Fractal Fract., Volume 9, Issue 6 (June 2025) – 68 articles

A clear method is provided to explain a new approach for solving systems of fractional Navier–Stokes equations (SFNSEs) using initial conditions (ICs) that rely on the Elzaki transform (ET). A few steps show the technique’s validity and utility for handling SFNSE solutions. For fractional derivatives, the Caputo sense is used. This method does not need discretization or limiting assumptions and may be used to solve both linear and nonlinear SFNSEs. By eliminating round-off mistakes, the technique reduces the need for numerical calculations. Using examples, the new technique’s accuracy and efficacy are illustrated. Full article

In this paper, we investigate a class of fuzzy partially differentiable models for Caputo–Hadamard-type Goursat problems with generalized Hukuhara difference, which have been widely recognized as having a significant role in simulating and analyzing various kinds of processes in engineering and physical sciences. By transforming the fuzzy partially differentiable models into equivalent integral equations and employing classical Banach and Schauder fixed-point theorems, we establish the existence and uniqueness of solutions for the fuzzy partially differentiable models. Furthermore, in order to overcome the complexity of obtaining exact solutions of systems involving Caputo–Hadamard fractional derivatives, we explore numerical approximations based on trapezoidal and Simpson’s rules and propose three numerical examples to visually illustrate the main results presented in this paper. Full article

A Bayesian network (BN) is an uncertainty processing model that simulates human cognitive thinking on the basis of probability theory and graph theory. Its network topology is a directed acyclic graph (DAG) that can be manually constructed through expert knowledge or automatically generated through data learning. However, the acquisition of expert knowledge faces problems such as excessively high labor costs, limited expert experience, and the inability to solve large-scale and highly complex DAGs. Moreover, the current data learning methods also face the problems of low computational efficiency or decreased accuracy when dealing with large-scale and highly complex DAGs. Therefore, we consider mining fragmented knowledge from the data to alleviate the bottleneck problem of expert knowledge acquisition. This generated fragmented knowledge can compensate for the limitations of data learning methods. In our work, we propose a new binary stochastic fractal search (SFS) algorithm to learn DAGs. Moreover, a new feature selection (FS) method is proposed to mine fragmented knowledge. This fragmented prior knowledge serves as a soft constraint, and the acquired expert knowledge serves as a hard constraint. The combination of the two serves as guidance and constraints to enhance the performance of the proposed SFS algorithm. Extensive experimental analysis reveals that our proposed method is more robust and accurate than other algorithms. Full article

Since polynomial regression models are generally quite reliable for data that can be handled using a linear system, it is important to note that in some cases, they may suffer from overfitting during the training phase. This can lead to negative values of the coefficient of determination $R^2$ when applied to unseen data. To address this issue, this work proposes the partial implementation of fractional operators in polynomial regression models to construct a fractional regression model. The aim of this approach is to mitigate overfitting, which could potentially improve the $R^2$ value for unseen data compared to the conventional polynomial model, under the assumption that this could lead to predictive models with better performance. The methodology for constructing these fractional regression models is presented along with examples applicable to both Riemann–Liouville and Caputo fractional operators, where some results show that regions with initially negative or near-zero $R^2$ values exhibit remarkable improvements after the application of the fractional operator, with absolute relative increases exceeding $800\%$ on unseen data. Finally, the importance of employing sets in the construction of the fractional regression model within this methodological framework is emphasized, since from a theoretical standpoint, one could construct an uncountable family of fractional operators derived from the Riemann–Liouville and Caputo definitions that, although differing in their formulation, would yield the same regression results as those shown in the examples presented in this work. Full article

Evolution equations with fractional-time derivatives and singular memory kernels are used for modeling phenomena exhibiting hereditary properties, as they effectively incorporate memory effects into their formulation. Time-fractional partial integro-differential equations (FPIDEs) represent a significant class of such evolution equations and are widely used in diverse scientific and engineering fields. In this study, we use the $\mathrm{sinc}$-collocation and iterative Laplace transform methods to solve a specific FPIDE with a weakly singular kernel. Specifically, the $\mathrm{sinc}$-collocation method is applied to discretize the spatial domain, while a combination of numerical techniques is utilized for temporal discretization. Then, we prove the convergence analytically. To compare the two methods, we provide two examples. We notice that both the $\mathrm{sinc}$-collocation and iterative Laplace transform methods provide good approximations. Moreover, we find that the accuracy of the methods is influenced by fractional order $\alpha \in (0,1)$ and the memory-kernel parameter $\beta \in (0,1)$. We observe that the error decreases as $\beta$ increases, where the kernel becomes milder, which extends the single-value study of $\beta =1/2$ in the literature. Full article

This study is organized to introduce the concept of center–radius $\left(cr\right)$-ordered fuzzy number-valued convex mappings. Based on this class of mappings, we have initiated the idea of fuzzy number-valued extended $cr$-ℏ convex mappings incorporating control mapping ℏ. Furthermore, several potential new classes of convexity will be provided to discuss its generic nature. Also, some essential properties, criteria, and detailed characterizations through Jensen’s and Hermite–Hadamard-like inequalities are provided, incorporating Riemann–Liouville fractional operators, which are defined by $\rho$-level mappings. To validate the proposed fractional bounds through simulations, we consider both triangular and trapezoidal fuzzy numbers. Our results are based on totally ordered fuzzy-valued mappings, which are new and generic. The under-consideration class also includes a blend of new classes of convexity, which are controlled by non-negative mapping ℏ. In previous studies, the researchers have focused on different partially ordered relations. Full article

The time-fractional generalized Burger–Fisher equation (TF-GBFE) is utilized in many physical applications and applied sciences, including nonlinear phenomena in plasma physics, gas dynamics, ocean engineering, fluid mechanics, and the simulation of financial mathematics. This mathematical expression explains the idea of dissipation and shows how advection and reaction systems can work together. We compare the homotopy perturbation transform method and the new iterative method in the current study. The suggested approaches are evaluated on nonlinear TF-GBFE. Two-dimensional (2D) and three-dimensional (3D) figures are displayed to show the dynamics and physical properties of some of the derived solutions. A comparison was made between the approximate and accurate solutions of the TF-GBFE. Simple tables are also given to compare the integer-order and fractional-order findings. It has been verified that the solution generated by the techniques given converges to the precise solution at an appropriate rate. In terms of absolute errors, the results obtained have been compared with those of alternative methods, including the Haar wavelet, OHAM, and q-HATM. The fundamental benefit of the offered approaches is the minimal amount of calculations required. In this research, we focus on managing the recurrence relation that yields the series solutions after a limited number of repetitions. The comparison table shows how well the methods work for different fractional orders, with results getting closer to precision as the fractional-order numbers get closer to integer values. The accuracy of the suggested techniques is greatly increased by obtaining numerical results in the form of a fast-convergent series. Maple is used to derive the approximate series solution’s behavior, which is graphically displayed for a number of fractional orders. The computational stability and versatility of the suggested approaches for examining a variety of phenomena in a broad range of physical science and engineering fields are highlighted in this work. Full article

This research proposes a fractional-order adaptive neural control scheme using an optimized backstepping (OB) approach to address strict-feedback nonlinear systems with uncertain control directions and predefined performance requirements. The OB framework integrates both fractional-order virtual and actual controllers to achieve global optimization, while a Nussbaum-type function is introduced to handle unknown control paths. To ensure convergence to desired accuracy within a prescribed time, a fractional-order dynamic-switching mechanism and a quartic-barrier Lyapunov function are employed. An input-to-state practically stable (ISpS) auxiliary signal is designed to mitigate unmodeled dynamics, leveraging classical lemmas adapted to fractional-order systems. The study further investigates a decentralized control scenario for large-scale stochastic nonlinear systems with uncertain dynamics, undefined control directions, and unmeasurable states. Fuzzy logic systems are employed to approximate unknown nonlinearities, while a fuzzy-phase observer is designed to estimate inaccessible states. The use of Nussbaum-type functions in decentralized architectures addresses uncertainties in control directions. A key novelty of this work lies in the combination of fractional-order adaptive control, fuzzy logic estimation, and Nussbaum-based decentralized backstepping to guarantee that all closed-loop signals remain bounded in probability. The proposed method ensures that system outputs converge to a small neighborhood around the origin, even under stochastic disturbances. The simulation results confirm the effectiveness and robustness of the proposed control strategy. Full article

The complementary dual of entropy has received significant attention in the literature. Due to the emergence of many generalizations and extensions of entropy, the need to generalize the complementary dual of uncertainty arose. This article develops the residual cumulative generalized fractional extropy as a generalization of the residual cumulative complementary dual of entropy. Many properties, including convergence, transformation, bounds, recurrence relations, and connections with other measures, are discussed. Moreover, the proposed measure’s order statistics and stochastic order are examined. Furthermore, the dynamic design of the measure, its properties, and its characterization are considered. Finally, nonparametric estimation via empirical residual cumulative generalized fractional extropy with an application to blood transfusion is performed. Full article

This study enhances the classical deterministic SIR model by incorporating soliton-like dynamics and gradient-induced diffusion, effectively capturing the complex spatiotemporal patterns of disease transmission within nomadic populations. The proposed model incorporates an advection–diffusion mechanism that modulates the spatial gradients in infection dynamics, transitioning from highly localized infection peaks to distributed infection fronts. We discussed the role of diffusion coefficients in shaping the spatial distribution of susceptible, infected, and recovered populations, as well as the impact of gradient-induced advection in mitigating epidemic intensity. Numerical simulations demonstrate the effects of varying key parameters such as transmission rates, recovery rates, and advection–diffusion coefficients on the epidemic’s progression. The soliton-like dynamics ensure the stability of infection waves over time, specifying targeted intervention strategies such as localized quarantines and vaccination campaigns. This model underscores the critical importance of spatial heterogeneity and mobility patterns in managing infectious diseases. The applicability of the model has been tested using the AIDS data from the last 25 years. Full article

The acidification modification treatment of coal is a key technical means to improve the permeability of coal seams and enhance the efficiency of coalbed methane extraction. Yet, current acidic fracturing fluids are highly corrosive, corroding downhole pipelines and contaminating groundwater. By compounding environmentally friendly and non-polluting acidic fracturing fluids and combining fractal theory and the Frenkel–Halsey–Hill (FHH) model, this paper systematically investigates their effects on the pore structure, permeability, and mechanical properties of coal bodies. It was found that the complex acid treatment significantly reduced the surface fractal dimension D₁ and spatial fractal dimension D₂ of the coal samples and optimized pore connectivity, thus improving gas transport efficiency. Meanwhile, a static splitting test and digital image analysis showed that the fracture evolution pattern of the treated coal samples changed from a centralized strain extension of the original coal to a discrete distribution, peak stress and strain were significantly reduced, and permeability was significantly increased. These findings can offer dramatic support for the optimal optimization of acidic fracturing fluids. Full article

The Hurst phenomenon is regarded as an intrinsic characteristic of many natural processes closely related to high uncertainty and long-term persistence. Temperature and precipitation are the two important meteorological factors characterizing the climate conditions of different regions. Analyzing the Hurst phenomenon in precipitation and temperature are crucial for understanding the long-term dynamics of our climate system. This study examines the annual mean temperature (AMT) and annual total precipitation (ATP) series for regions across all the land areas of the world, using both gridded climate data and ground station records. The results demonstrate that, in most regions, the Hurst exponent of AMT is higher than that of ATP, particularly with larger spatial scales of averaging. Like ATP, the Hurst exponents of AMT also increase with the spatial scale of averaging. Unlike AMT, ATP is more controlled by local meteorological conditions which tend to weaken its long-term persistence. Moreover, the cumulative departure from the mean series of ATP is much more variable across different regions, whereas those of AMT for different regions are more similar. What is identified for the first time in this study is the strong similarity in the cumulative departure from the mean patterns of regionally averaged and individual stations’ ATP and AMT series over many regions of the world. At most of these regions and stations where such similarities are identified, more than half have confirmed that AMT is the Granger cause of ATP variations. Moreover, the fluctuation functions obtained in multifractal detrended cross-correlation analysis exhibit approximately linear behavior in the log–log spaces across all regions at both global and continental scales, indicating that ATP and AMT series are long-range cross-correlated. Full article

In this paper, we apply Grünwald–Letnikov-type fractional-order calculus to simulate the growth of Serbia’s gross domestic product (GDP). We also compare the fractional-order model’s results with those of a similar integer-order model. The significance of variables is assessed by the Akaike Information Criterion (AIC). The research demonstrates that the Grünwald–Letnikov fractional-order model provides a more accurate representation compared to the standard integer-order model and performs very accurately in predicting GDP values. Full article

This study investigates the magnetic fields produced by a three-solenoid system configuration using both traditional numerical solvers and fractional integral methods. We focus on the role of mesh resolution in influencing simulation accuracy, examining coils with dimensions 80 mm × 160 mm and a radius of 15.5 mm, each carrying a current of 200 A. Magnetic field behavior is analyzed along a line parallel to the central axis at a distance equal to half the solenoid’s radius. The fractional integral formulation employed provides a refined understanding of field variations, especially in off-axis regions. Comparisons with the Poisson solver highlight consistency across methods and suggest pathways for further optimization. The results support the potential of fractional approaches in advancing electromagnetic field modeling, particularly in accelerator and beamline applications. Full article

In this study, we analyze a time-fractional spatiotemporal $SIR$ model in a specific area $\Omega$. Taking into account the available resources, vaccines are allocated to region ${\omega }_1\subseteq \Omega$ and treatments to region ${\omega }_2\subseteq \Omega$, which may or may not coincide. Our objective is to minimize infections and costs by implementing an optimal regional control strategy. We establish the existence of optimal controls and related solutions, providing a characterization of optimal control in terms of state and adjoint functions. We employ the forward–backward sweep method to solve the associated optimality system numerically. The findings indicate that a combined strategy of vaccination and treatment is more effective in reducing disease transmission from adjacent regions. Full article

The deformation control of surrounding rock in the combustion air zone is crucial for the safety and efficiency of underground coal gasification (UCG) projects. Coal-bearing sandstone, a common surrounding rock in UCG chambers, features a brittle structure composed mainly of quartz, feldspar, and clay minerals. Its mechanical behavior under high-temperature and dynamic loading is complex and significantly affects rock stability. To investigate the deformation and failure mechanisms under thermal–dynamic coupling, this study conducted uniaxial impact compression tests using a high-temperature split Hopkinson pressure bar (HT-SHPB) system. The focus was on analyzing mechanical response, energy dissipation, and fragmentation characteristics under varying temperature and strain rate conditions. The results show that the dynamic elastic modulus, compressive strength, fractal dimension of fragments, energy dissipation density, and energy consumption rate all increase initially with temperature and then decrease, with inflection points observed at 400 °C. Conversely, dynamic peak strain first decreases and then increases with rising temperature, also showing a turning point at 400 °C. This indicates a shift in the deformation and failure mode of the material. The findings provide critical insights into the thermo-mechanical behavior of coal-bearing sandstone under extreme conditions and offer a theoretical basis for designing effective deformation control strategies in underground coal gasification projects. Full article

In this study, we define the $k$-Cullen, $k$-Cullen–Lucas, and Modified $k$-Cullen sequences, and certain terms in these sequences are given. Then, we obtain the Binet formulas, generating functions, summation formulas, etc. In addition, we examine the relations among the terms of the $k$-Cullen, $k$-Cullen–Lucas, Modified $k$-Cullen, Cullen, Cullen–Lucas, Modified Cullen, $k$-Woodall, $k$-Woodall–Lucas, Modified $k$-Woodall, Woodall, Woodall–Lucas, and Modified Woodall sequences. The generating functions were derived and analyzed, especially for cases where Fibonacci numbers were assigned to parameter $k.$ Graphical representations of the generating functions and their logarithmic transformations revealed interesting growth trends and convergence behavior. Further, by multiplying the generating functions with exponential expressions such as $e^k$, we explored the self-similar nature and mirrored dynamics among the sequences. Specifically, it was observed that the Modified Cullen sequence exhibited a symmetric and inverse-like resemblance to the Cullen and Cullen–Lucas sequences, suggesting the presence of deeper structural dualities. Additionally, indefinite integrals of the generating functions were computed and visualized over a range of Fibonacci-indexed k values. These integral-based graphs further reinforced the phenomenon of symmetry and self-similarity, particularly in the Modified Cullen sequence. A key insight of this study is the discovery of a structural duality between the Modified Cullen and standard Cullen-type sequences, supported both algebraically and graphically. This duality suggests new avenues for analyzing generalized recursive sequences through generating function transformations. This observation provides new insight into the structural behavior of generalized Cullen-type sequences. Full article

This paper investigates the synchronization control problem for delayed fractional-order neural networks (DFONNs) with mismatched parameters. A novel synchronization behavior termed quasi-Mittag-Leffler projective synchronization (QMLPS) is studied. The core contribution of this work lies in the following: (1) The time delay and mismatched parameters between driven and response systems are considered, which is more general. Both static controllers and adaptive controllers are designed to synchronize the DFONNs. (2) The synchronization errors are estimated, and the rate of convergence is clarified description. By using the Lyapunov stability theory and some significant fractional-order differential inequalities, some sufficient conditions for DFONNs are derived under two kinds of control methods; furthermore, the bound of synchronization errors is estimated by the Mittag-Leffler function. Quantitative numerical simulations have demonstrated the superiority of our controller. Compared to existing results, the QMLPS introduced in this paper is more general, incorporating many existing synchronization concepts. The numerical simulation section verifies the effectiveness of the theoretical results, providing several types of synchronization behaviors of the controlled system under both mismatched and matched parameter conditions, and it also demonstrates the accuracy of the theoretical estimation of synchronization error bounds. Full article

This paper addresses the existence of solutions to a Hadamard fractional differential equation of arbitrary order on an infinite interval, subject to integral boundary conditions that incorporate both Riemann–Stieltjes integrals and Hadamard fractional derivatives. Due to the presence of nontrivial solutions in the associated homogeneous boundary value problem, the problem is classified as resonant. The Mawhin continuation theorem is utilized to derive the main findings. Full article

The present work proposes a novel fractional-order multifunction filter topology in current-mode (CM), which is designed based on the Modified Current Feedback Operational Amplifier (MCFOA). The proposed design simultaneously generates fractional-order low-pass (FO-LPF), high-pass (FO-HPF), and band-pass (FO-BPF) outputs while utilizing an optimized set of essential active and passive elements, thereby ensuring simplicity, cost efficiency, and compatibility with integrated circuits (ICs). The fractional-order feature allows precise control over the transition slope between the passband and the stopband, enhancing design flexibility. PSpice simulations validated the filter’s theoretical performance, confirming a 1 kHz cut-off frequency, making it suitable for VLF applications such as military communication and submarine navigation. Monte Carlo analyses demonstrate robustness against parameter variations, while a low THD, a wide dynamic range, and low power consumption highlight its efficiency for high-precision, low-power applications. This work offers a practical and adaptable approach to fractional-order circuit design, with significant potential in communication, control, and biomedical systems. Full article

Repeated mining under insufficiently collapsed gobs is a complex process in underground mining and is associated with safety hazards such as ground collapse and subsidence. The effect of layer spacing on the fracture network evolution and fluid transport mechanisms in rock strata during this process has not been systematically studied. In this work, the discrete element method was employed to analyze the fracture development and seepage evolution of surrounding rocks in the Nanliang coal mine across varying layer spacings (5, 20, 35, 50, and 65 m). A systematic evaluation of the rock mass integrity was conducted through damage coefficient quantification. The key findings revealed that an increase in the layer spacing progressively reduced the damage coefficients in both the overburden strata above the goaf and in the interlayer formations ahead of the working face, accompanied by reduced fracture propagation intensity. Shear failure mechanisms dominated throughout the mining process. Fractal characteristics of the fractures intensified with the advance of the working face, while the hydraulic conductivity and interstitial pressure in the interlayer strata exhibited declining trends with reduced attenuation rates. Our findings provide critical insights for ensuring the safety and improving the efficiency of repeated mining under insufficiently collapsed gobs. Full article

This study presents, for the first time, a new class of interval-valued superquadratic stochastic processes and examines their core properties through the lens of the center-radius total order relation on intervals. These processes serve as a powerful tool for modeling uncertainty in stochastic systems involving interval-valued data. By utilizing their intrinsic structure, we derive sharpened versions of Jensen-type and Hermite–Hadamard-type inequalities, along with their fractional extensions, within the framework of mean-square stochastic Riemann–Liouville fractional integrals. The theoretical findings are validated through extensive graphical representations and numerical simulations. Moreover, the applicability of the proposed processes is demonstrated in the domain of information theory by constructing novel stochastic divergence measures and Shannon’s entropy grounded in interval calculus. The outcomes of this work lay a solid foundation for further exploration in stochastic analysis, particularly in advancing generalized integral inequalities and formulating new stochastic models under uncertainty. Full article

The study of fractional differential equations is gaining increasing significance due to their wide-ranging applications across various fields. Different methods, including fixed-point theory, variational approaches, and the lower and upper solutions method, are employed to analyze the existence and uniqueness of solutions to fractional differential equations. This paper investigates the existence and uniqueness of solutions to a class of nonlinear fractional differential equations involving mixed Caputo–Riemann fractional derivatives with integral initial conditions, set within a Banach space. Sufficient conditions are provided for the existence and uniqueness of solutions based on the problem’s parameters. The results are derived by constructing the Green’s function for the initial value problem. Schauder’s fixed-point theorem is used to prove existence, while Banach’s contraction mapping principle ensures uniqueness. Finally, an example is given to demonstrate the practical application of the results. Full article

This study introduces an innovative analytical solution to the time-fractional Cattaneo heat conduction equation, which models photothermal transport in metallic thin films subjected to short laser pulse irradiation. The model integrates the Caputo fractional derivative of order 0 < p ≤ 1, addressing non-Fourier heat conduction characterized by finite wave speed and memory effects. The equation is nondimensionalized through suitable scaling, incorporating essential elements such as a newly specified laser absorption coefficient and uniform initial and boundary conditions. A hybrid approach utilizing the finite Fourier cosine transform (FFCT) in spatial dimensions and the Laplace transform in temporal dimensions produces a closed-form solution, which is analytically inverted using the two-parameter Mittag–Leffler function. This function inherently emerges from fractional-order systems and generalizes traditional exponential relaxation, providing enhanced understanding of anomalous thermal dynamics. The resultant temperature distribution reflects the spatiotemporal progression of heat from a spatially Gaussian and temporally pulsed laser source. Parametric research indicates that elevating the fractional order and relaxation time amplifies temporal damping and diminishes thermal wave velocity. Dynamic profiles demonstrate the responsiveness of heat transfer to thermal and optical variables. The innovation resides in the meticulous analytical formulation utilizing a realistic laser source, the clear significance of the absorption parameter that enhances the temperature amplitude, the incorporation of the Mittag–Leffler function, and a comprehensive investigation of fractional photothermal effects in metallic nano-systems. This method offers a comprehensive framework for examining intricate thermal dynamics that exceed experimental capabilities, pertinent to ultrafast laser processing and nanoscale heat transfer. Full article

Sandstone reservoirs exhibit strong heterogeneity and complex microscopic pore structures, presenting challenges for quantitative characterization. This study investigates the Chang 8 tight sandstone reservoir in the Jiyuan, Ordos Basin through analyses of its physical properties, high-pressure mercury injection (HPMI), casting thin sections (CTS), and scanning electron microscopy (SEM). Deep learning techniques were employed to extract the geometric parameters of the pores from the SEM images. Fractal geometry was applied for the combined quantitative characterization of pore parameters and fractal dimensions of the tight sandstone. This study also analyzed the correlations between the fractal dimensions, sample properties, pore structure, geometric parameters, and mineral content. The results indicate that the HPMI-derived fractal dimension (D_MIP) reflects pore connectivity and permeability. D_MIP gradually increases from Type I to Type III reservoirs, indicating deteriorating pore connectivity and increasing reservoir heterogeneity. The average fractal dimensions of the small and large pore-throats are 2.16 and 2.52, respectively, indicating greater complexity in the large pore-throat structures. The SEM-derived fractal dimension (D_SEM) reflects the diversity of pore shapes and the complexity of the micro-scale geometries. As the reservoir quality decreases, the pore structure becomes more complex, and the pore morphology exhibits increased irregularity. D_MIP and D_SEM values range from 2.21 to 2.49 and 1.01 to 1.28, respectively, providing a comprehensive quantitative characterization of multiple pore structure characteristics. The fractal dimension shows negative correlations with permeability, porosity, median radius, maximum mercury intrusion saturation, mercury withdrawal efficiency, and sorting factor, while showing a positive correlation with median and displacement pressures. Among these factors, the correlations with the maximum mercury intrusion saturation and sorting factor are the strongest (R² > 0.8). Additionally, the fractal dimension is negatively correlated with pore circularity and major axis length, but positively correlated with pore perimeter, aspect ratio, and solidity. A higher proportion of circular pores and fewer irregular or long-strip pores correspond to lower fractal dimensions. Furthermore, mineral composition influences the fractal dimension, showing negative correlations with feldspar, quartz, and chlorite concentrations, and a positive correlation with carbonate content. This study provides new perspectives for the quantitative characterization of pore structures in tight sandstone reservoirs, enhances the understanding of low-permeability formation reservoir performance, and establishes a theoretical foundation for reservoir evaluation and exploration development in the study area. Full article

This paper constructs a fundamental mathematical model to depict the therapeutic effects of two drugs on breast cancer patients. The model is described by fractional order differential equations with two control variables. Two scenarios are considered: the constant control and the optimal control. For the constant control scenario, the existence and uniqueness of the solution of the system are proved by using the fixed point theorem and combining with the Caputo–Fabrizio fractional derivative; then, the sufficient conditions for the existence and stability of the system’s equilibriums are derived. For the optimal control scenario, the optimal control solution is obtained by using the Pontryagin’s maximum principle. To further validate the effectiveness of the theoretical results, numerical simulations were conducted. The results show that the parameters have significant sensitivity to the dynamic behavior of the system. Full article

In this research, we consider the inverse problem for the wave equation under an unknown initial condition. A generalized solution to the direct problem was formulated, its correctness was established, and the stability assessment was obtained. The inverse problem was reduced to an optimization problem, where the objective function was minimized using gradient methods, including the accelerated Nesterov algorithm. The conjugate problem was constructed, and the functional gradient was computed, while the existence of the Frechet derivative was proved. For the first time, the quaternion Fourier transform (QFT) was applied to the numerical solution of a direct problem, making it possible to analyze multidimensional wave processes more efficiently. A computational experiment was carried out, which demonstrated that if there is insufficient additional information, the restoration of the initial condition is incomplete. The introduction of the second boundary condition makes it possible to significantly improve the accuracy and stability of the solution. The results confirm the importance of an integrated approach and the availability of sufficient a priori information when solving inverse problems. Full article

Chaotic behavior and memory-dependent dynamics in fractional-order brushless DC motors (FOBLDCMs) pose significant challenges for robust and stable control design, particularly when physical constraints such as actuator saturation and rate limitations are present. Existing control frameworks often neglect these nonlinear limitations, resulting in performance degradation and potential instability in practical applications. Motivated by these challenges, this paper presents a comprehensive Lyapunov-based stability and control synthesis framework for FOBLDCMs within the fractional-order (FO) range $0<v<1$. The proposed methodology employs indirect, direct, and composite Lyapunov functions to derive sufficient stability conditions under four scenarios: unconstrained input, saturation-only, rate-limited-only, and combined constraints. For each case, a family of stabilizing controllers is designed to explicitly handle the respective limitations. To the best of our knowledge, this is the first study to rigorously address both saturation and rate limitations in the control design of FO chaotic systems. Numerical simulations confirm that the proposed controllers significantly improve performance over existing methods. Specifically, the unconstrained controller achieves a notable reduction in control energy (from $2.72\times {10}^5$ to $1.83\times {10}^5$), a 26.3% decrease in maximum control effort, and enhanced or comparable tracking accuracy, as indicated by lower ISE and RMSE values. These results highlight the robustness and practical applicability of the proposed control framework for real-world FO electromechanical systems. Full article

This paper proposes a weak Galerkin (WG) finite element method for solving a multi-dimensional evolution equation with a weakly singular kernel. The temporal discretization employs the backward Euler scheme, while the fractional integral term is approximated via a piecewise constant function method. A fully discrete scheme is constructed by integrating the WG finite element approach for spatial discretization. $L^2$-norm stability and convergence analysis of the fully discrete scheme are rigorously established. Numerical experiments are conducted to validate the theoretical findings and demonstrate optimal convergence order in both spatial and temporal directions. The numerical results confirm that the proposed method achieves an accuracy of the order $O\left(\tau +h^{k+1}\right)$, where $\tau$ and h represent the time step and spatial mesh size, respectively. This work extends previous studies on one-dimensional problems to higher spatial dimensions, providing a robust framework for handling evolution equations with a weakly singular kernel. Full article

This study investigates the coupled effects of waste rock-to-tailings ratio (WTR) and curing temperature on the pore structure and mechanical performance of cemented tailings waste rock backfill (CTRB). Four WTRs (6:4, 7:3, 8:2, 9:1) and curing temperatures (20–50 °C) were tested. Low-field nuclear magnetic resonance (NMR) was used to characterize pore size distributions and derive fractal dimensions ($D_a$, $D_b$, $D_c$) at micropore, mesopore, and macropore scales. Uniaxial compressive strength (UCS) and elastic modulus (E) were also measured. The results reveal that (1) the micropore structure complexity was found to be a key indicator of structural refinement, while excessive temperature led to pore coarsening and strength reduction. $D_a$ = 2.01 reaches its peak at WTR = 7:3 and curing temperature = 40 °C; (2) at this condition, the UCS and E achieved 20.5 MPa and 1260 MPa, increasing by 45% and 38% over the baseline (WTR = 6:4, 20 °C); (3) when the temperature exceeded 40 °C, $D_a$ dropped significantly (e.g., to 1.51 at 50 °C for WTR = 7:3), indicating thermal over-curing and micropore coarsening; (4) correlation analysis showed strong negative relationships between total pore volume and mechanical strength ($R$ = −0.87 for ${\mathsf{\delta }}_a\mathrm{v}\mathrm{s}.\mathrm{U}\mathrm{C}\mathrm{S}),$ and a positive correlation between $D_a$ and UCS (R = 0.43). (5) multivariate regression models incorporating pore volume fractions, $T_2$ relaxation times, and fractal dimensions predicted $\mathrm{U}\mathrm{C}\mathrm{S}$ and E with R² > 0.98; (6) the hierarchical sensitivity of fractal dimensions follows the order micro-, meso-, macropores. This study provides new insights into the microstructure–mechanical performance relationship in CTRB and offers a theoretical and practical basis for the design of high-performance backfill materials in deep mining environments. Full article