Fractal and Fractional

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

In this article, we introduce and study a novel class of polynomial $\phi$-contractions, which simultaneously generalizes classical polynomial contractions and $\phi$-contractions within a unified framework. We establish generalized fixed point theorems that encompass some results in the existing literature. Furthermore, we derive explicit error estimates and convergence rates for the associated Picard iteration, providing practical insights into the speed of convergence. Several illustrative examples, including higher-degree polynomial contractions, demonstrate the scope and applicability of our results. As an application, we prove existence and uniqueness results for solutions of a class of fractional logistic growth equations, highlighting the relevance of our theoretical contributions to nonlinear analysis and applied mathematics. Full article

This paper deals not only with pointwise and uniform convergence but also Y-valued fractional approximation results by univariate symmetrized neural network (SNN) operators on Banach space $\left(Y,∥.∥\right)$. Moreover, our main motivation in this work is to compare the convergence results obtained by classical neural network (NN) operators and symmetric neural network (SNN) operators and try to convert them into numerical examples and graphs through computer programming language Python codes. As a result of this experimental study conducted under the regime of certain parameters, the convergence speed and results of SNN operators are superior to those of classical NN operators. Full article

Fractal theory of particle size distribution (PSD) is a widely used approach in soil science. However, fractal studies on sandstone PSDs are scarce in sedimentology and geology. Taking littoral sandstones in the Carboniferous Donghetang Formation of the Hade Oilfield as an example, fractal dimensions of 115 fine sandstone and 150 silty sandstone PSDs are calculated and compared with particle size compositions and traditional statistical parameters in this paper. The results show that fractal dimension values in fine sandstones, 1.69–2.17 averaged at 1.99, are usually lower than that in silty sandstones, 2.12–2.73 averaged at 2.37. Fractal dimension and sandy content of littoral sandstones show a strong negative linear relationship. Significant logarithmic correlations are implied between fractal dimension and silty and clayey contents of littoral sandstones, which is different from linear relations in soil PSDs. The relationships between fractal dimension and mean, sorting, and skewness of silty sandstone PSDs are better than those of fine sandstones. Fractal dimension and kurtosis of silty sandstones and fine sandstones exhibit weak negative and positive linear relationships, respectively. Fractal dimension values in lower-shoreface facies, 2.05–2.47 averaged at 2.33, are generally higher than that in upper-shoreface facies, 1.79–2.30 averaged at 2.11. Fractal dimension values in bar and beach microfacies are commonly lower than those in trough microfacies. Combined with additional sedimentary information from various clastic deposits, the fractal dimension can serve as a new depositional environment indicator. Full article

In this paper, we introduce a general fractional master equation involving regularized general fractional derivatives with Sonin kernels, and we discuss its physical characteristics and mathematical properties. First, we show that this master equation can be embedded into the framework of continuous time random walks, and we derive an explicit formula for the waiting time probability density function of the continuous time random walk model in form of a convolution series generated by the Sonin kernel associated with the kernel of the regularized general fractional derivative. Next, we derive a fractional diffusion equation involving regularized general fractional derivatives with Sonin kernels from the continuous time random walk model in the asymptotical sense of long times and large distances. Another important result presented in this paper is a concise formula for the mean squared displacement of the particles governed by this fractional diffusion equation. Finally, we discuss several mathematical aspects of the fractional diffusion equation involving regularized general fractional derivatives with Sonin kernels, including the non-negativity of its fundamental solution and the validity of an appropriately formulated maximum principle for its solutions on the bounded domains. Full article

In this paper, we are studying a class of nonlinear fractional difference equations with time-varying delays in Banach space. By means of mathematical induction and the Picard iteration method, we first obtain the existence result of this fractional difference system. Under some new criteria along with the Schauder’s fixed point theorem, we then derive the attractivity conclusions. Subsequently, with the aid of Grönwall’s inequality, we prove that the system is globally attractive. Finally, we give two examples to prove the validity of our theorems. Full article

This study combines artificial intelligence (AI) with mathematical modeling to improve the forecasting of the water cycle mechanism. The proposed model simulates the development of global temperature, precipitation, and water availability, integrating key climate parameters that control these dynamics. Using a system of fractional-order differential equations in the fractal–fractional sense of derivatives, the model captures interactions between solar radiation, the greenhouse effect, evaporation, and runoff. The deep learning framework is trained on extensive climate datasets, allowing it to refine predictions and identify complex patterns within the water cycle. By applying AI techniques alongside mathematical modeling, this procedure provides valuable insights into climate change and water resource administration. The model’s predictions can contribute to assessing future climate states, optimizing environmental policies, and designing sustainable water management strategies. Furthermore, the hybrid methodology improves decision-making by offering data-driven solutions for climate adaptation. The findings illustrate the effectiveness of AI-driven models in addressing global climate challenges with improved precision. Full article

In this paper, we investigated a three-dimensional incompressible fractional rotating magnetohydrodynamic (FrMHD) system by reformulating the Cauchy problem into its equivalent mild formulation and working in critical homogeneous Sobolev spaces. For this, we first established the existence and uniqueness of a global mild solution for small and divergence-free initial data. Moreover, our approach is based on proving sharp bilinear convolution estimates in critical Sobolev norms, which in turn guarantee the uniform analyticity of both the velocity and magnetic fields with respect to time. Furthermore, leveraging the decay properties of the associated fractional heat semigroup and a bootstrap argument, we derived algebraic decay rates and established the long-time dissipative behavior of FrMHD solutions. These results extended the existing literature on fractional Navier–Stokes equations by fully incorporating magnetic coupling and Coriolis effects within a unified fractional-dissipation framework. Full article

This paper introduces a novel version of the Gronwall inequality specifically related to the Hilfer–Hadamard proportional fractional derivative. By utilizing Picard’s method of successive approximations along with the definition of Mittag–Leffler functions, we derive a representation formula for the solution of the Hilfer–Hadamard proportional fractional differential equation featuring constant coefficients, expressed in the form of the Mittag–Leffler kernel. We establish the uniqueness of the solution through the application of Banach’s fixed-point theorem, leveraging several properties of the Mittag–Leffler kernel. The current study outlines optimal stability, a new Ulam-type concept based on classical special functions. It aims to improve approximation accuracy by optimizing perturbation stability, offering flexible solutions to various fractional systems. While existing Ulam stability concepts have gained interest, extending and optimizing them for control and stability analysis in science and engineering remains a new challenge. The proposed approach not only encompasses previous ideas but also emphasizes the enhancement and optimization of model stability. The numerical results, presented in tables and charts, are provided in the application section to facilitate a better understanding. Full article