Fractal and Fractional

This work investigates the fractional Dirac system that has transmission conditions, and its boundary condition contains an eigenparameter. Defining a convenient inner product space and a new operator that has the same eigenvalues as the considered problem, we demonstrate that the fractional Dirac system is symmetric in this space. Thus, we have reached some remarkable results for the spectral characteristics of the operator. Furthermore, in the next section of the study, the existence of solutions was examined. Full article

This paper focuses on the application of fractional calculus techniques in the identification and control of multivariable (multiple input—multiple output) systems (MIMO). By considering a previously reported experimental set-up similar to a greenhouse, this study proposes the open-loop identification of fractional order transfer functions relating to the controlled and manipulated variables, which were validated by experimental data. Afterward, the theoretical analysis of Fractional-order Proportional and Integral (FOPI) closed-loop control for this MIMO system was carried out. An important aspect concerns the use of Particle Swarm Optimization (PSO) metaheuristic algorithm for optimization tasks, both in parameter estimation and controller tuning. Moreover, comparisons with integer order models and controllers (IOPID-IMC) were performed. The results demonstrate the superior performance and robustness of the FOPI-PSO fractional control, which achieves up to 79.6% reduction in ITAE and 72.1% reduction in ITSE criteria. Without the need for explicit decouplers, the decentralized FOPI-PSO control structure demonstrated effective handling of interactions between the temperature and humidity control loops, simplifying the control design while maintaining performance. The fractional-order controllers exhibited robustness to measurement noise, as evidenced by stable and precise control responses in the presence of experimental uncertainties. Additionally, the optimized tuning of FOPI controllers implicitly compensated for disturbances and setpoint changes without requiring additional feedforward mechanisms. This study contributes to a better understanding of fractional calculus applications in designing FO–MIMO systems and provides a practical framework for addressing the identified gaps in the field. Full article

In this study, we introduce an innovative approach for addressing fractional partial differential equations (fPDEs) by combining Monte Carlo-based physics-informed neural networks (PINNs) with the cuckoo search (CS) optimization algorithm, termed PINN-CS. There is a further enhancement in the application of quasi-Monte Carlo assessment that comes with high efficiency and computational solutions to estimates of fractional derivatives. By employing structured sampling nodes comparable to techniques used in finite difference approaches on staggered or irregular grids, the proposed PINN-CS minimizes storage and computation costs while maintaining high precision in estimating solutions. This is supported by numerous numerical simulations to analyze various high-dimensional phenomena in various environments, comprising two-dimensional space-fractional Poisson equations, two-dimensional time-space fractional diffusion equations, and three-dimensional fractional Bloch–Torrey equations. The results demonstrate that PINN-CS achieves superior numerical accuracy and computational efficiency compared to traditional fPINN and Monte Carlo fPINN methods. Furthermore, the extended use to problem areas with irregular geometries and difficult-to-define boundary conditions makes the method immensely practical. This research thus lays a foundation for more adaptive and accurate use of hybrid techniques in the development of the fractional differential equations and in computing science and engineering. Full article

In the Southeastern Ordos Basin, the Chang 2 low-permeability sandstone reservoir of the Triassic Yanchang Formation is a typical heterogeneous reservoir. Quantitatively characterizing and analyzing its complex pore throat structure has become crucial for enhancing storage and production in the study area. The pore throat structure is a key factor influencing reservoir properties. To achieve this, a comprehensive suite of analytical techniques was employed, including cast thin section (CTS), scanning electron microscopy (SEM), cathodoluminescence (CL), X-ray diffraction (XRD), and mercury intrusion capillary pressure (MICP). This study quantitatively characterizes the pore size distribution of reservoirs in the Southeast Ordos Basin. Based on fractal theory, it clarifies the complexity of the pore throat structure and the degree of microscopic heterogeneity at different scales. Finally, this study reveals the correlation between fractal dimensions and storage and permeability capacities and analyzes the controlling factors. The findings indicate that the predominant lithotype in the study area is fine-grained feldspar sandstone, which develops pore types such as intergranular pores, dissolution pores, and microfractures. Based on the shapes of mercury injection curves and pore throat structural parameters, and in conjunction with SEM images, the samples are categorized into three types. Type I samples exhibit good pore throat connectivity and are characterized by a lattice model. Type II samples are characterized by a tubular pore throat model. Type III samples have poor pore throat connectivity and are characterized by an isolated model. The pore throat network of low-permeability sandstone is primarily composed of micropores (pore throat radius r < 0.1 μm), mesopores (0.1 < r < 1.0 μm), and macropores (r > 1.0 μm). The complexity of the reservoir pore throat structure was quantitatively characterized by fractal theory. The total fractal dimension (D) of all the samples is between 2 and 3, which indicates that the reservoir has capillary fractal characteristics. The average fractal dimension of micropores (D1) is 2.57, while that for mesopores (D2) and macropores (D3) is slightly higher, at an average of 2.68. This suggests that micropores have higher self-similarity and homogeneity. The fractal dimensions D1, D2, and D3 of the three types of reservoirs all exhibit a negative correlation with porosity and permeability. This shows that the more complex the pore throat structure is, the worse the storage and seepage capacity of the reservoir. For type I samples, the correlation of D3 with pore throat structural parameters such as entry pressure, skewness, and maximum mercury saturation is better than that of D2 and D1. For type II and type III samples, D2 shows a significant correlation with pore throat structural parameters. This indicates that the heterogeneity and complexity of mesopores are key factors influencing the pore throat structure of poor-quality reservoirs. Different mineral compositions have varying effects on the fractal characteristics of pore structures. Quartz, feldspar, and clay exert both negative and positive dual impacts on reservoir quality by altering the pore throat structure and the diagenetic processes. The mineral content exhibits a complex quadratic relationship with the fractal dimension. Moreover, micropores are more significantly influenced by the mineral content. The study of the relationship between the fractal dimension and physical properties, pore throat structural parameters, and mineral composition can improve the understanding of the reservoir quality of low-permeability reservoirs. This provides a theoretical basis for exploration and improving the recovery rate in the study area. Full article

The fractal characteristics of soil microstructures, including the size, distribution, and dynamic evolution process of soil particles, cracks, and intergranular pores, are important factors influencing the macroscopic physical properties of soils. Quantitative characterization, qualitative analysis, and the impact of fractal characteristics on macroscopic properties have been important research directions in recent years. This paper summarizes the research status on soil microstructure fractal characteristics, elaborating on the kinds of soil fractal characteristics, the calculation methods of fractal characteristic parameters, and the influence of fractal characteristics on macroscopic properties. Based on existing research results, this paper proposes that future research on soil microstructure fractal characteristics should focus on the following aspects: (1) advancing fractal characteristic research towards higher precision and multi-dimensionality to reveal the internal relations between soil fractal characteristics and macroscopic physical properties; (2) strengthening interdisciplinary collaboration to promote theoretical innovation in fractal analysis and build a more comprehensive system for studying the evolution of soil fractal characteristics; and (3) a close integration with engineering tests to promote the application and transformation of research results, providing valuable references for optimizing construction schemes and improving the service performance of engineering structures. Full article

To effectively mitigate resonance in dual-inertia servo inverter systems with a lightweight flexible shaft or coupling, the precise modeling of the dual-mass mechanism is essential. This paper proposes a fractional-order modeling and identification methodology tailored for a dual-mass loading permanent magnet synchronous motor (PMSM) servo inverter system. By extending the traditional integer-order model to a more precise fractional-order one, the accuracy of resonance capture can be enhanced within the dual-inertia mechanism. Model parameters are identified using an output error approach combined with the Levenberg–Marquardt (LM) algorithm for fractional-order identification. To validate the effectiveness of this proposed methodology, a PMSM servo inverter experimental platform was developed, and identification experiments were conducted on this platform. The experimental results demonstrate that the proposed fractional-order modeling and parameter identification method significantly improves the system characterization accuracy of the dual-inertia servo inverter system. Full article

Uniaxial compression tests were conducted on coal samples subjected to different dry–wet cycling treatments to investigate the damage and degradation mechanisms of coal samples under the dry–wet cyclic action of acidic, high-salinity solutions. The damage process of the coal samples was monitored in situ using acoustic emission (AE). The degradation evolution of the mechanical parameters and macroscopic failure modes with the number of cycles was analyzed. Based on the AE ringing parameters, the RA-AF distribution and the AE fractal dimension’s variation characteristics were studied. Additionally, scanning electron microscopy (SEM) was used to observe the microstructure of the coal samples. The results showed that with the increase in the number of dry–wet cycles, both the peak strength and elastic modulus of the coal samples exhibited varying degrees of degradation, and the failure mode gradually shifted from tensile failure to shear failure. AE ringing counts decreased progressively, while the proportion of shear cracks based on the RA-AF classification increased. At the same time, the mean AE fractal dimension of the coal samples increased, and the fractal dimension decreased with an increase in AE ringing counts. The sharp drop in fractal dimensions could serve as an early warning signal for a major failure in the coal samples. Furthermore, under the influence of dry–wet cycling in acidic, high-salinity solutions, defects such as pores and cracks in the microstructure of the coal samples became more pronounced, and the degradation effect continuously intensified. Full article

Based on the uniaxial compression tests and in situ CT scanning experiments of lignite with different dehydration times and the fractal theory, this paper qualitatively and quantitatively investigated the influence of the dehydration effect on the evolution of pore–fractures and the mechanical behavior of lignite under uniaxial compression conditions. The results show that the dehydration effect significantly affects the pre-peak deformation and post-peak failure behavior of lignite but has no significant impact on its peak strength. The pore–fracture parameters, such as the fractal dimension, surface porosity, and fracture volume, of three samples all exhibit an evolutionary pattern of “continuous decrease in the compaction and elastic stages–gradual increase in the plastic stage–sharp growth in the post-peak stage” with the dynamic evolution of the pore–fractures. However, the dehydration effect leads to an increase in the intensity of pore–crack evolution and a nonlinear rise in all the parameters characterizing the pore–crack complexity during uniaxial compression, which, in turn, leads to an increment in the fluctuation of the above evolutionary trends. The mechanism underlying the differential influence of the dehydration effect on the macroscopic mechanical behavior of lignite is follows: The dehydration effect non-linearly and positively affects the initial pore–fracture structure of lignite, thereby non-linearly and positively promoting the evolution of pore–fractures during the loading process. Nevertheless, since it fails to weaken the micro-mechanical properties of lignite and cannot form effective through-going fractures, it has no significant impact on the uniaxial compressive strength of the coal samples. The findings of this study can provide some references for the support design and deformation control of underground lignite roadways. Full article

The study of vibration equations of large membranes is crucial in various scientific and engineering fields. Analyzing the vibration equations of bridges, roofs, and spacecraft structures helps in designing structures that resist excessive oscillations and potential failures. Aircraft wings, parachutes, and satellite components often behave like large membranes. Understanding their vibration characteristics is essential for stability, efficiency, and durability. Studying large membrane vibration involves solving partial differential equations and eigenvalue problems, contributing to advancements in numerical methods and computational physics. In this paper, the Elzaki transformation decomposition method and the Shehu transformation decomposition method, along with inverse Elzaki and inverse Shehu transformations, are used to investigate the fractional vibration equation of a large membrane. The solutions are obtained in terms of Mittag–Leffler functions. Full article

This work integrates the fast Alikhanov method with a compact scheme to solve the time-fractional Kuramoto–Sivashinsky (KS) equation with the generalized Burgers’ type nonlinearity. Initially, the Alikhanov algorithm, designed to handle the Caputo fractional derivative on non-uniform time grids, effectively avoids the initial singularity. Additionally, the combination of the Alikhanov method with the sum-of-exponentials (SOE) technique significantly reduces both computational cost and memory requirements. By discretizing the spatial direction using a compact finite difference method, a fully discrete scheme is developed, achieving fourth-order convergence in the spatial domain. Stability and convergence are analyzed through energy methods. Several numerical examples are provided to validate the theoretical framework, demonstrating that the proposed algorithm is accurate, stable, and efficient. Full article

Against the backdrop of increasingly severe global climate challenges, various industries are in urgent need of developing materials that can both improve performance and reduce carbon emissions. In this study, carbon dioxide nanobubble water (CO₂NBW) was evaluated as an innovative additive for cemented backfill materials (CBMs), and its optimization effect on the mechanical properties and microstructure of the materials was explored. The effects of different concentrations of CO₂NBW on stress–strain behavior, compressive strength, and microstructure were studied by uniaxial compression tests and scanning electron microscopy (SEM) analysis. The results show that with changes in CO₂NBW concentration and fractal dimension, the uniaxial compressive strength (UCS), peak strain, and elastic modulus of the specimens first increase and then decrease. At the optimal concentration level (C = 3) and fractal dimension (2.4150–2.6084), UCS reaches a peak value of 24.88 MPa, which is significantly higher than the initial value (C = 1). The peak strain and elastic modulus also reach maximum values of 0.01231 and 3.005 GPa, respectively. When the fractal dimension was between 2.4150 and 2.6084, the microstructural optimization effect of CO₂NBW on CBM was most significant, which was reflected in the compactness of the internal pore structure and the thoroughness of the hydration degree. In addition, based on the close correlation between peak strain and elastic modulus and UCS, a damage constitutive model of CBM specimens considering the influence of CO₂NBW concentration and fractal dimension was constructed. The study also found that the damage of CBM specimens is normally distributed with strain, and the accumulated damage in the plastic deformation stage dominates the total damage. Full article

Multifractal analysis has been used in the exploration of soil grain size distributions (GSDs) in environmental and agricultural research. However, multifractal studies regarding the GSDs of sediments in braided delta-front are currently scarce. Open-source software designed for the realization of this technique has not yet been programmed. In this paper, the multifractal parameters of 61 GSDs from braided delta-front in the Paleogene Enping Formation in Huilu Low Uplift, Pearl River Mouth basin, are calculated and compared with traditional parameters. Multifractal generalized dimension spectrum curves are sigmoidal and decrease monotonically. Multifractal singularity spectrum curves are asymmetric, convex, and right-hook unimodal. The entropy dimension and singularity spectrum width ranges of silt-mudstones and gravelly sandstones are wider than those of fine and medium-coarse sandstones. The symmetry degree scopes from different lithologies are concentrated in distinguishing intervals. With the increase of grain sizes, the symmetry degree decreases overall. Both the symmetry degree and mean of GSDs are effective to distinguish the different lithologies from various depositional environments. A flexible and easy-to-use MATLAB (2021b)^® GUI (graphic user interface) package, MfGSD (Multifractal of GSD, V1.0), is provided to perform multifractal analysis on sediment GSDs. After raw GSDs imported into MfGSD, multifractal parameters are batch calculated and graphed in the interface. Then, all multifractal parameters can be exported to an Excel file, including entropy dimension, singularity spectrum, correlation dimension, symmetry degree of multifractal spectrum, etc. MfGSD is effective, and the multifractal parameters outputted from MfGSD are helpful to distinguish depositional environments of GSDs. MfGSD is open-source software that can be used to explore GSDs from various kinds of depositional environments, including water or wind deposits. Full article

In this paper, we focus on the multiplicity of positive solutions for a singular tempered fractional initial-boundary value problem with a p-Laplacian operator and a changing-sign perturbation term. By introducing a truncation function and combing with the properties of the solution of isomorphic linear equations, we transform the changing-sign tempered fractional initial-boundary value problem into a positive problem, and then the existence results of multiple positive solutions are established by the fixed point theorem in a cone. It is worth noting that the changing-sign perturbation term only satisfies the weaker Carathèodory conditions, which implies that the perturbation term ℏ can be allowed to have an infinite number of singular points; moreover, the value of the changing-sign perturbation term can tend to negative infinity in some singular points. Full article

Freeze–thaw cycles and the soil compaction coefficient ($\lambda \mathrm{c}$) have significant influence on the plastic strain for the foundation of underground structures in seasonal permafrost regions. Understanding the microstructural evolution of freeze–thawed soil is pivotal for assessing the long-term settlement of infrastructure foundation under repeated train loading. This study investigates the impacts of freeze–thaw cycles and $\lambda \mathrm{c}$ on the plastic strain and pore size distribution (PSD), as well as fractal characteristics, of soft clay via a set of cyclic triaxial tests and nuclear magnetic resonance (NMR) analyses. Fractal theory was adopted to analyze the heterogeneity of soil specimens. The results showed that an increase in $\lambda \mathrm{c}$ could efficiently alleviate the cumulative plastic strain. It also decreased the proportion of large pores and facilitated the generation of small and medium-sized pores. The analysis of the NMR test demonstrated that the freeze–thaw cycle led to the disruption of the soil’s microporous structure. Moreover, a higher value of $\lambda \mathrm{c}$ encouraged the formation of a more intricate and uniform pore structure. This, in turn, increased the fractal dimension, enhanced the structural heterogeneity, and thereby improved the soil’s structural complexity and its resistance to deformation. These findings underscore the significance of achieving optimal compaction levels to bolster soil stability under freeze–thaw conditions, provide valuable guidance for infrastructure design in permafrost regions, and help to ensure the durability and stability of transportation networks, such as railways and roads, over time. Full article

Self-organized criticality (SOC) describes natural systems spontaneously tuned at equilibrium yet capable of catastrophic events or avalanches. The cardiovascular system, characterized by homeostasis and vasovagal syncope, is a prime candidate for SOC. Power laws are the cornerstone for demonstrating the presence of SOC. This study aimed to provide evidence of power-law behavior in cardiovascular dynamics. We analyzed beat-by-beat blood pressure and heart rate data from seven healthy subjects in the head-up position over 40 min. Cardiovascular avalanches were quantified by their duration (in beats), and symbolic sequences were identified. Five types of distributions were assessed for power-law behavior: Gutenberg–Richter, classical Zipf, modified Zipf, Zipf of time intervals between avalanches, and Zipf of symbolic sequences. A three-stage approach was used to show power laws: (1) regression coefficient r > 0.95, (2) comparison with randomized data, and (3) Clauset’s statistical test for power law. Numerous avalanches were identified (13.9 ± 0.8 per minute). The classical and modified Zipf distributions met all the criteria (r = 0.99 ± 0.00 and 0.98 ± 0.01, respectively), while the others showed partial agreement, likely due to the limited data duration. These findings reveal that Zipf’s distributions of cardiovascular avalanches strongly support SOC, shedding light on the organization of this complex system. Full article

In this work, we introduce an innovative analytical–numerical approach to solving nonlinear fractional differential equations by integrating the homotopy perturbation method with the new integral transform. The Kawahara equation and its modified form, which is significant in fluid dynamics and wave propagation, serve as test cases for the proposed methodology. Additionally, we apply the fractional new integral transform–homotopy perturbation method (FNIT-HPM) to a nonlinear system of coupled Burgers’ equations, further demonstrating its versatility. All calculations and simulations are performed using Mathematica 12 software, ensuring precision and efficiency in computations. The FNIT-HPM framework effectively transforms complex fractional differential equations into more manageable forms, enabling rapid convergence and high accuracy without linearization or discretization. By evaluating multiple case studies, we demonstrate the efficiency and adaptability of this approach in handling nonlinear systems. The results highlight the superior accuracy of the FNIT-HPM compared to traditional methods, making it a powerful tool for addressing complex mathematical models in engineering and physics. Full article

This study introduces a novel one-dimensional Fractional Time–Space Stochastic Advection–Diffusion Equation that revolutionizes the modeling of moisture transport within atmospheric boundary layers adjacent to oceanic surfaces. By synthesizing fractional calculus, advective transport mechanisms, and pink noise stochasticity, the proposed model captures the intricate interplay between temporal memory effects, non-local turbulent diffusion, and the correlated-fluctuations characteristic of complex ocean–atmosphere interactions. The framework employs the Caputo fractional derivative to represent temporal persistence and the fractional Laplacian to model non-local turbulent diffusion, and incorporates a stochastic term with a $1/f$ power spectral density to simulate environmental variability. An efficient numerical solution methodology is derived utilizing complementary Fourier and Laplace transforms, which elegantly converts spatial fractional operators into algebraic expressions and yields closed-form solutions via Mittag–Leffler functions. This method’s application to a benchmark coastal domain demonstrates that stronger advection significantly increases the spatial extent of conditions exceeding fog formation thresholds, revealing advection’s critical role in moisture transport dynamics. Numerical simulations demonstrate the model’s capacity to reproduce both anomalous diffusion phenomena and realistic stochastic variability, while convergence analysis confirms the numerical scheme’s robustness against varying noise intensities. This integrated fractional stochastic framework substantially advances atmospheric moisture modeling capabilities, with direct applications to meteorological forecasting, coastal climate assessment, and environmental engineering. Full article

Slope deformation poses significant risks to infrastructure, ecosystems, and human safety, making early and accurate predictions essential for mitigating slope failures and landslides. In this study, we propose a novel approach that integrates a fractional-order grey model (FOGM) with particle swarm optimization (PSO) to determine the optimal fractional order, thereby enhancing the model’s accuracy, even with limited and fluctuating data. Additionally, we employ a k-means clustering technique to account for both temporal and spatial variations in multi-point monitoring data, which improves the model’s ability to capture the relationships between monitoring points and increases prediction relevance. The model was validated using displacement data collected from 12 monitoring points on a slope located in Qinghai Province near the Yellow River, China. The results demonstrate that the proposed model outperforms the traditional statistical model and artificial neural networks, achieving a significantly higher coefficient of determination $R^2$ up to 0.9998 for some monitoring points. Our findings highlight that the model maintains robust performance even when confronted with data of varying quality—a notable advantage over conventional approaches that typically struggle under such conditions. Overall, the proposed model offers a robust and data-efficient solution for slope deformation prediction, providing substantial potential for early warning systems and risk management. Full article

To address the detrimental effects of high mud content on cemented sand and gravel (CSG) materials, this study focuses on CSG materials with different contents of alkali-activated rice husk ash (RHA). The microscopic enhancement mechanism of mechanical properties of CSG materials with alkali-activated RHA were studied through the experiments. Based on the changes in microstructure, pore sizes, and mineral composition, the microstructural features of CSG materials are quantitatively characterized using pore characteristic parameters and fractal dimensions, revealing the mechanism by which alkali-activated RHA improves the mechanical properties of CSG materials. The results indicate that alkali-activated RHA effectively enhances the mechanical properties of CSG materials. As the RHA content increases, the compressive strength and elastic modulus initially increase and then decrease, while the failure strain first decreases and then increases. The failure mode transitions from splitting failure to shear failure, with the optimal mechanical performance at 5% RHA content. Microscopic experimental analysis found that adding an appropriate amount of RHA can promote the generation of cementitious substances, improve the internal pore structure, and increase the fractal dimension. However, excessive RHA can adsorb moisture, inhibit part of the hydration reactions and reduce the fractal dimension. Under the action of alkali activators, the activity of RHA is enhanced, generating more cementitious materials and significantly improving the pore-filling effect within the material, especially affecting the capillary pores. Full article