Fractal and Fractional

This study employs the modified extended direct algebraic method (MEDAM) to investigate the generalized nonlinear fractional $(3+1)$-dimensional wave equation with gas bubbles. This advanced analytical framework is used to construct a comprehensive class of exact wave solutions and explore the associated dynamical characteristics of diverse wave structures. The analysis yields several categories of soliton solutions, including rational, hyperbolic (sech, tanh), and trigonometric (sec, tan) function forms. To the best of our knowledge, these soliton solutions have not been previously documented in the existing literature. By selecting appropriate standards for the permitted constraints, the qualitative behaviors of the derived solutions are illustrated using polar, contour, and two- and three-dimensional surface graphs. Furthermore, a stability analysis is performed on the obtained soliton solutions to ascertain their robustness and dynamical stability. The suggested analytical approach not only deepens the theoretical understanding of nonlinear wave phenomena but also demonstrates substantial applicability in various fields of applied sciences, particularly in engineering systems, mathematical physics, and fluid mechanics, including complex gas–liquid interactions. Full article

This work focuses on the finite-time synchronization control (FTSC) for fractional-order chaotic systems (FOCSs) subject to uncertain dynamics, unknown parameters and input nonlinearities. In the control law design, the uncertain dynamics of the FOCSs are addressed by using fuzzy logic systems (FLSs), while the unknown control direction caused by unknown input nonlinearity is handled through applying the Nussbaum gain function (NGF) method. Parameter adaptive laws are derived to estimate the unknown parameters of the given FOCSs, the parameter vectors of the FLSs, and unknown bounded constants, respectively. By integrating these parameter-adaptive laws with the FT backstepping control framework and FO Lyapunov direct method, an adaptive fuzzy FTSC strategy is developed. This strategy ensures that the synchronization error (SE) can converge to a small neighborhood of zero (SNoZ) within a FT and all signals of the closed-loop system (CLS) remain ultimately bounded. In the end, three simulation cases are utilized to demonstrate the efficiency of the proposed control method. Full article

In this paper, we consider the family of parameterized Mandelbrot-like sets generated as any point $c\in \mathbb{C}\setminus \left\{0\right\}$ of the complex plane belongs to any member of this family for a real parameter $t\ge 1$, provided that its corresponding orbit of 0 does not escape to infinity under iteration $f_c^n\left(0\right)={\left[f_c^{n-1}\left(0\right)\right]}^2+log{c}^t$; otherwise, it is not a member of this set. This classically means there is only a binary membership possibility for all points. Here, we call this type of fractal set a Mandelblog set, and then we introduce a membership function that assigns a degree to each c to be an element of a fuzzy Mandelblog set under the iterations, even if the orbits of the points are not limited. Moreover, we provide numerical examples and gray-scale graphics that illustrate the membership degrees of the points of the fuzzy Mandelblog sets under the effects of iteration parameters. This approach enables the formation of graphs for these fuzzy fractal sets by representing points that belong to the set as white pixels, points that do not belong as black pixels, and other points, based on their membership degrees, as gray-toned pixels. Furthermore, the membership function facilitates the direct proofs of the symmetry criteria for these fractal sets. Full article

In this paper, we study the local stability of an incommensurate fractional reaction–diffusion glycolysis model. The glycolysis process, fundamental to cellular metabolism, exhibits complex dynamical behaviors when formulated as a nonlinear reaction–diffusion system. To capture the heterogeneous memory effects often present in biochemical and chemical processes, we extend the classical model by introducing incommensurate fractional derivatives, where each species evolves with a distinct fractional order. We linearize the system around the positive steady state and derive sufficient conditions for local asymptotic stability by analyzing the eigenvalues of the associated Jacobian matrix under fractional-order dynamics. The results demonstrate how diffusion and non-uniform fractional orders jointly shape the stability domain of the system, highlighting scenarios where diffusion destabilizes homogeneous equilibria and others where incommensurate memory effects enhance stability. Numerical simulations are presented to illustrate and validate the theoretical findings. Full article

Extracting high-quality surface wave dispersion curves from crosscorrelation functions (CCFs) of ambient noise data is critical for seismic velocity inversion and subsurface structure interpretation. However, the non-uniform spatial distribution of noise sources may introduce spurious noise into CCFs, significantly reducing the signal-to-noise ratio (SNR) of empirical Green’s functions (EGFs) and degrading the accuracy of dispersion measurement and velocity inversion. To mitigate this issue, this study aims to develop an effective denoising approach that enhances the quality of CCFs and facilitates more reliable surface wave extraction. We propose a fractional Laplacian-based adaptive progressive denoising (FLAPD) method that leverages local gradient information and a fractional Laplacian mask to estimate noise variance and construct a fractional bilateral kernel for iterative noise removal. We applied the proposed method to the CCFs from 79 long-period seismic stations in Shandong, China. The results demonstrate that the denoising method enhanced the data quality substantially, increasing the number of reliable dispersion curves from 1094 to 2196, and allowing an increased number of temporal sampling points to be retrieved from previously low-SNR curves. This helps to expand the spatial coverage and results in a more accurate velocity inversion result than that without denoising. This study provides a robust denoising solution for ambient noise tomography in regions with complex noise source distributions. Full article

This study presents a novel softsign-function-based fractional-order proportional–integral–derivative (softsign-FOPID) controller optimized using the fungal growth optimizer (FGO) for the stabilization and precise position control of an unstable magnetic ball suspension system. The proposed controller introduces a smooth nonlinear softsign function into the conventional FOPID structure to limit abrupt control actions and improve transient smoothness while preserving the flexibility of fractional dynamics. The FGO, a recently developed bio-inspired metaheuristic, is employed to tune the seven controller parameters by minimizing a composite objective function that simultaneously penalizes overshoot and tracking error. This optimization ensures balanced transient and steady-state performance with enhanced convergence reliability. The performance of the proposed approach was extensively benchmarked against four modern metaheuristic algorithms (greater cane rat algorithm, catch fish optimization algorithm, RIME algorithm and artificial hummingbird algorithm) under identical conditions. Statistical analyses, including boxplot comparisons and the nonparametric Wilcoxon rank-sum test, demonstrated that the FGO consistently achieved the lowest objective function value with superior convergence stability and significantly better (p < 0.05) performance across multiple independent runs. In time-domain evaluations, the FGO-tuned softsign-FOPID exhibited the fastest rise time (0.0089 s), shortest settling time (0.0163 s), lowest overshoot (4.13%), and negligible steady-state error (0.0015%), surpassing the best-reported controllers in the literature, including the sine cosine algorithm-tuned PID, logarithmic spiral opposition-based learning augmented hunger games search algorithm-tuned FOPID, and manta ray foraging optimization-tuned real PIDD². Robustness assessments under fluctuating reference trajectories, actuator saturation, sensor noise, external disturbances, and parametric uncertainties (±10% variation in resistance and inductance) further confirmed the controller’s adaptability and stability under practical non-idealities. The smooth nonlinearity of the softsign function effectively prevented control signal saturation, while the fractional-order dynamics enhanced disturbance rejection and memory-based adaptability. Overall, the proposed FGO-optimized softsign-FOPID controller establishes a new benchmark in nonlinear magnetic levitation control by integrating smooth nonlinear mapping, fractional calculus, and adaptive metaheuristic optimization. Full article

Pore–throat structure and gas distribution are critical factors in evaluating the quality of tight sandstone reservoirs and hydrocarbon resource potential. Twelve tight sandstone samples from the Lower Permian Shihezi Formation in Hangjin Banner, Ordos Basin, were selected for CTS, X-ray diffraction, HPMI, and gas displacement NMR analyses. By converting the T₂ spectra into pore–throat distributions and applying fractal methods, we quantitatively analyzed the influences of multiple factors on gas distribution characteristics across different pore–throat sizes. The main results are as follows: All samples exhibit a three-stage pore–throat distribution, defining mesopores, micropores, and nanopores; quartz content mainly influences the fractal dimension of mesopores by enhancing structural stability and gas storage capacity, whereas clay minerals control the fractal characteristics of nanopores by increasing pore–throat complexity. An increase in clay mineral content increases the fractal dimension, indicating stronger reservoir heterogeneity and consequently poorer gas-bearing capacity. Larger pore–throat parameters (R_m, S_k, and S_max) correspond to lower fractal dimensions, indicating better connectivity and greater gas storage capacity. Among these factors, pore–throat parameters exert the most significant influence on the fractal dimensions of mesopores and micropores, jointly determining the overall connectivity and the upper limit of the reservoir’s gas-bearing capacity. The results demonstrate that larger pore–throat parameters and higher quartz content help reduce the fractal dimension and enhance the gas-bearing capacity of tight reservoirs. This research enhances understanding of pore–throat structures and gas-bearing capacity in low-permeability reservoirs and provides a theoretical basis for exploration, development, and enhanced recovery in the study area. Full article

This paper investigates the fixed-time synchronization of fractional-order proportional delay Hopfield neural networks (PDHNNs) with bounded parameter uncertainties. Unlike constant delay and bounded variable delay, proportional delay has time-varying and unbounded characteristics, which pose challenges for the synchronization control of primary–secondary fractional neural networks. To achieve fixed-time synchronization, we propose a new nonlinear multi-module feedback controller. It consists of three key functional modules: eliminating the impact of proportional delay on system stability; ensuring convergence within a fixed time frame without being limited by initial conditions; and expanding the selectable range of parameters. Combining the stability lemma and inequality techniques, synchronization criteria of PDHNNs are derived based on the construction of a Lyapunov function with a negative fractional derivative. The settling time can be effectively estimated, which depends on the control parameters and is independent of initial values. Two numerical experiments verify the effectiveness of the theorem and corollary in this study. Full article

In this study, a time-fractional extension of the classical Theis problem with an exponential source term is investigated in a confined aquifer. The governing equation is modeled using two different fractional derivatives—the Caputo and Atangana–Baleanu–Caputo (ABC) operators—to account for memory effects in groundwater flow. The Fractional Reduced Differential Transform Method (FRDTM) is applied to obtain approximate series solutions up to the fifth order. The impact of the fractional order $\alpha$, the nature of the fractional kernel and the localized source term on the hydraulic head are explored at different radial positions. The comparative analysis between the Caputo and Atangana–Baleanu–Caputo (ABC) models reveals how memory effects and operator choice significantly influence the hydraulic head response, offering insights into selecting suitable models for aquifers with varying recharge characteristics. Full article

In this article, mathematical modeling and stability analysis of Lyme disease and its transmission dynamics using Caputo fractional-order derivatives is presented. The compartmental model has been formulated to analyze the spread of Borrelia burgdorferi virus through tick vectors and mammalian hosts. The feasible region is established, and the boundedness of the model is verified. Analytically, the disease-free equilibrium and the basic reproduction number $\left({\Re }_0\right)$ has been determined to assess outbreak potential. By virtue of the fixed-point theory, the existence and uniqueness of solutions has been established. The numerical simulations are obtained via the Runge–Kutta 4 method, demonstrating the model’s ability to capture realistic disease progression. Finally, sensitivity analysis and control strategies (tick population reduction, host vaccination, public awareness, and early treatment) are evaluated, revealing that integrated control measures significantly reduce infection rates and enhance recovery. Full article

The Upper Permian marine shale of the inter-platform basin in the Nanpanjiang Basin are rich in organic matter, widely distributed, and relatively thick, indicating abundant resource potential for hydrocarbon exploration. To clarify the sedimentary condition and the variability of reservoir properties, the paleo-environment was reconstructed by using geochemical, mineralogical, rock-property, and pore-structure data. Building on a lithofacies classification, the development patterns of different shale lithofacies were revealed. Reservoir characteristics among lithofacies were compared using scanning electron microscopy (SEM), nuclear magnetic resonance (NMR), and low-temperature Nuclear Magnetic Resonance Cryoporometry (NMRC) experiments. A fractal analysis was performed based on NMR and NMRC data to quantify pore-scale heterogeneity, calculate fractal dimensions (D₁, D₂, and D_c), and evaluate the complexity of pore systems across lithofacies. Correlation analysis and redundancy analysis were applied to further explore the controlling factors of reservoir heterogeneity. The results showed that organic-rich shale in the Permian Linghao Formation occurred mainly in the 1st Member, with average total organic carbon (TOC) content of 2.57%, and the lower part of the 3rd Member (average TOC content 2.88%). In the 1st Member, high-carbon shale was deposited under humid conditions with intense weathering, abundant fine-grained clastic input from basin margins, strongly reducing (anoxic) bottom waters, vigorous phosphorus recycling, and moderate to low primary productivity. Using TOC and mineral composition, seven shale lithofacies were identified in the Linghao Formation, and their development patterns were established based on depositional paleo-environment characteristics and evolution. In the 1st Member, organic-rich shale was dominated by mixed lithofacies with moderate to high TOC. The paleo-environment exerted a primary control on reservoir properties, gas content, pore structure, and heterogeneity. The high-carbon lithofacies had the most favorable rock properties—higher porosity, greater pore volume, and higher gas content—and contained a larger proportion of well-developed organic pores. Fractal analysis revealed that seepage pores exhibited greater structural complexity than adsorption-related pores, with the high-carbon lithofacies showing the highest overall fractal dimensions and thus the strongest heterogeneity. Across the formation, higher clay content and TOC were the primary drivers of increased pore-scale heterogeneity, whereas greater feldspar and quartz contents tended to diminish it. Carbonates exerted a minor effect. Heterogeneity in adsorption pores exerted the strongest influence on differences among lithofacies. These results highlighted the utility of fractal analysis in quantitatively linking shale mineralogy and organic content to multiscale heterogeneity in inter-platform basin settings. Full article

In recent decades, the fractional derivative (FD) and memory-dependent derivative (MDD) have been borrowed for modifying the unsteady diffusion process. Yet, according to their definitions, only memory effects in temporal sense are reflected. To make spatial modifications, that is, a new version of MDD, the space-dependent derivative (SDD) is developed here. With the help of it, the heat diffusion process is remodeled in a more objective manner. The comparisons among the conventional model, spatial FD model, and SDD model show that the last one has the most expressive force, and it can be borrowed for considering other unsteady diffusion problems. Full article

We study the multifractal structure of irregular sets arising from Fibonacci-weighted sums of sequences of random variables. Focusing on Cantor-type subsets $K_{\epsilon }$ of the unit interval, we construct sequences of free and forced blocks, where the free blocks allow full binary branching and the forced blocks fix the digits, controlling the weighted averages. We prove that these sets can attain full Hausdorff and packing dimension while their Hausdorff measure can vanish. We prove that the packing measure of $K_{ϵ}$ depends sensitively on the growth of the forced blocks. Our construction illustrates the mechanism by which Fibonacci-type weights induce irregularity, providing a probabilistic counterpart to classical multifractal phenomena in dynamical systems. Full article

Amid growing threats of image data leakage and misuse, image encryption has become a critical safeguard for protecting visual information. However, many recent image encryption algorithms remain constrained by trade-offs between security, efficiency, and practicability. To address these challenges, this paper first proposes a novel two-dimensional variable fractional-order coupled quadratic hyperchaotic map (2D-VFCQHM), which incorporates a state-dependent dynamic memory effect, wherein the fractional-order is adaptively determined at each iteration by the mean of the system’s current state. This mechanism substantially enhances the complexity and unpredictability of the underlying chaotic dynamics. Building upon the superior hyperchaotic properties of the 2D-VFCQHM, we further develop a high-performance image encryption algorithm that integrates a novel fusion strategy within a dynamic vector-level diffusion-scrambling framework (IEA-VMFD). Comprehensive security analyses and experimental results demonstrate that the proposed algorithm achieves robust cryptographic performance, including a key space of $2^{298}$, inter-pixel correlation coefficients below 0.0018, ciphertext entropy greater than 7.999, and near-ideal plaintext sensitivity. Crucially, the algorithm attains an encryption speed of up to 126.2963 Mbps. The exceptional balance between security strength and computational efficiency underscores the practical viability of our algorithm, rendering it well-suited for modern applications such as telemedicine, instant messaging, and cloud computing. Full article

This paper proposes a new simple method for generating fractal-like aggregates in 2D real spaces. The idea is to use an initial fractal aggregate and simulate a Gravity-based attraction from a distant point (using a gravity attractor with a large mass, but without volume). A Diffusion-Limited Aggregation (DLA) procedure is then applied by considering a single particle situated in the gravity attractor, with a minimum distance $R_a$ for deciding between aggregation or no aggregation. The final aggregates obtained are completely new fractal-like aggregates (images or structures). We analyze the fractal-like generated images obtained with the proposed method, considering different configurations and parameters in the simulations, including different initial fractals, different minimum distances $R_a$, etc. We also analyze the fractal dimensions of some of the new aggregates constructed by the proposed Gravity-based DLA simulation method. Full article

The method of deriving fractional-order differential equations using Riesz fractional-order calculus is a new breakthrough, and it is possible to use the inverse scattering transform (IST) to solve the analytical solutions of the derived equations. However, there are still relatively few examples of using discrete Riesz fractional (DRF) order calculations. This article focuses on using discrete Riesz fractional-order calculations to derive a variable-coefficient fractional integrable semi-discrete nonlinear Schrödinger (vcfISDNLS) equation. On the one hand, we derive the vcfISDNLS equation through dispersion relation (DR) and DRF order calculations. On the other hand, we obtain the explicit expressions of the one-soliton solution, two-soliton solution, and three-soliton solution of this equation without reflection potentials (RPs). By deriving and solving the equation and displaying the obtained soliton solutions, possible evidential support can be provided for control engineering, automotive engineering, image processing, and optical communication systems. Full article

This paper presents a Crank–Nicolson mixed finite element method along with its reduced-order extrapolation model for a fourth-order nonlinear diffusion equation with Caputo temporal fractional derivative. By introducing the auxiliary variable $v=-{\epsilon }^2\Delta u+f(u)$, the equation is reformulated as a second-order coupled system. A Crank–Nicolson mixed finite element scheme is established, and its stability is proven using a discrete fractional Gronwall inequality. Error estimates for the variables u and v are derived. Furthermore, a reduced-order extrapolation model is constructed by applying proper orthogonal decomposition to the coefficient vectors of the first several finite element solutions. This scheme is also proven to be stable, and its error estimates are provided. Theoretical analysis shows that the reduced-order extrapolation Crank–Nicolson mixed finite approach reduces the degrees of freedom from tens of thousands to just a few, significantly cutting computational time and storage requirements. Numerical experiments demonstrate that both schemes achieve spatial second-order convergence accuracy. Under identical conditions, the CPU time required by the reduced-order extrapolation Crank–Nicolson mixed finite model is only $1/60$ of that required by the Crank–Nicolson mixed finite scheme. These results validate the theoretical analysis and highlight the effectiveness of the methods. Full article

In this study, we derive the Fokker–Planck equation for a colloidal particle subject to a harmonic trap and viscous forces under the influence of a magnetic field. We then extend the analysis to a charged colloid driven by both thermal and active noises in the same magnetic environment. Finally, the case of a charged colloid experiencing a harmonic trap together with thermal and active noises is investigated. Analytical solutions for the joint probability density are obtained in the limits of $t\ll \tau$, $t\gg \tau$, and $\tau =0$. For a colloid under a harmonic trap and magnetic field, the mean squared displacement exhibits a superdiffusive scaling proportional to $t^3$ in the short-time regime ($t\ll \tau$), while the mean squared velocity scales as $\sim t$ when $\tau =0$. For a charged colloid with thermal noise, the mean-squared displacement follows a superdiffusive form $\sim t^{2h+1}$ for $t\ll \tau$, and the mean squared velocity again scales linearly with time for $\tau =0$. When the active noise is included together with a harmonic trap, the characteristic time scale grows as $\sim t^4$ in the short-time regime, while the mean squared velocity becomes normally diffusive at $\tau =0$. In the long-time limit ($t\gg \tau$) and for $\tau =0$, the moments of the joint probability density under combined thermal and active noises scale as $\sim t^{4h+2}$, consistent with our analytical results. Notably, as $h\to 1/2$, the entropy of the joint probability density with thermal noise ${\zeta }_{\mathrm{th}}(t)$ coincides with that obtained for active noise ${\zeta }_{\mathrm{ac}}(t)$ in both $t\gg \tau$ and $\tau =0$ limits. Full article