Fractal Fract., Volume 9, Issue 9 (September 2025) – 66 articles

Aiming at the synchronization problem of fractional time-varying perturbation systems, an improved WRBF neural network was proposed based on the wavelet function and radial basis function (RBF). Then, the adaptive controller and updated law are derived based on the WRBF network. It is used to approximate functions and adjust the corresponding parameters in the controller. Based on Lyapunov and Barbalat stability theory, the synchronization of a fractional system with time-varying perturbation is proved effectively. Full article

The aim of this manuscript is to introduce the fractional integral inequalities of H-H types via multiplicative (Antagana-Baleanu) A-B fractional operators. We also provide the fractional version of the H-H type of the product and quotient of multiplicative superquadratic and multiplicative subquadratic functions via the same operators. Superquadratic functions, have stronger convexity-like behavior. They provide sharper bounds and more refined inequalities, which are valuable in optimization, information theory, and related fields. The use of multiplicative fractional operators establishes a nonlinear fractional structure, enhancing the analytical tools available for studying dynamic and nonlinear systems. The authenticity of the obtained results are verified by graphical and numerical illustrations by taking into account some examples. Additionally, the study explores applications involving special means, special functions and moments of random variables resulting in new fractional recurrence relations within the multiplicative calculus framework. These contributions not only generalize existing inequalities but also pave the way for future research in both theoretical mathematics and real-world modeling scenarios. Full article

DNA methylation is an epigenetic modification where a methyl group is added to a DNA molecule, typically at the cytosine base within a CpG dinucleotide. This process can influence gene expression without changing the underlying DNA sequence. Essentially, methylation can act like a switch that regulates which genes are active in a cell. DNA methylation (DNAm) models often describe the dynamic changes of methylation levels at specific DNA sites, considering methylation and demethylation processes. A common approach involves representing the methylation state as a continuous variable, and modelling its change over time or in response to various factors using differential equations. These equations can incorporate parameters such as the methylation and demethylation rates, factors like DNA replication, the influence of regulatory proteins, and other related parameters. Understanding DNAm dynamics in relation to age is crucial for elucidating ageing processes and developing biomarkers. This work introduces a theoretical framework for modelling DNAm dynamics using a fractional calculus approach, extending standard models based on the integer-order differential equations. The proposed fractional-calculus representation of the methylation process, defined by the fractional-order differential equation and its solution based on the Mittag–Leffler function, provides improved results compared to the standard model that uses a first-order differential equation, which contains an exponential function in its solution, in terms of the comparison criteria (sum of absolute errors, sum of squared errors, mean absolute percentage error, R-squared, and adjusted R-squared). Moreover, the Mittag–Leffler model provides a more general representation of DNAm dynamics, making the standard exponential model only one specific case. Full article

A class of boundary value problems for fractional differential systems involving variable-order derivatives is considered. Such problems can be transformed into some boundary value problems for nonlinear Caputo fractional differential systems. Here, the relations between linear Caputo fractional differential equations and their corresponding linear integral equations are investigated, and the results demonstrate that a proper Lipschitz-type condition is needed for studying nonlinear Caputo fractional differential equations. Then, an existence and uniqueness result is established in some vector subspaces by Banach’s fixed-point theorem and ${\parallel ·\parallel }_e$ norm. In addition, two examples are presented to illustrate the theoretical conclusions. Full article

Brain tumors usually appear as masses formed by localized abnormal cell proliferation. Although complete removal of tumors is an ideal treatment goal, this process faces many challenges due to the aggressive nature of malignant tumors and the need to protect normal brain tissue. Therefore, early diagnosis is crucial to mitigate the harm posed by brain tumors. In this study, the classification accuracy is improved by improving the ResNet50 model. Specifically, the image is preprocessed and enhanced firstly, and the image is denoised by fractional calculus; then, transfer learning technology is adopted, the ECA attention mechanism is introduced, the convolutional layer in the residual block is optimized, and the multi-scale convolutional layer is fused. These optimization measures not only enhance the model’s ability to grasp the overall details but also improve its ability to recognize micro and macro features. This allows the model to understand data features more comprehensively and process image details more efficiently, thereby improving processing accuracy. In addition, the improved ResNet50 model is combined with EfficientNetB0 to further optimize performance and improve classification accuracy by utilizing EfficientNetB0’s efficient feature extraction capabilities through feature fusion. In this study, we used a brain tumor image dataset containing 5712 training images and 1311 validation images. The optimized ResNet50 model achieves a verification accuracy of 98.78%, which is 3.51% higher than the original model, and the Kappa value is also increased by 4.7%. At the same time, the lightweight design of the EfficientNetB0 improves performance while reducing uptime. These improvements can help diagnose brain tumors earlier and more accurately, thereby improving patient outcomes and survival rates. Full article

A class of initial boundary value problems is here considered for a one-dimensional diffusion equation with a general time-fractional derivative with the Sonin kernel. One of the boundary conditions is in a general non-classical form, which includes no-nlocal cases of integral or multi-point boundary conditions. The problem is studied here by applying spectral projection operators to convert it to a system of relaxation equations in generalized eigenspaces. The uniqueness of the solution is established based on the uniqueness property of the spectral expansion. An algorithm is given for constructing the solution in the form of spectral expansion in terms of the generalized eigenfunctions. Estimates for the time-dependent components in this expansion are established and applied to prove the existence of a solution in the classical sense. The obtained results are applied to a particular case in which the specified boundary conditions lead to two sequences of eigenvalues, one of which consists of triple eigenvalues. Full article

(1) Background: It is important to know, radiologically, the relationship of Mandibular third molars (M3) to the mandibular canal to minimize postoperative complications by causing damage to the inferior alveolar vessels and nerve during extraction. This study aimed to evaluate the usability of various image analyses or high-risk radiographic findings in determining the relationship of M3s to the mandibular canal on Digital Panoramic Radiography (DPR). (2) Methods: DPRs of 60 patients with bilateral mandibular M3s in the dental arch, determined one of them to be related to the mandibular canal unilaterally by Cone Beam Computed Tomography (CBCT), were included. The high-risk radiological signs of M3s and Fractal Analysis (FA) and Histogram Analysis (HA) measurements of the trabecular bone around the M3s’ roots were compared. The Independent t-test, Kolmogorov–Smirnov, Mann–Whitney U, and Chi-Square tests were used for statistical analyses. (3) Results: DPR signs, such as radiolucency and bifurcation at the root apex, discontinuity of the mandibular canal cortex, and superimposition of the tooth root and mandibular canal, were observed statistically significantly more frequently for mandibular canal-related M3s (p < 0.05). As an objective image analysis, Lacunarity showed a statistically significant difference between related and unrelated M3s for measurements made inside and outside the mandibular canal (p < 0.05). (4) Conclusions: This study demonstrated that the discontinuity of the mandibular canal cortex and Lacunarity measured on DPR could help determine the relationship of the mandibular M3s to the mandibular canal. Full article

Defect detection in lithium-ion battery (LIB) welding presents unique challenges, including scale heterogeneity, subtle texture variations, and severe class imbalance. We propose a multi-scale convolutional framework that integrates EfficientNet-B0 for lightweight representation learning, PANet for cross-scale feature aggregation, and a YOLOv8 detection head augmented with multi-head attention. Parallel dilated convolutions are employed to approximate self-similar receptive fields, enabling simultaneous sensitivity to fine-grained microstructural anomalies and large-scale geometric irregularities. The approach is validated on three datasets including RIAWELC, GC10-DET, and an industrial LIB defects dataset, where it consistently outperforms competitive baselines, achieving 8–10% improvements in recall and F1-score while preserving real-time inference on GPU. Ablation experiments and statistical significance tests isolate the contributions of attention and multi-scale design, confirming their role in reducing false negatives. Attention-based visualizations further enhance interpretability by exposing spatial regions driving predictions. Limitations remain regarding fixed imaging conditions and partial reliance on synthetic augmentation, but the framework establishes a principled direction toward efficient, interpretable, and scalable defect inspection in industrial manufacturing. Full article

To reveal the influence mechanism of shear creep behavior of the weak interlayer (carbonaceous mud shale) from a microscopic perspective, acoustic emission (AE) technology was introduced to conduct shear creep tests to capture micro-fracture acoustic signals and analyze the microscopic damage evolution laws. The results indicate that, as normal stress increased, shear creep strain decayed exponentially, while the steady state creep rate increased gradually. Additionally, the peak value and cumulative value of the AE ringing count rate also increased gradually. The AE b-value had a staged pattern of “fluctuation adjustment → stable increase → abrupt decline”. The sudden drop in the b-value could serve as a precursor feature of creep failure. The higher the normal stress, the earlier the sudden drop in b-value and the larger the Δb value. The damage variable was defined based on the AE ringing count rate, and a new creep damage model was constructed by combining fractional-order theory. The model can uniformly describe the creep damage law of carbonaceous mud shale under different normal stresses. The reliability of the model was verified through experimental data. The research results provide a theoretical basis for long-term stability analysis of mine slopes containing weak interlayers. Full article

In this paper, we propose a new class of self-mappings, referred to as polynomial Kannan contractions, which extend the classical Kannan contractions by incorporating higher-order polynomial distance terms with variable coefficient functions. Unlike polynomial contractions, polynomial Kannan contractions are not necessarily continuous. We establish fixed point results for such mappings under suitable conditions on the coefficient functions, in addition to presenting the error estimates for the associated Picard iteration. Furthermore, we provide some supported numerical examples to show that our extensions are proper and significant. As an application, we show that our results ensure the existence and uniqueness of solutions for a certain class of fractional differential equations. Full article

Tight sandstone reservoirs generally exhibit poor physical properties and characterization of microscopic pore structure is crucial for evaluating reservoir quality and fluid flow behavior. Fractal dimension provides an effective means to quantify the complexity and heterogeneity of pore structures in such reservoirs. This study investigates tight sandstone reservoirs of the Kongdian Formation in the Nanpi Slope, Cangdong Sag, using cast thin sections, scanning electron microscopy (SEM), high-pressure mercury injection (HPMI), and constant-rate mercury injection (CRMI) experiments. We establish a full-range fractal model to characterize pore-throat distributions and elucidate the correlation between fractal dimensions and reservoir properties, alongside factors influencing pore-structure heterogeneity. Key findings include that (1) pore types are predominantly residual intergranular pores, intergranular dissolution pores, and clay mineral intercrystalline pores, with throats primarily consisting of sheet-like and curved sheet-like types, exhibiting strong pore-structure heterogeneity; (2) full-range fractal dimensions D₁, D₂ and D₄ effectively characterize the heterogeneity of pore structure, where higher D₁ and D₂ values correlate with increased macro–mega pore and micro-fine throat abundance, respectively, indicating enhanced pore connectivity and superior flow capacity, while elevated D₄ reflects greater nano throat complexity, degrading reservoir properties and impeding hydrocarbon flow; (3) compared to conventional methods splicing HPMI and CRMI data at 0.12 μm, the fractal-derived integration point more accurately resolves full-range pore-throat distributions, revealing significant disparities in pore-throat size populations; (4) the fractal dimensions D₁, D₂, and D₄ are collectively governed by clay mineral content, average throat radius, displacement pressure, and tortuosity. Full article

We present a hybrid parallel scheme for efficiently solving Caputo time-fractional partial differential equations (CTFPDEs) with integer-order spatial derivatives on multicore CPU and GPU platforms. The approach combines a second-order spatial discretization with the $L1$ time-stepping scheme and employs MATLAB parfor parallelization to achieve significant reductions in runtime and memory usage. A theoretical third-order convergence rate is established under smooth-solution assumptions, and the analysis also accounts for the loss of accuracy near the initial time $t=t_0$ caused by weak singularities inherent in time-fractional models. Unlike many existing approaches that rely on locally convergent strategies, the proposed method ensures global convergence even for distant or randomly chosen initial guesses. Benchmark problems from fractional biological models—including glucose–insulin regulation, tumor growth under chemotherapy, and drug diffusion in tissue—are used to validate the robustness and reliability of the scheme. Numerical experiments confirm near-linear speedup on up to four CPU cores and show that the method outperforms conventional techniques in terms of convergence rate, residual error, iteration count, and efficiency. These results demonstrate the method’s suitability for large-scale CTFPDE simulations in scientific and engineering applications. Full article

In this study, we examine the error bounds related to Milne-type inequalities and a widely recognized Newton–Cotes method, originally developed for three-times-differentiable convex functions within the context of Jensen–Mercer inequalities. Expanding on this foundation, we explore Milne–Mercer-type inequalities and their application to a more refined class of three-times-differentiable s-convex functions. This work introduces a new identity involving such functions and Jensen–Mercer inequalities, which is then used to improve the error bounds for Milne-type inequalities in both Jensen–Mercer and classical calculus frameworks. Our research highlights the importance of convexity principles and incorporates the power mean inequality to derive novel inequalities. Furthermore, we provide a new lemma using Caputo–Fabrizio fractional integral operators and apply it to derive several results of Milne–Mercer-type inequalities pertaining to $(\alpha ,m)$-convex functions. Additionally, we extend our findings to various classes of functions, including bounded and Lipschitzian functions, and explore their applications to special means, the q-digamma function, the modified Bessel function, and quadrature formulas. We also provide clear mathematical examples to demonstrate the effectiveness of the newly derived bounds for Milne–Mercer-type inequalities. Full article

This paper undertakes a rigorous analytical exposition of the approximate controllability of a novel class of Sobolev-type stochastic impulsive differential inclusions, incorporating the Atangana–Baleanu fractional derivative in the Caputo configuration under the influence of Wiener process and Poissonian discontinuities. The system’s analytical landscape is further enriched by the incorporation of Clarke sub-differentials, facilitating the treatment of nonsmooth, nonconvex, and multivalued dynamics. The inherent complexity arising from the confluence of fractional memory, stochastic perturbations, and impulsive phenomena necessitates the deployment of a sophisticated apparatus from variational analysis, measurable selection theory, and multivalued fixed point frameworks within infinite-dimensional Banach spaces. This study delineates rigorous sufficient conditions, ensuring controllability under such hybrid influences, thereby generalizing classical paradigms to encompass nonlocal and discontinuous dynamical regimes. A precisely articulated exemplar is included to validate the theoretical constructs and demonstrate the operational efficacy of the proposed analytical methodology. Full article

The root causes forcing the differential pore-throat performances and crude oil recoverability in heterogeneous shale lithofacies of saline-lacustrine fine-grained mixed sedimentary sequences are still debated. Especially application cases of fractal theory in characterizing pore-throat heterogeneity are still lacking and the significance of differential multifractal distribution patterns on reservoir assessment remains controversial. This present study focuses on the shale-oil reservoirs in saline-lacustrine fine-grained mixed depositional sequences of the Middle Permian Lucaogou Formation (southern Junggar Basin, NW China), and presents a set of new results from petrographical investigation, field-emission scanning electron microscopy (FE-SEM) imaging, fluid injection experiments (low-pressure N₂ adsorption and high-pressure mercury intrusion porosimetry (HMIP)), nuclear magnetic resonance (NMR) spectroscopy and T₁-T₂ mapping, directional spontaneous imbibition, as well as contact angle measurements. Our results demonstrated that the investigated lithofacies are mainly divided into a total of five lithofacies categories: felsic siltstones, sandy dolomitic sandstones, dolarenites, micritic dolomites, and dolomitic mudstones, respectively. More importantly, the felsic siltstone and sandy dolomitic siltstones can be identified as the most advantageous lithofacies categories exhibiting the strongest movable oil-bearing capacity owing to an acceptable complexity and heterogeneity of mesopore-throat structures, as evidenced by the corresponding moderate fractal dimension of mesopores (D₂) from HMIP and apparently lower fractal dimension of movable fluids’ pores (D₂) from NMR results. Particularly noteworthy is the relatively poor shale-oil movability recognized in the dolarenites, micritic dolomites, and dolomitic mudstones due to heterogeneous and unfavorable pore-throat systems, even though an acceptable micro-connectivity and a more oleophilic interfacial wettability prevails in crucial dolomitic components. Finally, a comprehensive and conceptual model is established for an effective and characteristic parameter system for assessing differential reservoir petrophysical properties, interfacial wettability, and shale-oil movability concerning heterogeneous lithofacies categories. Our achievements can serve as an analog for investigating saline-lacustrine mixed shale-oil reservoirs to gain a more comprehensive understanding of differential recoverability of dessert reservoir intervals, and to guide the assessment of “sweet spots” distribution and optimization of engineering technique schemes for commercial exploitation. Full article

Three-dimensional human pose estimation from monocular video remains challenging for clinical gait analysis due to high computational cost and the need for temporal consistency. We present Pose3DM, a bidirectional Mamba-based state-space framework that models intra-frame joint relations and inter-frame dynamics with linear computational complexity. Replacing transformer self-attention with state-space modeling improves efficiency without sacrificing accuracy. We further incorporate fractional-order total-variation regularization to capture long-range dependencies and memory effects, enhancing temporal and spatial coherence in gait dynamics. On Human3.6M, Pose3DM-L achieves 37.9 mm MPJPE under Protocol 1 (P1) and 32.1 mm P-MPJPE under Protocol 2 (P2), with 127 M MACs per frame and 30.8 G MACs in total. Relative to MotionBERT, P1 and P2 errors decrease by 3.3% and 2.4%, respectively, with 82.5% fewer parameters and 82.3% fewer MACs per frame. Compared with MotionAGFormer-L, Pose3DM-L improves P1 by 0.5 mm and P2 by 0.4 mm while using 60.6% less computation: 30.8 G vs. 78.3 G total MACs and 127 M vs. 322 M per frame. On AUST-VisGait across six gait patterns, Pose3DM consistently yields lower MPJPE, standard error, and maximum error, enabling reliable extraction of key gait parameters from monocular video. These results highlight state-space models as a cost-effective route to real-time gait assessment using a single RGB camera. Full article

This study investigates the impact of long-term memory stochastic forcing on the ensemble mean dynamics of the El Niño–Southern Oscillation (ENSO) by incorporating a fractional Ornstein-Uhlenbeck (FOU) process as external forcing into an ENSO spatiotemporal oscillator (STO) conceptual model. Unlike the classic Ornstein-Uhlenbeck (OU) process, the FOU process exhibits a slow power-law decay in auto-correlation and can trigger an anomalous growth rate that persistently influences the divergence/dissipation of the ensemble mean system. Furthermore, theoretical and quantitative analyses verify that the anomalous growth rate is determined by the ensemble mean of the FOU process, reflecting the cumulative nature of long-memory forcing. Crucially, the anomalous growth rate is positive and thus induces system divergence when the FOU process exhibits a positive ensemble mean, whereas it is negative and induces dissipation when the ensemble mean is negative. Also, the anomalous growth rate intensifies with decreasing fractional order. On the other hand, FOU forcing can also modulate the phase of the ensemble mean oscillation for ENSO. When the phases of the natural and forced oscillations are in phase alignment, the phase of the ensemble mean oscillation leads relative to the natural oscillation. Conversely, when the phases of the natural and forced oscillations are in anti-phase alignment, the phase of the ensemble mean oscillation lags behind relative to the natural oscillation. Future work would address the case when the external forcing has a spatial structure and seek observational validation of the identified growth rate signatures and phase shifts. Full article

This paper provides a thorough examination of the Actinium radioactive decay series, which converts Uranium-235 into the stable Lead-207 isotope via a succession of alpha, beta, and gamma decays. For the first time, the series is modeled using fractional calculus, employing two innovative analytical methods: the Sumudu Residual Power Series Method (SRPSM) and the Temimi Ansari Method (TAM). The study discusses the well-posedness of the fractional-order model in the Caputo sense within a Banach space setting. These fractional models capture complex, non-ideal decay behaviors more accurately than traditional exponential models. Mathematica is used to do numerical computations for four different Actinium series scenarios. The results are tabulated and visually depicted to show how radionuclide concentrations change over time. The findings demonstrate that SRPSM and TAM effectively simplify the complex differential equations governing nuclear decay, offering enhanced precision and flexibility. This work provides a robust framework for modeling the Actinium series, with potential applications in nuclear physics, radiometric dating, and radiation safety studies. Full article

This study presents a compact, wideband fractal antenna fabricated using silver nanoparticles (AgNPs) and screen-printing technology. The antenna consists of a decagonal monopole patch and a mesh ground plane, both printed on a transparent polyethylene terephthalate (PET) substrate. The proposed antenna has a compact size of 18 × 16 × 0.55 mm³, achieved by stacking two PET layers joined using double-sided tape. The antenna covers both C- and X-bands, with measured optical transmittance of 68.1% and radiation efficiency of 72%. The simulated −10 dB bandwidth (without bending) spans 4–10.8 GHz and 11.2–12.5 GHz, while the measured −10 dB bandwidth is 3.8–11.2 GHz without bending, 3–11.4 GHz at 30° bending, and 3–11.2 GHz at 45° bending, confirming that there was stable performance under flexure. The conductive patterns were formed using silver nanoparticle paste with a sheet resistance of 0.2 Ω/sq, followed by annealing in a vacuum oven at 140 °C for 20 min. The proposed antenna was tested under 30° and 45° bending, and the measured S₁₁ remained stable, confirming flexibility. The use of a flexible, optically transparent PET substrate enables installation on curved or see-through surfaces. Combining compact size, wideband performance, cost-effective fabrication, and optical transparency, the antenna demonstrates strong potential for application in X-band radar, C-band satellite communications, and S-band Wi-Fi. Full article

Nonlinear switched systems, which combine multiple subsystems with a switching rule, have garnered significant research interest due to their complex stability properties. In this paper we consider the case where the switching times, the switching rule and the family of functions defining the subsystems are given initially. Note that the switching rule could be such that it is not activated at any initially given switching time. When the switching rule is activated, then a subsystem from the given family is chosen. We study the case where the nonlinear subsystems consist of fractional differential equations. To be more concise we apply generalized Caputo fractional derivatives with respect to other functions. Lyapunov functions are used to analyze stability, and several sufficient conditions are obtained. The influence of the switching rule on the stability property of the solutions is discussed and illustrated with examples. It is shown that, in spite of the solutions of some of the subsystems being unstable, the zero solution of the switched system could be stable. Full article

Automatic control in biomedicine has attracted the attention of clinicians to mitigate the side effects resulting from drug overdoses administered to patients. To provide the most optimal and accurate results, the computer-controlled systems in biomedical engineering require more advanced tuning procedures that tackle patient variability and ensure the robustness of the control system. This has been enhanced over the past two decades through the replacement of standard PID controllers with fractional-order controllers. However, most of the developed fractional-order control methods address only the robustness with respect to gain variations. In this study, a novel fractional-order control algorithm that is robust to time constant variations is developed. The control algorithm is designed for second-order plus dead time systems. A graphical solution is chosen to solve the nonlinear system of equations for the proposed approach. Three biomedical applications are employed as case studies. The first one consists in the control of the bispectral index in general anesthesia, the second one refers to the blood glucose level control for diabetic patients, and finally, the third one tackles computerized control in chemotherapy. The closed-loop simulation results validate the efficiency of the tuning method according to the accepted values of the performance specifications in the scientific literature. Full article

In light of the computational efficiency bottleneck and inadequate regional feature representation in traditional global data approximation methods, this paper introduces the concept of non-uniform partition to transform global continuous approximation into multi-region piecewise approximation. Additionally, we propose an image representation algorithm based on linear canonical transformation and non-uniform partitioning, which enables the regional representation of sub-signal features while reducing computational complexity. The algorithm first demonstrates that the two-dimensional linear canonical transformation series has a least squares solution within each region. Then, it adopts the maximum likelihood estimation method and the scale transformation characteristics to achieve conversion between the nonlinear and linear expressions of the two-dimensional linear canonical transformation series. It then uses the least squares method and the recursive method to convert the image information into mathematical expressions, realize image vectorization, and solve the approximation coefficients in each region more quickly. The proposed algorithm better represents complex image texture areas while reducing image quality loss, effectively retains high-frequency details, and improves the quality of reconstructed images. Full article

Against the backdrop of global climate change, accurate prediction of carbon emissions is crucial for formulating effective emission reduction policies. Utilizing data from the China Energy Statistical Yearbook and the Shandong Statistical Yearbook between 2010 and 2022, this study estimates carbon emissions in Shandong Province from 2016 to 2022 using the carbon emission factor method and projects future trends through the fractional-order accumulated grey model FAGM(1,1). The forecast results indicate that both total carbon emissions and per capita carbon emissions in Shandong will follow a trajectory characterized by ‘slow increase-peak-steady decline’, while carbon emission intensity is expected to decrease consistently year by year. Based on these projections, this study proposes that Shandong should accelerate the optimization of its energy supply structure to establish a clean and low-carbon energy system, promote green transformation and upgrading of industries to cultivate new economic growth drivers, and enhance policy-market coordination mechanisms to strengthen institutional incentives and constraints. These findings provide a scientific basis for Shandong to achieve its carbon peak and carbon neutrality goals and also offer methodological references for other industrialized provinces facing similar challenges. Full article

This paper discusses the asymptotic controllability of fractional-order Sobolev-type perturbed stochastic control systems with Brownian motion and nonlocal fractional-order Sobolev stochastic conditions. A new set of sufficient conditions is established using the theory of semigroups together with iterative methods, with some advancements on the Brownian motion properties. Our main results are obtained assuming that the associated linear system is controllable. A most essential example is brought to illustrate the analysis obtained. Full article

The theory of integral inequalities has a wide range of applications in physics and numerical computation, and plays a fundamental role in mathematical analysis. The present study delves into the attractive domain of Hermite–Hadamard–Mercer (H–H–M)-type inequalities having a special emphasis on Wright’s general functions, referred to as Raina’s functions in the scientific literature. The main goal of our progressive study is to use Raina’s Fractional Integrals to derive two useful lemmas for second-differentiable functions. Using the derived lemmas, we proved a large number of fractional integral inequalities related to trapezoidal and midpoint-type inequalities where those that are twice differentiable in absolute values are convex. Some of these results also generalize findings from previous research. Next, we provide applications to error estimates for trapezoidal and midpoint quadrature formulas and to analytical evaluations involving modified Bessel functions of the first kind and q-digamma functions, and we show the validity of the proposed inequalities in numerical integration and analysis of special functions. Finally, the results are well-supported by numerous examples, including graphical representations and numerical tables, which collectively highlight their accuracy and computational significance. Full article

In precision agriculture, semantic segmentation enhances the crop yield by enabling precise disease monitoring, targeted herbicide application, and accurate crop–weed differentiation. This enhances yield; reduces the overuse of herbicides, water, and fertilizers; lowers labor costs; and promotes sustainable farming. Deep-learning-based methods are particularly effective for crop and weed segmentation, and achieve potential results. Typically, segmentation is performed using homogeneous data (the same dataset is used for training and testing). However, previous studies, such as crop and weed segmentation in a heterogeneous data environment, using heterogeneous data (i.e., different datasets for training and testing) remain inaccurate. The proposed framework uses patch-based augmented limited training data within a heterogeneous environment to resolve the problems of degraded accuracy and the use of extensive data for training. We propose an attention-driven and hierarchical feature fusion network (AHFF-Net) comprising a flow-constrained convolutional block, hierarchical multi-stage fusion block, and attention-driven feature enhancement block. These blocks independently extract diverse fine-grained features and enhance the learning capabilities of the network. AHFF-Net is also combined with an open-source large language model (LLM)-based pesticide recommendation system made by large language model Meta AI (LLaMA). Additionally, a fractal dimension estimation method is incorporated into the system that provides valuable insights into the spatial distribution characteristics of crops and weeds. We conducted experiments using three publicly available datasets: BoniRob, Crop/Weed Field Image Dataset (CWFID), and Sunflower. For each experiment, we trained on one dataset and tested on another by reversing the process of the second experiment. The highest mean intersection of union (mIOU) of 65.3% and F1 score of 78.7% were achieved when training on the BoniRob dataset and testing on CWFID. This demonstrated that our method outperforms other state-of-the-art approaches. Full article

This paper considers a new nonlocal fractional differential quasi-variational inequality (NFDQVI) comprising a fractional differential equation with a nonlocal condition and a time-dependent quasi-variational inequality in Hilbert spaces. Qualitative properties of the solution for the time-dependent parameterized quasi-variational inequality are investigated, which improve some known results in the literature. Moreover, the unique existence of the solution and Hyers–Ulam stability are obtained for the novel NFDQVI under mild conditions. Finally, the obtained abstract results for NFDQVI are applied to analyze the unique solvability and stability, addressing a time-dependent multi-agent optimization problem and a time-dependent price control problem. Full article

Background: Traumatic brain injury (TBI) disrupts hippocampal neurogenesis and dendritic structure. Objective: The objective was to assess whether fractal and multifractal analyses can sensitively quantify dendritic complexity changes in newly formed dentate gyrus neurons following TBI and hyperbaric oxygen therapy (HBO). Methods: Adult rats underwent sham surgery with HBO (SHBO), lesion-induced TBI (L), or lesion-induced TBI with HBO (LHBO). Dendritic morphology was evaluated using Euclidean, monofractal, and multifractal metrics. Results: Lesioned animals exhibited marked reductions in dendritic complexity across multiple metrics compared to both HBO-treated groups. HBO treatment partially restored complexity to near-sham levels, with multifractal spectra revealing subtle structural differences between SHBO and LHBO. Conclusions: Fractal and multifractal analyses provide sensitive tools for detecting TBI-induced morphological changes and therapeutic effects. Our findings support HBO as a potential neuroprotective intervention and demonstrate the utility of mathematical modeling in evaluating therapeutic efficacy in neurotrauma. Full article

Reservoirs and caprocks overlap with each other in heterogeneous carbonate rocks. The sealing capacity of caprocks and their controlling factors are not clear, which restricts the prediction, exploration, and development of carbonate hydrocarbon reservoirs. We selected core samples from the Ordovician reservoirs and caprocks in the Tarim Basin, China, for scanning electron microscopy, thin section, breakthrough pressure (BP), high-pressure mercury intrusion porosimetry (HMIP), and nitrogen adsorption method (N₂GA). The experimental results show that the reservoir and caprock can be distinguished by BP. The BP of the reservoir is less than 3.0 MPa, and the BP of the caprock is less than 3.0 Mpa. We analyzed the heterogeneity characteristics and differences in reservoirs and caprocks with different lithologies from the perspectives of monofractal and multifractal. The results indicate that the differences in pore structure of grainstone, dolomite, and micrite/argillaceous limestone result in significant heterogeneity differences between samples. The correlation analysis between the fractal parameters and BP indicates that the characteristics of reservoir microporous structures have a decisive impact on BP (correlation coefficient > 0.7). The pore structure of the carbonate reservoir–caprock system exhibits self-similarity. The heterogeneity of the caprock has no significant control effect on BP (correlation coefficient < 0.3), while the higher the heterogeneity of the reservoir, the greater the BP. The sealing capacity of the caprock depends on the heterogeneity differences in pore types and pore structures between the reservoirs and caprocks. When both the reservoir and the caprock are grainstone, the micropores in the reservoirs and caprocks are dispersed but evenly distributed, and little heterogeneous differences can achieve sealing. When the lithology of reservoirs and caprocks is different, the enhancement of heterogeneity differences in micropores will improve the sealing capacity of the caprock. In summary, fractal dimension is an effective method for studying the heterogeneous structure and sealing capacity of pore–throat in carbonate caprocks. This study proposes a new perspective that the difference between the heterogeneity of micropore structures of reservoirs and caprocks affects the sealing capacity of carbonate rocks, and provides a new explanation and model for the sealing mode of carbonate rock caprocks. Full article