Fractal and Fractional

Motivated by the strong seasonality of East Asian meteorology and its control on pollution episodes characterized by fluctuation level, we model the season-resolved climatology of the regional PM_2.5 connectivity over the Korean Peninsula. Using daily AirKorea data for 2016–2020, we (i) remove daily climatology and the peninsula-wide background (empirical orthogonal function; EOF1) to obtain residual signals; (ii) compute the sign-preserving multifractal detrended cross-correlation coefficient MFDCCA-$\rho \left(q,s\right)$; (iii) apply iAAFT surrogate significance across scales; and (iv) construct signed, weighted networks aggregated over short (5–15 d) and mid (15–30 d) bands for DJF/MAM/JJA/SON. Our analysis targets the seasonal climatology of fluctuation-level ($q$-dependent) connectivity by pooling seasons across years; this approach increases statistical robustness at 5–30-day scales and avoids diluting season-specific organization. We find negligible connectivity for $q<0$ (small fluctuations) but dense, seasonally organized networks for $q>0$ (strongest in winter–spring and at 15–30 days). After removing the EOF1, positive subgraphs form assortative regional backbones, while negative subgraphs reveal a northwest–southeast anti-phase dipole; the connectivity around Baengnyeongdo (B) highlights a transboundary sentinel role in cool seasons. These results demonstrate that a season-resolved, fluctuation-level framework effectively isolates regional connectivity that would otherwise be masked in annual aggregates or by the peninsula-wide background. Full article

This paper deals with a new coupled system of integro-differential equations involving the generalized proportional Caputo derivatives equipped with nonlocal four-point boundary conditions. Sufficient criteria for the existence and uniqueness of solutions for the studied system are derived based on Krasnoselskii’s and Banach fixed-point theorems, respectively. Applications are constructed with three different cases to illustrate the main results. Full article

In this paper, we evaluate a general class of finite integrals involving the error function, generalized Mittag-Leffler functions, and incomplete Aleph functions. The main result provides a unified framework that extends several known formulas related to the incomplete Gamma, I-, and H-functions. Under suitable conditions, these results reduce to many classical special cases. We discuss convergence conditions that justify the validity of the obtained formulas and include explicit corollaries that highlight connections with earlier results in the literature. To illustrate applicability, we present numerical examples and graphs, demonstrating the behavior of the error function integral and Mittag-Leffler functions for specific parameter values. These integrals arise naturally in fractional calculus, probability theory, viscoelasticity, and anomalous diffusion, underscoring the importance of the present work in both mathematical analysis and applications. Full article

Gradient accumulation enables training large-scale deep learning models under GPU memory constraints by aggregating gradients across multiple microbatches before parameter updates. Standard gradient accumulation treats all microbatches uniformly through simple averaging, implicitly assuming that all stochastic gradient estimates are equally reliable. This assumption becomes problematic in non-convex optimization where gradient variance across microbatches is high, causing some gradient estimates to be noisy and less representative of the true descent direction. In this paper, FracGrad is proposed, a simple weighting scheme for gradient accumulation that biases toward recent microbatches via a power-law schedule derived from a discretized Riemann–Liouville integral. Unlike uniform summation, FracGrad reweights each microbatch gradient by $w_i\left(\alpha \right)={\displaystyle \frac{{(N-i+1)}^{\alpha }-{(N-i)}^{\alpha }}{{\sum }_{j=1}^N[{(N-j+1)}^{\alpha }-{(N-j)}^{\alpha }]}}$, controlled by $\alpha \in (0,1]$. When $\alpha =1$, standard accumulation is recovered. In experiments on mini-ImageNet with ResNet-18 using up to $N=32$ accumulation steps, the best FracGrad variant with $\alpha =0.1$ improves test accuracy from 16.99% to 31.35% at $N=16$. Paired t-tests yield $p\approx 2\times {10}^{-6}$. Full article

This study models customer patterns using fractal analysis and examines how touchpoints and customer journey mapping shape value to enhance sustainable organizational and customer performance. The analysis is made at cell-level using mathematical modelling and tactical indicators within a top-three retail organization with national presence. This study reveals that only half of the customers engage with multiple touchpoints, with specific departments experiencing low interaction levels. The company’s customer concentration is well-balanced, reducing dependency on a single client, and mutual dependency suggests robust customer–company sustainable relationships. Results underline the significance of the proposed tools in evaluating the value of the customers for the company. Practical implications include guiding managers to use modelling in strengthening customer touchpoints to enhance qualitative relationships and profitability. This research contributes to customer relationship management and fractal analysis literature, demonstrating their role in maximizing bi-directional value and performance for success. Full article

Organic pores serve as crucial storage spaces for shale gas, whose morphology and structure vary significantly among different types of organic matter, directly influencing the storage and seepage capacity of shale gas. The Upper Permian shale in the Western Hubei Trough formed in diverse sedimentary facies and has undergone multiple geological activities, resulting in strong heterogeneity of organic pores across different strata and regions. To figure out the heterogeneous characteristics of organic pores and the forming reason, the occurrence state of organic matter, pore morphology, and structural parameters (pore size, specific surface area, pore volume, and fractal dimension) of the Upper Permian shale in Western Hubei, have been discussed in detail, based on the data of field emission scanning electron microscopy and low-temperature nitrogen adsorption experiments conducted on the extracted organic matter. On this basis, fractal dimension theory was applied to discuss the heterogeneity of organic pores in different layers, and the reason for heterogeneity has been analyzed in detail. The results indicate that the occurrence mode of organic matter in different layers presents various characteristics: in the Gufeng Formation, the organic matters distribute primarily dispersed in flocculent state; at the bottom of Wujiaping Formation, they occur as isolated individuals, while the organic matters turn into discontinuous laminated distribution in the middle and upper Wujiaping Formation; in the Dalong Formation, the organic matters show continuous parallel banded distribution. Moreover, the morphology and structural parameters of organic pores exhibit obvious changes from the Gufeng Formation to the Dalong Formation: (a) the pore morphology shows the changed trend as extremely complex-simple-complex; (b) the specific surface area and pore volume follow the trend as large-small-large; (c) the pore size distribution displays in the pattern of bimodal-unimodal-bimodal; (d) the data of fractal dimension show the variation of high–low–high. Overall, the various sedimentary environments during the Upper Permian shale depositional period determined the differences in organic sources, which dominated the heterogeneity of organic pores in shale. These data clarify the development and variation characteristics of organic matter pores under different depositional environments, providing a theoretical basis for shale gas exploration and development during the transition from marine to marine–continental facies. Full article

In this paper, we introduce the notion of fuzzy exponential contractions within the framework of fuzzy metric spaces. These mappings, which involve point-dependent exponential terms, are studied under the assumptions of either fuzzy continuity or the weaker condition of fuzzy Picard continuity. We establish corresponding existence and uniqueness theorems, and we further demonstrate the scope of the theory through illustrative examples and by applying it to prove an existence and uniqueness result for a class of nonlinear fractional differential equations. Full article

In the observation of tight sandstone cores, the variations in the hydrocarbon charging usually can be observed in the same geological age reservoirs, which manifest as differential oil staining on the core surface. In order to clarify the micro total pore–throat size distribution (TPSD) characteristics and oil content differences of different oil-bearing tight reservoirs, we drilled two types of oil-bearing cores in the Chang6 formation of Xiasiwan Oilfield, conducted casting thin section (CTS), scanning electron microscopy (SEM), and X-ray diffraction (XRD) to qualitatively and quantitatively analyze petrological and pore–throat characteristics. The TPSD of different oil-bearing cores were quantitatively characterized and compared by combining high-pressure mercury injection (HPMI) and constant rate mercury injection (CRMI). Meanwhile, we quantitatively evaluated the complexity of the pore–throat structure based on fractal theory. Our results reveal significant difference in the clay mineral contents between the two types of cores, despite both being classified as arkose. Due to higher contents of illite, calcite, and chlorite, the pores of oil-smelling sandstone are obviously affected by cementation. The result of TPSD characteristics shows that the oil-appearing sandstone samples exhibit well-developed big pores and throats, displaying bimodal distribution, and three-stage fractal characteristics in the TPSD curves. Conversely, oil-smelling sandstone samples manifesting a left-skewed bimodal, pore space contribution of the samples is more dependent on pores and throats smaller than 0.12 μm. The TPSD curves exhibit three-stage and four-stage fractal characteristics. Therefore, the differences in oil-bearing properties between the two types of cores are attributed to variations in mineral composition, diagenesis, clay mineral content, pore types, pore–throat size distribution (PSD), and pore–throat complexity. Our results provide crucial guidance for subsequent reservoir quality assessment in this study area and the development of tight sandstone reservoirs with similar geological characteristics. Full article

We study stochastic variants of the Kairat-II and Kairat-X equations in (3 + 1) dimensions, two canonical models in soliton theory. Random fluctuations are incorporated through a Wiener process, yielding a multiplicative stochastic embedding of the wave fields. By combining the enhanced direct algebraic technique with the new projective Riccati equation approach, we obtain closed-form stochastic soliton solutions and analyze how noise modulates their amplitude and localization. The solutions are illustrated with consistent 3D surface plots (mean field vs. sample paths) and 2D time traces to highlight wave geometry and variability. In addition, we employ the energy balance approach to separate kinetic and potential contributions and to verify an energy balance relation for the derived solutions, thereby clarifying their physical plausibility and stability under noise. The results provide exact, easily verifiable benchmarks for stochastic nonlinear wave models and a practical template for incorporating randomness into nonlinear dispersive systems. Full article

In this paper, we present a collocation algorithm for numerically treating the time-fractional Kuramoto–Sivashinsky equation (TFKSE). Certain orthogonal polynomials, which are expressed as combinations of Chebyshev polynomials, and their shifted polynomials are introduced. Some new theoretical formulas regarding these polynomials have been developed, including their operational matrices of both integer and fractional derivatives. The derived formulas will be the foundation for designing the proposed numerical algorithm, which relies on converting the governing problem with its underlying conditions into a nonlinear algebraic system, which can be solved using Newton’s iteration technique. A rigorous error analysis for the proposed combined Chebyshev expansion is presented. Some numerical examples are given to ensure the applicability and efficiency of the presented algorithm. These results demonstrate that the proposed algorithm attains superior accuracy with fewer expansion terms. Full article

The nonlocality (capable of associating target dynamics across multiple time moments) and memory properties (able to retain historical trajectories) of fractional calculus serve as the core theoretical approach to resolving the “dynamic information association deficiency” in UAV mobile target search. This paper proposes the Fractional-order Black-winged Kite Algorithm (FOBKA), which transforms the search problem into an adaptability function optimization model aimed at “maximizing target capture probability” based on Bayesian theory. Addressing the limitations of the standard Black-winged Kite Algorithm (BKA), the study incorporates fractional calculus theory for enhancement: A fractional-order operator is embedded in the migration behavior phase, leveraging the memory advantage of fractional-orders to precisely capture the temporal span, spatial position, and velocity evolution of targets, thereby enhancing global detection capability and convergence accuracy. Simultaneously, population individuals are initialized using motion-encoding, and the attack behavior phase combines alternating updates with a Lévy flight mechanism to balance local exploration and global search performance. To validate FOBKA’s superiority, comparative experiments were conducted against eight newly proposed meta-heuristic algorithms across six distinct test scenarios. Experimental data demonstrate that FOBKA significantly outperforms the comparison algorithms in convergence accuracy, operational robustness, and target capture probability. Full article

We first investigate the dynamical behavior of an active Brownian particle influenced by a viscoelastic memory effect characterized by a power-law kernel, under the effects of thermal and active noises. We then analyze the dynamics of an active Brownian particle confined in a harmonic trap in the presence of the same noise sources. To derive the Fokker–Planck equation for the joint probability density of the active particle, we obtain analytical solutions for the joint probability density and its moments using double Fourier transforms in the limits $t\ll \tau$, $t\gg \tau$, and $\tau =0$. As a result, the mean squared displacement of an active Brownian particle driven by thermal noise exhibits a super-diffusive scaling of $t^{2h+1}$ in the short-time regime ($t\ll \tau$). In contrast, for a particle in a harmonic trap driven by active noise, the mean squared velocity scales linearly with $t$ when $\tau =0$. Moreover, the higher-order moments of an active Brownian particle in a harmonic trap with thermal noise scale with $t^{4h+2}$ in the long-time limit ($t\gg \tau$) and for $\tau =0$, consistent with our analytical results. Full article

Fractional calculus in discrete-time is a recent field that has drawn much interest for dealing with multidisciplinary systems. A result of this tremendous potential, researchers have been using constant and variable-order fractional discrete calculus in the modelling of financial and economic systems. This paper explores the emergence of chaotic and regular patterns of the fractional four-dimensional (4D) discrete economic system with constant and variable orders. The primary aim is to compare and investigate the impact of two types of fractional order through numerical solutions and simulation, demonstrating how modifications to the order affect the behavior of a system. Phase space orbits, the 0-1 test, time series, bifurcation charts, and Lyapunov exponent analysis for different orders all illustrate the constant and variable-order systems’ behavior. Moreover, the spectral entropy (SE) and $C_0$ complexity exhibit fractional-order effects with variations in the degree of complexity. The results provide new insights into the influence of fractional-order types on dynamical systems and highlight their role in promoting chaotic behavior. Additionally, two types of control strategies are devised to guide the states of a 4D fractional discrete economic system with constant and variable orders to the origin within a specified amount of time. MATLAB simulations are presented to demonstrate the efficacy of the findings. Full article

Macroeconomic mathematical models are practical instruments structured to carry out theoretical analyses of macroeconomic developments. In this manuscript, the Caputo-like fractional operator of variable order is used to introduce and investigate the mechanism of the discrete macroeconomic model. The nature of the dynamics was established, and the emergence of chaos using a distinct variable fractional order, especially the stability of the trivial solution, is examined. The findings reveal that the variable-order discrete macroeconomic model displays chaotic dynamics employing bifurcation, the Largest Lyapunov exponent ($L{E}_{max}$), the phase portraits, and the 0–1 test. Furthermore, a complexity analysis is performed to demonstrate the existence of chaos and quantify its complexity using $C_0$ complexity and spectral entropy ($SE$). These studies show that the suggested variable-order fractional discrete macroeconomic model has more complex features than integer and constant fractional orders. Finally, MATLAB R2024b simulations are run to exemplify the outcomes of this study. Full article

Medical image segmentation, driven by the intrinsic fractal characteristics of biological patterns, plays a crucial role in medical image analysis. Recently, universal image segmentation, which aims to build models that generalize robustly to unseen anatomical structures and imaging modalities, has emerged as a promising research direction. To achieve this, previous solutions typically follow the in-context learning (ICL) framework, leveraging segmentation priors from a few labeled in-context references to improve prediction performance on out-of-distribution samples. However, these ICL-based methods often overlook the quality of the in-context set and struggle with capturing intricate anatomical details, thus limiting their segmentation accuracy. To address these issues, we propose VG-SAM, which employs a multi-scale in-context retrieval phase and a visual in-context guided segmentation phase. Specifically, inspired by the hierarchical and self-similar properties in fractal structures, we introduce a multi-level feature similarity strategy to select in-context samples that closely match the query image, thereby ensuring the quality of the in-context samples. In the segmentation phase, we propose to generate multi-granularity visual prompts based on the high-quality priors from the selected in-context set. Following this, these visual prompts, along with the semantic guidance signal derived from the in-context set, are seamlessly integrated into an adaptive fusion module, which effectively guides the Segment Anything Model (SAM) with powerful segmentation capabilities to achieve accurate predictions on out-of-distribution query images. Extensive experiments across multiple datasets demonstrate the effectiveness and superiority of our VG-SAM over the state-of-the-art (SOTA) methods. Notably, under the challenging one-shot reference setting, our VG-SAM surpasses SOTA methods by an average of $6.61\%$ in DSC across all datasets. Full article

The study of economic maps has consistently attracted researchers due to their rich dynamics and practical relevance. A deeper understanding of these systems enables the development of more effective control strategies. In this work, we examine the influence of the fractional order $\upsilon$ with the Caputo fractional difference on an economic market map. The primary contribution is the comprehensive analysis of how both commensurate and incommensurate fractional orders affect the stability and complexity of the map. Numerical investigations, including phase portraits, largest Lyapunov exponents, and bifurcation analysis, reveal that the system undergoes a cascade of period-doubling bifurcations before transitioning into chaos. To further characterize the dynamics, complexity is evaluated using the 0–1 test and $C_0$ complexity, both confirming chaotic behavior. Furthermore, two-dimensional control schemes are introduced and theoretically validated to both stabilize the chaotic economic market map and achieve synchronization with a combined response map. The theoretical and numerical results are validated through MATLAB 2016 simulations. Full article

The current study examines optical soliton solutions in a complicated system of metamaterial-based optical solutions coupled with extremely dispersive couplers. The conformable fractional derivative (CFD) influences the nonlinear refractive index, which is governed by a parabolic equation. Some soliton solutions are extracted, like bright, singular solitons, and singular periodic ones; also, Weierstrass elliptic doubly periodic, and several other exact solutions are systematically revealed by the study using the modified extended direct algebraic method. The findings shed important light on the many solitons in these intricate systems and the interactions between nonlinearity, dispersion, and metamaterial properties. The findings have significance beyond advancing our theoretical understanding of soliton behavior in metamaterial-based optical couplers; they might influence the advancement and development of optical communication technologies and systems. Complementary 2D and 3D representations show how stability parameters change throughout various dynamical regimes and confirm solution consistency. In order to comprehend the complex nonlinear phenomena of this system and its possible practical applications, this paper offers a comprehensive theoretical framework. Full article

Shale gas production dynamics display strong heterogeneity and nonlinearity, posing challenges to conventional analytical methods. This study applies fractal theory to analyze long-term production data from 178 global shale gas fields by calculating Hurst exponent (H) and fractal dimension (D). Results show 60.7% of fields are suitable for fractal analysis, with H values of 0.64–0.92 and D values of 1.0–1.9, indicating significant long-term memory and structural complexity. Cluster analysis reveals two distinct production patterns: stable trend (13.0%) and fluctuating trend (87.0%). Key innovations include: (1) extending fractal theory from static reservoir characterization to dynamic production analysis; (2) establishing a fractal-based production classification system; and (3) defining applicability conditions for fractal analysis in shale gas evaluation. This study demonstrates fractal theory’s effectiveness as a quantitative tool for production dynamics analysis, EUR estimation, and development strategy optimization. Full article