Search Results (157)

This paper focuses on the invariant properties of fractal operators and aims to achieve the axiomatization of the theory of fractal operators. Building upon the derivative and integral theorems of the Laplace transform, we redefine the time differential operator and demonstrate that the newly defined operator exhibits form invariance under the Laplace transform. This property is further generalized to encompass broader classes of operators, including non-rational and fractional fractal operators. Inspired by Klein’s concept of “invariance under transformation groups”, we propose a postulate asserting the “form invariance of operators under the Laplace transform group”. Based on this postulate, we clarified the algebraic operational rules of fractal operators and constructed a rigorously axiomatized theory of fractal operators. Full article

Background: This paper develops a novel, interdisciplinary framework for optimizing high-speed rail (HSR) freight logistics hubs in the Ottawa–Quebec City corridor, addressing critical gaps in geospatial mismatches, static optimization limitations, and narrow sustainability scopes found in the existing literature. Methods: The research methodology integrates a hybrid graph neural network-reinforcement learning (GNN-RL) architecture that encodes 412 nodes into a dynamic graph with adaptive edge weights, fractal accessibility (α = 1.78) derived from fractional calculus (α = 0.75) to model non-linear urban growth patterns, and a multi-criteria sustainability evaluation framework embedding shadow pricing for externalities. Methodologically, the framework is validated through global sensitivity analysis and comparative testing against classical optimization models using real-world geospatial, operational, and economic datasets from the corridor. Results: Key findings demonstrate the framework’s superiority. Empirical results show an obvious reduction in emissions and lower logistics costs compared to classical models, with Pareto-optimal hubs identified. These hubs achieve the most GDP coverage of the corridor, reconciling economic efficiency with environmental resilience and social equity. Conclusions: This research establishes a replicable methodology for mid-latitude freight corridors, advancing low-carbon logistics through the integration of GNN-RL optimization, fractal spatial analysis, and sustainability assessment—bridging economic viability, environmental decarbonization, and social equity in HSR freight network design. Full article

The Yang local fractional setting provides the generalized framework to explore the non-differentiable mappings considering the local properties. Due to the dominance of these concepts, mathematicians have investigated multiple problems, including mathematical modelling, optimization, and inequalities. Incorporating these useful concepts, this study aims to derive Weddle-type integral inequalities within the context of fractal space. To achieve the intended results, we establish a new local fractional identity. By using this identity, the convexity property, the bounded property of mappings, the L-Lipschitzian property of mappings, and other famous inequalities, we develop numerous upper bounds. Additionally, we provide 2D and 3D graphical representations and numerous applications, which show the significance of our main findings. To the best of our knowledge, this is the first study concerning error inequalities of Weddle’s quadrature formulation within the fractal space. Full article

Small-diameter vascular grafts often fail due to thrombosis and compliance mismatch. Decellularized plant scaffolds are a biocompatible, sustainable alternative. Leatherleaf viburnum leaves provide natural architecture and mechanical integrity suitable for tissue-engineered vessels. However, the persistence of immunogenic plant biomolecules and limited degradability remain barriers to clinical use. This study tested whether mild heat treatment improves scaffold biocompatibility without compromising mechanical performance. Decellularized leatherleaf viburnum scaffolds were treated at 30–40 °C in 5% NaOH for 15–60 min and then evaluated via tensile testing, burst pressure analysis, scanning electron microscopy, histology, and in vitro assays with white blood cells and endothelial cells. Scaffold properties were compared to those of untreated controls. Heat treatment did not significantly affect scaffold thickness but decreased fiber area fraction and diameter across all anatomical layers. Scaffolds treated at 30–35 °C for ≤30 min retained >90% of tensile strength and achieved burst pressures ≥820 mmHg, exceeding physiological arterial pressures. Heat treatment reduced surface fractal dimension while increasing entropy and lacunarity, producing a smoother but more heterogeneous microarchitecture. White blood cell viability increased up to 2.5-fold and endothelial cell seeding efficiency improved with treatment duration, with 60 min producing near-confluent monolayers. Mild alkaline heat treatment therefore improved immune compatibility and endothelialization while preserving mechanical integrity, offering a simple, scalable modification to advance plant-derived scaffolds for grafting. Full article

This study introduces a new class of coupled differential systems described by fractal–fractional Caputo derivatives with both constant and state-dependent delays. In contrast to traditional delay differential equations, the proposed framework integrates memory effects and geometric complexity while capturing adaptive feedback delays that vary with the system’s state. Such a formulation provides a closer representation of biological and physical processes in which delays are not fixed but evolve dynamically. Sufficient conditions for the existence and uniqueness of solutions are established using fixed-point theory, while the stability of the solution is investigated via the Hyers–Ulam (HU) stability approach. To demonstrate applicability, the approach is applied to two illustrative examples, including a predator–prey interaction model. The findings advance the theory of fractional-order systems with mixed delays and offer a rigorous foundation for developing realistic, application-driven dynamical models. Full article

We introduce a class of triply coupled systems of differential equations with fractal–fractional Caputo derivatives and time-dependent delays. This framework captures long-memory effects and complex structural patterns while allowing delays to evolve over time, offering greater realism than constant-delay models. The existence and uniqueness of solutions are established using fixed point theory, and Hyers–Ulam stability is analyzed. A numerical scheme based on the Adams–Bashforth method is implemented to approximate solutions. The approach is illustrated through a numerical example and applied to a three-species food-chain model, comparing scenarios with and without time-dependent delays to demonstrate their impact on system dynamics. Full article

In this study, we consider a fractional-order extension of the Jeffreys equation (also known as the dual-phase-lag equation) by introducing the Reimann–Liouville fractional integral, of order $0<\nu <1$, to the Jeffreys constitutive law, where for $\nu =1$ it corresponds to the conventional Jeffreys equation. The kinetical behaviors of the fractional equation such as non-negativity of the propagator, mean-squared displacement, and the temporal amplitude are investigated. The fractional Langevin equation, or the fractional damped oscillator, is a special case of the considered integrodifferential equation governing the temporal amplitude. When $\nu =0$ and $\nu =1$, the fractional differential equation governing the temporal amplitude has the mathematical structure of the classical linear damped oscillator with different coefficients. The existence of a real solution for the new temporal amplitude is proven by deriving this solution using the complex integration method. Two forms of conditional closed-form solutions for the temporal amplitude are derived in terms of the Mittag–Leffler function. It is found that the proposed generalized fractional damped oscillator equation results in underdamped oscillations in the case of $0<\nu <1$, under certain constraints derived from the non-fractional case. Although the nonfractional case has the form of classical linear damped oscillator, it is not necessary for its solution to have the three common types of oscillations (overdamped, underdamped, and critical damped), unless a certain condition is met on the coefficients. The obtained results could be helpful for analyzing thermal wave behavior in fractals, heterogeneous materials, or porous media since the fractional-order derivatives are related to the porosity of media. Full article

Blast-furnace staves serve as critical protective components in ironmaking, requiring synergistic optimization of slag-coating behavior and self-protection capability to extend furnace lifespan and reduce energy consumption. Traditional integer-order heat transfer models, constrained by assumptions of homogeneous materials and instantaneous heat conduction, fail to accurately capture the cross-scale thermal memory effects and non-local diffusion characteristics in multiphase heterogeneous blast-furnace systems, leading to substantial inaccuracies in predicting dynamic slag-layer evolution. This review synthesizes recent advancements across three interlinked dimensions: first, analyzing design principles of zonal staves and how refractory material properties influence slag-layer formation, proposing a “high thermal conductivity–low thermal expansion” material matching strategy to mitigate thermal stress cracks through optimized synergy; second, developing a mechanistic model by introducing the Caputo fractional derivative to construct a non-Fourier heat-transfer framework (i.e., a heat-transfer model that accounts for thermal memory effects and non-local diffusion, beyond the instantaneous heat conduction assumption of Fourier’s law), which effectively describes fractal heat flow in micro-porous structures and interfacial thermal relaxation, addressing limitations of conventional models; and finally, integrating industrial case studies to validate the improved prediction accuracy of the fractional-order model and exploring collaborative optimization of cooling intensity and slag-layer thickness, with prospects for multiscale interfacial regulation technologies in long-life, low-carbon stave designs. Full article

The article presents research on concrete creep in bridge structures, focusing on the influence of concrete mix composition and the use of advanced rheological models with fractional-order derivatives. Laboratory tests were performed on nine mixes varying in blast furnace slag content (0%, 25%, and 75% of cement mass) and air-entrainment. The results were used to calibrate fractal rheological models—Kelvin–Voigt and Huet–Sayegh—where the viscous element was replaced with a fractal element. These models showed high agreement with experimental data and improved the accuracy of creep prediction. Comparison with Eurocode 2 revealed discrepancies up to 64%, especially for slag-free concretes used in prestressed bridge structures. The findings highlight the important role of mineral additives in reducing creep strains and the need to consider individual mix characteristics in design calculations. In the context of modern bridge construction technologies, such as balanced cantilever or incremental launching, reliable modeling of early-age creep is particularly important. The proposed modeling approach may enhance the precision of long-term structural behavior analyses and contribute to improved safety and durability of concrete infrastructure. Full article

This paper focuses on the numerical modeling of drug diffusion in biological tissues using fractional time-dependent parabolic equations with non-local boundary conditions. The model includes a Caputo fractional derivative to capture the non-local effects and memory inherent in biological processes, such as drug absorption and transport. The theoretical framework of the problem is based on the work of Alhazzani, et al.,which demonstrates the solution’s goodness, existence, and uniqueness. Building on this foundation, we present a robust numerical method designed to deal with the complexity of fractional derivatives and non-local interactions at the boundaries of biological tissues. Numerical simulations reveal how fractal order and non-local boundary conditions affect the drug concentration distribution over time, providing valuable insights into drug delivery dynamics in biological systems. The results underscore the potential of fractal models to accurately represent diffusion processes in heterogeneous and complex biological environments. Full article

In this paper, we first introduce a parametric identity for generalized differentiable functions using a generalized fractal–fractional integral operators. Based on this identity, we establish several variants of parameterized inequalities for functions whose local fractional derivatives in absolute value satisfy generalized convexity conditions. Furthermore, we demonstrate that our main results reduce to well-known Ostrowski- and Simpson-type inequalities by selecting suitable parameters. These inequalities contribute to finding tight bounds for various integrals over fractal spaces. By comparing the classical Hölder and Power mean inequalities with their new generalized versions, we show that the improved forms yield sharper and more refined upper bounds. In particular, we illustrate that the generalizations of Hölder and Power mean inequalities provide better results when applied to fractal integrals, with their tighter bounds supported by graphical representations. Finally, a series of applications are discussed, including generalized special means, generalized probability density functions and generalized quadrature formulas, which highlight the practical significance of the proposed results in fractal analysis. Full article

Fractional calculus plays a central role in modeling memory-dependent processes and complex dynamics across various fields, including control theory, fluid mechanics, and bioengineering. This study introduces an efficient and stable fractional-order iterative method based on the Caputo derivative for solving nonlinear equations. By employing a Taylor series expansion, a local convergence analysis shows that for $\gamma \in (0,1]$, the method achieves a convergence order of $2\gamma +1$. To address challenges related to memory effects and instability in existing approaches, the proposed scheme incorporates parameter optimization through chaos and bifurcation analysis. Dynamical plane analysis reveals that parameter values within chaotic regimes lead to divergence, while those in stable regions converge uniformly. The method’s performance is evaluated using a set of nonlinear models drawn from biomedical engineering, including enzyme kinetics with inhibition, extended glucose–insulin regulation, drug dose–responses, and lung volume–pressure dynamics. Comparative results demonstrate that the proposed approach outperforms existing methods in terms of iteration count, residual error, CPU time, convergence order, fractal behavior, and memory efficiency. These findings underscore the method’s applicability to complex systems characterized by nonlinearity and memory effects in scientific and engineering contexts. Full article

Water pollution from industrial and domestic sewage demands the accurate modeling of wastewater treatment processes. While the Lawrence–McCarty model is widely used for activated sludge systems, its integer-order formulation cannot fully capture the fractal characteristics of microbial aggregation. This study proposed a fractal Lawrence–McCarty model (FLMM) by incorporating local fractional derivatives (α = ln2/ln3) to describe microbial growth dynamics on Cantor sets. Theoretical analysis reveals that the FLMM exhibits Mittag-Leffler-type solutions, which naturally generate step-wise growth curves—consistent with the phased behavior (lag, rapid growth, and stabilization) observed in real sludge systems. Compared with classical models, the FLMM’s fractional-order structure provides a more flexible framework to represent memory effects and spatial heterogeneity in microbial communities. These advances establish a mathematical foundation for future experimental validation and suggest potential improvements in predicting nonlinear biomass accumulation patterns. 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

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