Search Results (71)

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

Currently, there is a demand for effective flocculant systems that can be used without adverse impact on the environment and health. However, the challenge is to find the minimum dose to achieve the same efficacy as conventional flocculants. One technique involves using a mixture of natural and synthetic flocculants, the synergistic effects of which can enhance treatment efficiency. Thus, this work provides an approach using a low-cost chitosan (CH56)–polyaluminum chloride (PAC) mixture as a flocculant system for river water. Therefore, water quality was monitored in the Tamazula and Humaya rivers, which are sources of water for potabilization plants. According to the results of flocculation tests, the use of the mixture required a lower dosage (0.75 mg L⁻¹ of CH56 with 1 mg L⁻¹ of PAC; 0.75 mg L⁻¹ of CH56 with 2 mg L⁻¹ of PAC) than that used with individual flocculants (3 mg L⁻¹ of CH56; 5 mg L⁻¹ of PAC). Conveniently, the mixture produced larger and more compact flocs, favoring sedimentation kinetics and thus flocculation. Fractal dimension (F_D) and lacunarity (Λ) from microscopy images were used as indicators of the quality of the flocs formed. In general, CH56 and the mixture performed better than PAC, and the mixture allowed the best removal of the model microplastic (polystyrene). Flocculant mixtures reduced the concentration of copper ions by 58%, of tetracycline by 22%, of microplastics by 80%, and of bacteria by >90%. Hence, the authors believe that this work offers valuable information that could be used for potabilization plants aiming to reduce the dose of PAC and introduce chitosan into their coagulation–flocculation process. 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

Methanol, a renewable and sustainable fuel, provides an effective strategy for reducing greenhouse gas emissions when synthesized through carbon dioxide hydrogenation integrated with carbon capture technology. The incorporation of hydrogen into methanol-fueled engines enhances combustion efficiency, mitigating challenges such as pronounced cycle-to-cycle variations and cold-start difficulties. A simulation framework was developed using Python 3.13 and the Cantera 3.1.0 library to model the combustion system of a four-stroke spark-ignited (SI) methanol–hydrogen engine. This framework integrates a fractal turbulent combustion model with chemical reaction kinetics, complemented by early flame development and near-wall combustion models to address limitations during the initial and terminal combustion phases. The model was validated by using experimental data measured from a spark-ignited methanol engine. The effects of varying Hydrogen Energy Rates (HER) on engine power performance, combustion characteristics, and emissions (like formaldehyde and carbon monoxide) were subsequently analyzed under different operating loads, whilst the knock limit boundaries were established for different operational conditions. Findings demonstrate that increasing HER improves the engine power output and thermal efficiency, shortens the combustion duration, and reduces the formaldehyde and carbon monoxide emissions. Nevertheless, under high-load conditions, higher HER increases the knocking tendency, which constrains the maximum permissible HER decreasing from approximately 40% at 15% load to 20% at 100% load. The model has been developed into a Python library and will be open-sourced on Github. Full article

In the present work, the adsorption of vitamin B12 (VB12) on sugarcane-derived activated carbon (AC) was investigated with the purpose of developing a hybrid material able to degrade highly toxic and recalcitrant chlordecone (CLD) for remediating the severe environmental issue of pesticide pollution of water and soil. The AC used is made from locally accessible sugarcane bagasse. The experimental kinetic and isothermic data of VB12 adsorption on AC were modeled using multiple models, including Pseudo-Order 1 (Lagergren), Pseudo-Order 2, Elovich, and Brouers–Sotolongo models for the kinetics. The isotherms models used were Langmuir, Freundlich, Hill–Sips, Brouers–Sotolongo (BS), Brouers-Gaspard (BG), General Brouers–Sotolongo (GBS), and Redlich–Peterson (RP) models. The results showed that the BG model is the most suitable to satisfactorily describe the adsorption of VB12 on the studied AC, involving a heterogeneous adsorption mechanism onto a heterogeneous surface and providing the maximum adsorption capacity, a convenient tool to estimate the saturation level of adsorbate (i.e., chlordecone (CLD)) onto the adsorbent (AC). Voltammetric studies confirm the interaction between VB12 and the AC. Finally, molecular modeling is used to provide atomic insights showing the entrapment of VB12 inside the porous system to form a new hybrid material. The calculations provide the conformations with the best binding energy in the GaudiMM environment. Full article

Background: Rheumatoid arthritis is a chronic autoimmune disease that leads to severe disability and requires improved therapeutic strategies to optimize anti-inflammatory treatment. This study aimed to address this challenge by developing and characterizing an extended-release polymer matrix tablet containing ketoprofen and a ketoprofen–β-cyclodextrin complex with enhanced therapeutic properties. The objective was to improve inflammation management and therapeutic outcomes using a novel delivery system based on the inclusion of the active substance in cyclodextrin complexes. Methods: Tablets were formulated using ketoprofen and ketoprofen–β-cyclodextrin complexes combined with hydrophilic polymers such as Carbopol^® 971P NF, Kollidon^® VA 64, and Methocel^TM K4M. The complexes were obtained via the coprecipitation method to improve bioavailability. The kinetics of the release of ketoprofen, ketoprofen–β-cyclodextrin complex (2:1), and ketoprofen–β-cyclodextrin complex (1:1) from the tablets were investigated in vitro in artificial gastric and intestinal fluids, and drug release profiles were established. Advanced mathematical models were used to describe the nonlinear behavior of the drug–polymer systems. Results: The inclusion of ketoprofen in the β-cyclodextrin complexes was confirmed, revealing distinct release profiles. Tablets (K-3 F-3) containing the 1:1 complex showed rapid release (96.2% in 4–7 h), while tablets (K-1 F-4) containing free ketoprofen released 76% over 9–11 h. Higher polymer concentrations slowed the release due to gel barrier formation. Pharmacotechnical and stability tests supported their suitability as extended-release forms. A multifractal modeling approach described the release dynamics, treating the polymer–drug matrix as a complex system, with release curves characterized by variations in the fractal dimension and resolution. Conclusions: Specific hydrophilic polymer combinations effectively prolonged ketoprofen release. The developed matrix tablets, which were evaluated via in vitro studies and mathematical modeling, show promise for improving therapeutic outcomes and patient compliance during rheumatoid arthritis treatment. Full article

This study reported a one-spot preparation of magnetic composite carbon (MCC@Fe) from microcrystalline cellulose (MC). The pure cellulose was impregnated in iron (III) chloride solution and carbonized at 650 °C. The MCC@Fe composite adsorbent underwent various characterization techniques. XRD identified nanostructured Fe₃O₄ particles with an average crystallite size of 34.3 nm embedded in the core subunits of the material. FESEM images indicated a rough and irregular surface, with some cavities along its surface, incorporating Fe₃O₄ nanoparticles, while EDS analysis confirmed the presence of elements like Fe, C, and O. Notably, combining thermal and chemical treatments produces a composite with more pores and a high specific surface area (500.0 m² g⁻¹) compared to MC (1.5 m²/g). VSM analysis confirmed the magnetic properties (0.76 emu/g), while the Hydrophobic Index (HI) showed that MCC@Fe was hydrophobic (HI 1.395). The adsorption studies consisted of kinetic, mass transfer, equilibrium, and thermodynamics studies. Kinetic study of the adsorption of paracetamol on MCC@Fe composite proved to be rapid, and the time necessary for covering 95% of the surface (t_0.95) was lower than 27 min following the fractal-like pseudo-first-order model (FPFO). Liu’s isotherm proved to be the most appropriate for understanding the adsorption equilibrium. Remarkably, the maximum sorption capacity (Q_max) of paracetamol was 34.78 mg g⁻¹ at 45 °C. The ΔH° value (+27.00 kJ/mol) and the negative ΔG° values were consistent with the physisorption mechanism and favorable process. Furthermore, the mass transfer mechanism showed that the transfer is governed by the intraparticle diffusion model, with surface diffusion being the rate-limiting step when considering the Biot number greater than 100. This research displayed a single-route production of inexpensive magnetic nano adsorbents capable of efficiently eliminating paracetamol from aqueous environments. Full article

Background: Diabetes is a growing global health crisis that requires effective therapeutic strategies to optimize treatment outcomes. This study aims to address this challenge by developing and characterizing extended-release polymeric matrix tablets containing metformin hydrochloride (M-HCl), a first-line treatment for type 2 diabetes, and honokiol (HNK), a bioactive compound with potential therapeutic benefits. The objective is to enhance glycemic control and overall therapeutic outcomes through an innovative dual-drug delivery system. Methods: The tablets were formulated using hydrophilic polymers, such as Carbopol^® 71G NF and Noveon^® AA-1. The release kinetics of M-HCl and HNK were investigated through advanced mathematical models, including fractal and multifractal dynamics, to capture the non-linear and time-dependent release processes. Traditional kinetic models (zero-order, first-order, Higuchi equations) were also evaluated for comparison. In vitro dissolution studies were conducted to determine the release profiles of the active ingredients under varying polymer concentrations. Results: The study revealed distinct release profiles for the two active ingredients. M-HCl exhibited a rapid release phase, with 80% of the drug released within 4–7 h depending on polymer concentration. In contrast, HNK demonstrated a slower release profile, achieving 80% release after 9–10 h, indicating a greater sensitivity to polymer concentration. At shorter intervals, drug release followed classical kinetic models, while multifractal dynamics dominated at longer intervals. Higher polymer concentrations resulted in slower drug release rates due to the formation of a gel-like structure upon hydration, which hindered drug diffusion. The mechanical properties and stability of the matrix tablets confirmed their suitability for extended-release applications. Mathematical modeling validated the experimental findings and provided insights into the structural and time-dependent factors influencing drug release. Conclusions: This study successfully developed dual-drug extended-release matrix tablets containing metformin hydrochloride and honokiol, highlighting the potential of hydrophilic polymers to regulate drug release. The findings emphasize the utility of advanced mathematical models for predicting release kinetics and underscore the potential of these formulations to improve patient compliance and therapeutic outcomes in diabetes management. Full article

The Rank-Ordered Multifractal Analysis (ROMA) is a tool designed to characterize scale (in)variance and multifractality based on rank ordering the fluctuations in “groups” characterized by the same mono-fractal behavior (Hurst exponent). A range-limited structure-function analysis provides the mono-fractal index for each rank-ordered range of fluctuations. We discuss here two examples of multi-scale solar wind turbulence and complexity where ROMA is applied on the following: (a) data collected by Ulysses spacecraft in the fast solar wind, outside the ecliptic, between 25 and 31 January 2007, at roughly 2.5 Astronomical Units (AU) from the Sun, in the Southern heliosphere, at latitudes between −76.5 and −77.3 degrees, and (b) slow solar wind data collected in the ecliptic plane by Venus Express spacecraft, at 0.72 AU, on 28 January 2007. The ROMA spectrum of fast solar wind derived from ULYSSES data shows a scale-dependent structure of fluctuations: (1) at the smallest/kinetic range of scales (800 to 3200 km), persistent fluctuations are dominant, and (2) at the inertial range of scales (10⁴ to 2 × 10⁵ km), anti-persistent fluctuations are dominant, but less clearly developed and possibly indicative for the development of instabilities with cross-over behavior. The ROMA spectrum of the slow solar wind derived from Venus Express data, suggests a different structure of turbulence: (1) fully developed multifractal turbulence across scales between 5 × 10⁴ and 4 × 10⁵ km, with the Hurst index changing from anti-persistent to persistent values for the larger amplitude magnetic fluctuations; (2) at the smallest scales (400 to 6400 km), fluctuations are mainly anti-persistent, and the ROMA spectrum indicates a tendency towards mono-fractal behavior. Full article

Fractal scale-free structures are widely observed across a range of natural and synthetic systems, such as biological networks, internet architectures, and social networks, providing broad applications in the management of complex systems and the facilitation of dynamic processes. The global mean first-passage time (GMFPT) for random walks on the underlying networks has attracted significant attention as it serves as an important quantitative indicator that can be used in many different fields, such as reaction kinetics, network transport, random search, pathway elaboration, etc. In this study, we first introduce two degree-dependent random walk strategies where the transition probability is depended on the degree of neighbors. Then, we evaluate analytically the GMFPT of two degree-dependent random walk strategies on fractal scale-free tree structures by exploring the connection between first-passage times in degree-dependent random walk strategies and biased random walks on the weighted network. The exact results of the GMFPT for the two degree-dependent random walk strategies are presented and are compared with the GMFPT of the classical unbiased random walk strategy. Our work not only presents a way to evaluate the GMFPT for degree-dependent biased random walk strategies on general networks but also offers valuable insights to enrich the controlling of chemical reactions, network transport, random search, and pathway elaboration. Full article

Since it is a limiting nutrient element in rivers and lakes, the effective removal of phosphorus is key to alleviating eutrophication. In this study, the one-pot solvothermal method was adopted to prepare an environmentally friendly Ca-Fe-La composite. This is an amorphous material with a large specific surface area of 278.41 m² g⁻¹. The effects of coexisting anions and pH on the phosphate removal performance were explored. Phosphate adsorption mechanisms were revealed by various characterization techniques. The phosphate adsorption obeyed the fractal-like pseudo-second-order (PSO) kinetic model, implying that the overall adsorption system was highly heterogeneous. In this work, the maximum adsorption capacity predicted by the Langmuir model was 93.0 mg g⁻¹ (as PO₄³⁻-P). The phosphate-loaded Ca-Fe-La composite could be used as a slow-release fertilizer, achieving waste management and resource utilization. The presence of SO₄²⁻, CO₃²⁻ and HCO₃⁻ anions inhibited the phosphate adsorption significantly. It was unfavorable for phosphate removal at a high pH value. Inner-sphere complexation and electrostatic attraction were mainly responsible for phosphate adsorption onto the Ca-Fe-La composite. Full article

In this research, the doping of SrTiO₃ with Mn⁴⁺ was performed in order to evaluate the potential application as a photocatalyst for the degradation of organic dye pollutants. Since photocatalytic activity depends on grain microstructure, fractal analysis was used to estimate the Hausdorff dimension to provide a more thorough investigation of Mn@SrTiO₃ morphology. Structural analysis by infrared spectroscopy indicated the incorporation of Mn⁴⁺ into the SrTiO₃ lattice, while by using x-ray diffraction, the crystallite size of 44 nm was determined. The photocatalytic activity test performed on complex ethyl violet organic dye revealed potential for Mn@SrTiO₃ application in water treatment. Based on fractal regression analysis, a good estimate was obtained for the reconstruction of grain shape, with a Hasudorff dimension of 1.13679, which was used to find the best kinetics model for the photodegradation reaction. The experimental data showed a nearly linear fit with fractal-like pseudo-zero order. These findings and applications of fractal dimensions could contribute to future characterizations of photocatalysts, providing a deeper understanding of surface properties and their influence on photocatalytic activity. Full article

After hydraulic fracturing, unconventional reservoirs frequently encounter challenges related to limited effective proppant support distance and suboptimal proppant placement. Due to the strong heterogeneity of offshore reservoirs, which causes varying fracture roughnesses depending on different lithologies, a systematic study of the relationship between roughness and proppant transport could optimize operational parameters. This study incorporates the box dimension method for fractal dimension analysis to quantify roughness in auto-correlated Gaussian distributed surfaces created by true triaxial tests. Combined with the numerical analysis of (computational fluid dynamics) CFD-DEM (discrete element method) for bidirectional coupling, the laws of proppant deposition and transport processes within fractures with different roughnesses are obtained through comparative verification simulations. The results show that for rougher fractures of shale, the proppants are transported farther, but at JRC_52, (joint roughness coefficient), where there may be plugging in curved areas, there is a risk of near-well blockages. Compared to the smooth model, fluctuations in JRC_28 (tight sandstone) drastically increase turbulent kinetic energy within the fracture, altering particle transport dynamics. Moreover, smaller proppants (d/w ≤ 0.3) exhibit better transport capacity due to gravity, but the conductivity of the proppant is limited when the particles are too small. A d/w of 0.4 is recommended to guarantee transport capacity and proppant efficiency near the well. Additionally, proppants injected sequentially from small to large in shale fractures offer optimal propping effects, and can take advantage of the better transport capacity of smaller proppants in rough fractures. The large proppant (d/w = 0.8) is primarily deposited by gravity and forms a sloping sand bed, which subsequently ensures the aperture of the fractures. This research provides a fresh perspective on the influence of fracture roughness on proppant transport in offshore unconventional reservoirs and offers valuable considerations for the order of proppant injection. Full article

This study presents a novel method to determine the kinetics of spherical agglomeration in-line via monitoring the stirrer torque. In order to confirm that the torque can be used to characterise the process phases, samples were taken at different times during the agglomeration process and the morphology of agglomerates is examined using X-ray microtomography. It can be shown that if the stirrer torque is decreasing, the fast growth regime starts. Additionally, the influence of binding-liquid concentration as well as the primary particle shape was studied. Both have an influence on the process kinetics as well as on the different growth mechanisms of agglomerates during spherical agglomeration. These two growth mechanisms, distribution and immersion, can be identified by the morphological parameters sphericity, packing density and fractal dimension. In particular, the concept of fractal dimension was not used so far on agglomerates obtained from spherical agglomeration. Full article