adsorption capacity and development potential of deep CBM. Full article

This paper investigates the well-posedness of analytical solutions to fractional quadratic differential equations, which involve generalized fractional integrals with respect to other functions. The nonlinear components f and h depend on spatial variables and the general fractional integral, respectively. We use the operator equation $T_1\omega T_2\omega +T_3\omega =\omega$ to investigate the existence of solutions, marking the first study of its kind. Using an auxiliary function and Boyd and Wang’s fixed-point theorem, the uniqueness and continuous dependence of the solution are obtained. In particular, we applied nonlinear functional analysis to investigate Hyers-Ulam and Hyers-Ulam-Rassias stabilities for fractional quadratic integral equations. New results are provided for specific values of the parameter z, and a fundamental inequality is formulated to ensure the existence of maximal and minimal solutions. Some examples are given to illustrate our results. Full article

Multiple-stream deep learning (DL) models are typically used for multiple-modality datasets, with each model extracting favorable features from its own modality dataset. Through feature fusion, multiple-stream models can generally achieve higher recognition rates. While feature engineering is indispensable for machine learning models, it is generally omitted for DL. However, feature engineering can be regarded as an important supplement to DL, especially when using small datasets with rich characteristics. This study aims to utilize limited existing resources to improve the overall performance of the considered models. Therefore, I choose a single-modality dataset—the Chest X-Ray dataset—as my original dataset. For ease of evaluation, I take 16 pre-trained models as basic models for the development of multiple-stream models. Based on the characteristics of the Chest X-Ray dataset, three characteristic datasets are generated from the original dataset, including the Hurst exponent dataset (corresponding to a fractal dimension dataset), as inputs to the multiple-stream models. For comparison, various multiple-stream models are developed based on the same dataset. The experimental results show that, with feature engineering, the accuracy can be raised from 91.67% (one-stream) to 94.52% (two-stream), 94.73% (three-stream), and 94.79% (four-stream), while, without feature engineering, it can be increased from 91.67% to 92.35%, 93.49%, and 93.66%, respectively. In the future, the simple yet effective methodology proposed in this study can be widely applied to other datasets, in order to effectively promote the overall performance of models in scenarios characterized by limited resources. Full article

This article presents a computational analysis of approximate solutions for the time-fractional nonlinear Kawahara problem (KP) and the modified Kawahara problem (modified KP). This study utilizes the natural homotopy transform scheme (NHTS), which integrates the natural transform (NT) with the homotopy perturbation scheme (HPS). We derive the algebraic expression of nonlinear terms through the implementation of HPS. The fractional derivatives are considered in the Caputo form. Numerical results and visualizations present the practical interest and effectiveness of the fractional derivatives. The accuracy of the approximate results, coupled with their precise outcomes, emphasizes the reliability of the method. These findings demonstrate that NHTS is a robust and effective approach for solving time-fractional problems through series expansions. Full article

In this work, the novel general formula for a time-dependent nuclear reactor system of equations with delayed neutron effect has been formulated using a fractional calculus model. We explore the properties of this model, including two analytical approximation methods, the Temimi–Ansari method (TAM) and the Sumudu residual power series method (SRPSM), for solving the equation. These methods allow for the computation of approximate solutions at specific points. This is particularly useful for partial differential equations (PDEs) arising in various fields like physics, engineering, and finance. This work is hoped to improve the advancement of nuclear modeling and simulation, providing researchers and engineers with a powerful mathematical tool for studying the complex dynamics of these critical energy systems. Full article

This study focuses on a novel parameterized Wigner distribution, which is an organic integration of the free metaplectic Wigner distribution and the $\mathbf{K}$-Wigner distribution. We style this as the free metaplectic $\mathbf{K}$-Wigner distribution (FMKWD) and investigate its uncertainty principles and related applications. We establish a crucial equivalence relation between the uncertainty product in time-FMKWD and free metaplectic transformation (FMT)-FMKWD domains and those in two FMT domains, from which we derive two types of orthogonality conditions: an orthonormality condition; and two sub-types of minimum or maximum eigenvalue commutativity conditions on the FMKWD. Finally we separately formulate an uncertainty inequality in FMKWD domains for real-valued functions, three kinds of uncertainty inequalities in orthogonal FMKWD domains, an uncertainty inequality in orthonormal FMKWD domains, and four kinds of uncertainty inequalities in the minimum or maximum eigenvalue commutative FMKWD domains for complex-valued functions. The time-frequency resolution of the FMKWD is compared with those of the free metaplectic Wigner distribution, $\mathbf{K}$-Wigner distribution, and N-dimensional Wigner distribution to demonstrate its superiority in super-resolution analysis. For applications, the uncertainty inequalities derived are used to estimate the bandwidth in FMKWD domains, and the FMKWD is applied to detect noisy linear frequency-modulated signals. Full article

This paper investigates the fractional-order characteristics of the stator and rotor windings of a synchronous generator. Utilizing mechanism-based modeling methodology, it pioneers the derivation of the fractional-order voltage equations for a synchronous generator across both the three-phase stationary coordinate system (A, B, C) and the synchronous rotating coordinate system (d, q, 0). Through simplifying assumptions and rigorous derivations, a 2 + α (α ∈ (0, 2)) order synchronous generator model is formulated. This paper develops a digital simulation model of a fractional-order single-machine infinite bus system and analyzes the impact of the order α on the synchronous generator system’s dynamic performance through disturbance simulation experiments. Experimental results demonstrate that under conventional disturbances, increasing α from 0.8 to 1.2 reduces the system oscillation period and frequency while enhancing mechanical oscillation suppression, whereas decreasing α to 0.8 accelerates the generator terminal voltage response, lowers electromagnetic power overshoot, and improves excitation control effectiveness. Full article

This research examines a fractional partial advection–dispersion model, incorporating both mobile and immobile components, employing the Hilbert reproducing algorithm under an appropriate Neumann constraint condition. To effectively formulate the model while adhering to the specified constraints, two suitable Hilbert spaces are constructed, with the time-fractional Caputo derivative being utilized in the model’s formulation. Alongside the convergence analysis, a derived approximate solution formula is presented, and a systematic computational algorithm is developed to effectively implement the solution methodology. Numerical applications related to the proposed model are presented, complemented by tables and graphical illustrations. In conclusion, significant results are analyzed, and directions for future research are outlined. Full article

With the rapid expansion of the internet and accelerated information dissemination, rumors pose a significant threat to social stability. Effective rumor control requires a thorough understanding of propagation mechanisms. This study develops a Caputo fractional-order IDSR rumor propagation model with time delays. The equilibrium points are identified, and the local asymptotic stability of the system at the positive equilibrium is analyzed. Additionally, the conditions for Hopf bifurcation and its impact on the rumor dynamics are examined. Numerical simulations validate the theoretical findings. Full article

Finite-time synchronization depends on the initial conditions of the system in question. However, the initial conditions of an actual system are often difficult to estimate or even unknown. Therefore, a more valuable and pressing problem is fixed-time synchronization (FTS). This paper addresses the issue of FTS for a specific class of fractional-order memristive fuzzy neural networks (FOMFNNs) that include both leakage and transmission delays. We have designed two distinct discontinuous control methodologies that account for these delays: a state feedback control scheme and a fractional-order adaptive control strategy. Leveraging differential inclusion theory and fractional-order differential inequalities, we derive several novel algebraic conditions that are independent of delay. These conditions ensure the FTS of drive–response FOMFNNs in the presence of leakage and transmission delays. Additionally, we provide an estimate for the upper bound of the settling time required to achieve FTS. Finally, to validate the feasibility and applicability of our theoretical findings, we present two numerical examples which are accompanied by simulations. Full article

The static shift effect is a distortion in electromagnetic data that severely impacts exploration results. Traditional static effect correction methods are often ineffective, prone to overcorrection or undercorrection, and make it difficult to accurately assess the applicability of the correction. Furthermore, some correction processes require additional data, which increases correction costs. This paper first presents the theoretical foundation for correcting static shift effects in the electric field components using magnetic field component information. Based on time-frequency electromagnetic exploration technology, a method is proposed to correct static shift effects in the electric field by using simultaneously collected magnetic field data, aiming to address the distortion issues caused by static shift effects in the electric field and apparent resistivity. The method is validated through both theoretical models and field data, demonstrating its excellent correction performance. Additionally, the paper introduces the use of the multifractal spectrum analysis algorithm to analyze profile measurement points and study the fractal dimension characteristics of static shift effects, providing an effective way to evaluate the appropriateness and potential overcorrection of the correction. Finally, the multifractal features of field data are discussed, validating the ability of the multifractal spectrum to identify subsurface electrical complexity. Full article

In this paper, the authors consider a problem with comprehensive properties in terms of form and content in the space ${\mathcal{L}}_2\left[\left(a,b\right)\times \left(\mathrm{c},\mathrm{d}\right)\right]\times C\left[0,T\right],T<1$. In terms of time form, we assume that the time phase delay is implicitly contained in a nonlinear differential integral equation. The positional part is considered in two dimensions, and the position’s kernel is a general singular kernel, many different forms of which will be derived. In terms of content, all of the previously established numerical techniques are only appropriate for studying special cases of the kernel separately but are not suitable for studying the general kernel. This led to the use of the Toeplitz matrix method, which deals with the kernel in its extended nonlinear form and the special kernels will be studied as applications of the method. Moreover, this method has the advantage of converting all single integrals into regular integrals that can be easily solved. Additionally, the researchers examine the solution’s existence, uniqueness, and convergence in this paper. The error and its stability are also studied. At the end of the research, the authors studied some numerical applications of some of the singular kernels derived from the general kernel, examining the approximation error in each application separately. Full article

The stability of differential equations is one of the most important aspects to consider in dynamical system theory. Chaotic systems were classified according to stability as multi-stable systems; systems with a single stable equilibrium; bi-stable systems; and, recently, mega-stable systems. Mega-stability refers to the infinity countable nested attractors of a periodically forced non-autonomous system. Many researchers attempted to present a simple mega-stable system. In this paper, we investigated the mega-stability of periodically damped non-autonomous differential systems with the following different order cases: integer and fractional. In the case of the integer order, we generalize the mega-stable system, such that the velocity is multiplied by a trigonometrical polynomial, and we present the necessary and sufficient conditions to generated countable infinity nested attractors. In the case of the fractional order, we obtained that the fractional order of periodically damped non-autonomous differential systems has infinity countable nested unstable attractors for some orders. The mega-instability was illustrated for two examples, showing the order effect on the trajectories. In addition, and to further recent work presenting simple high dimensional mega-stable chaotic systems, we introduce a 4D mega-stable hyperchaotic system, examining chaotic and hyperchaotic behaviors through Lyapunov exponents and bifurcation diagrams. Full article

Taylor expansion is a remarkable tool with broad applications in analysis, science, engineering, and mathematics. In this manuscript, we derive a proof of generalized Taylor expansion for polynomials and write its particular case in symmetric quantum calculus. In addition, we define a novel type of calculus that is called symmetric $(\mathit{p},\mathit{q})$- or dual basic symmetric quantum calculus. Moreover, we derive a symmetric $(\mathit{p},\mathit{q})$-Taylor expansion for polynomials based on this calculus. After that, we investigate Taylor’s formulae through an example. Furthermore, we define symmetric definite $(\mathit{p},\mathit{q})$-integral and derive a fundamental law of symmetric $(\mathit{p},\mathit{q})$-calculus. Finally, we derive the symmetric $(\mathit{p},\mathit{q})$-Cauchy formula for integrals that enables us to construct the fractional perspectives of $(\mathit{p},\mathit{q})$-symmetric integrals. Full article