Search Results (18)

Predicting anomalous diffusion in quantum walks with non-Markovian environmental noise is computationally demanding. We introduce FracPINN, a fractional physics-informed neural network that embeds a fully differentiable, PyTorch-based Caputo PDE solver into a classical LSTM encoder. Rather than replacing classical predictors, FracPINN acts as a compact physics regularizer that constrains the inference with emergent fractional transport dynamics; every physics-informed loss component is strictly label-free, and the ground-truth exponent enters only through an explicitly supervised regression term. Evaluated on 3709 non-Markovian DTQW simulations with exponentially correlated Gaussian coin noise (filtered from 5000 raw samples to the physically admissible exponent range), FracPINN achieves a mean absolute error of $0.214$ and $R^2=0.682$ on held-out test data, outperforming classical baselines by $5.3\%$ in terms of the MAE overall, while adding only four interpretable physical parameters. Notably, gains concentrate in the sub-diffusive regime where memory effects dominate, with a $13.5\%$ MAE improvement over the baseline there, yet the normal and super diffusive accuracy remains intact. Once trained, the surrogate reduces inference from seconds of simulation to fractions of a millisecond per sample. These results show that our fractional PDE networks are most compelling as targeted physics refinements with minimal overhead within stable classical pipelines. Full article

Background/Objectives: The integration of artificial intelligence (AI) into pharmaceutical development has the potential to accelerate early-stage formulation design. In this study, large language models (ChatGPT (GPT-4o, OpenAI) and DeepSeek (DeepSeek-R1, DeepSeek AI) were evaluated as supportive tools for the design of sustained-release lornoxicam matrix tablets. Using constrained formulation prompts and a predefined excipient space, each model generated candidate formulations intended for direct compression, with the objective of producing sustained-release systems capable of mimicking the dissolution behaviour of a commercial reference product (LOROX OD 16 mg). Methods: The proposed formulations were prepared experimentally and evaluated for physicochemical properties, including weight variation, hardness, friability, and drug content, as well as in vitro dissolution performance over 24 h. Dissolution profiles were compared with the reference product using similarity (f₂) and difference (f₁) factors, and release behaviour was further characterized using kinetic models. Results: All formulations demonstrated sustained-release behaviour without evidence of dose dumping. One ChatGPT-generated formulation (F3C) met the regulatory criteria for dissolution similarity to the reference product (f₁ = 9.66, f₂ = 71.31), while the remaining formulations showed variable release behaviour with f₂ values ranging from 28.61 to 49.70. However, F3C exceeded the pharmacopeial friability limit marginally (1.108%), while DeepSeek formulations F5D and F6D exceeded pharmacopeial assay acceptance limits. Kinetic modelling indicated a range of transport mechanisms from anomalous diffusion to super Case II transport depending on polymer composition. Conclusions: Although both AI systems successfully generated experimentally viable formulations, prediction accuracy analysis showed high trend-level correlations between AI-predicted and experimental dissolution profiles. However, the magnitude of quantitative error was substantial, with RMSE values exceeding 17% and MAPE values ranging from approximately 38% to 60%. These findings indicate that the models captured general release trends but did not provide reliable quantitative dissolution predictions. Full article

This study develops a fractional Jeffreys heat conduction model to describe laser-induced thermoelastic distortions in thin optical materials under ultrafast surface heating. The framework employs three fractional parameters to characterize anomalous thermal transport modes: retarded conduction, accelerated conduction, and transitions between super- and sub-diffusive regimes. Thermo-optic effects are represented through a linear relation between temperature and refractive-index perturbation; however, a full optical-aberration decomposition is not claimed in this work. Numerical results demonstrate that anomalous heat transfer significantly affects temperature localization, heat-flux evolution, stress distributions, and OPD-based thermo-optic indicators in components subjected to ultrafast laser pulses. Quantitative optical indicators, including refractive-index variation, optical path difference, wavefront error, focal-length shift, and thermal-lens distortion, are derived from the computed temperature field to connect the thermal solution directly with thin-lens performance. Simulations combining Maple2024 and MATLAB R2023a quantify the coupled thermoelastic-optical response at picosecond time scales. Full article

Recently introduced, power Brownian motion and power Levy motion are versatile and practical anomalous-diffusion models. On the one hand, the power motions are easily constructed and are easily tracked. On the other hand, the power motions display an assortment of anomalous behaviors including: sub-diffusion and super-diffusion; aging and anti-aging; and persistence and anti-persistence. This paper investigates the power motions from a socioeconomic-inequality perspective. Using this perspective, key statistical and temporal behaviors of the power motions are interpreted and scored. In particular, the paper provides simple and explicit quantitative answers–which are based on socioeconomic inequality indices–to the following question: what is the ‘degree of anomaly’ of each of the power-motions’ anomalous behaviors? The socioeconomic approach presented in this paper may be applied (in future research) to additional anomalous-diffusion models. Full article

A study was carried out in Santiago de Chile, located in a geographic basin, on the sustainability and diffusion of the recent SARS-CoV-2 pandemic. Hourly measurements were used (carried out for 3.25 years in seven communes of the city) to quantify the accumulated sick (AS) population, urban meteorology variables (MVs) (temperature (T), relative humidity (RH), and magnitude of wind speed (WS)), and air pollution (P) (PM₁₀, PM_2.5, 0₃). Time series (TS) were constructed for each commune, which related AS to MVs, called AS/VM, and to P, noted AS/P. Chaos theory was applied to each TS, requiring the following variables: the Lyapunov exponent (λ > 0), the correlation dimension (D_C < 5), Kolmogorov entropy (S_K > 0), the Hurst exponent (H, such that 0 < H < 1), Lempel–Ziv complexity (LZ > 0), and information loss (<ΔI> < 0). Every TS complied with chaos theory. For each commune, C_K was calculated as a quotient between the sum of AS/T, AS/WS, and AS/RH entropies and the sum of AS/PM₁₀, AS/PM_2.5, and AS/O₃ entropies. The results show that the entropy for the AS/P ratio is lower than that of the AS/VM ratio in three of the seven communes, since between 2020 and early 2022, the population was confined, reducing pollution. The TS of the AS/P ratio is more persistent and complex. The predictability times of the ratios are comparable in four of the seven communes. The TS of the AS/MV ratios shows greater information loss and chaos. According to the calculated C_K values, it is possible to relate it to anomalous diffusion (sub/super-diffusion) and the context that favored the expansion of the pandemic: urban densification, pollution, urban meteorology, population density, etc. Using Fréchet heavy-tailed probability, the compatibility of the results with C_K is verified. Full article

Given that pectin is a well-known substance used for drug delivery, we aimed to obtain and further examine the efficacy of interpolyelectrolyte complexes based on citrus or apple pectin and the Eudragit^® EPO for using these carriers in oral drug delivery. To characterize the physicochemical properties of these compounds, turbidity, gravimetry, viscosity, elementary analysis, FTIR spectroscopy, and DSC analysis were utilized. Diffusion transport characteristics were evaluated to assess the swelling ability of the matrices and the release of diclofenac sodium. To examine the release parameters, mathematical modeling was performed by using the Korsmayer–Peppas and Logistic equations as well. During the turbidity study, stoichiometry compositions were selected for the developed IPECs EPO/PecA and EPO/PecC at pH values = 4.0, 5.0, 6.0, and 7.0. The FTIR spectra of the complexes were characterized by an increase in the intensity of the bands at 1610 cm⁻¹ and 1400 cm⁻¹. According to the DSC analysis, IPEC has a certain Tg = 57.3 °C. The highest release rates were obtained for IPEC EPO/PecC_1 and EPO/PecC_4. The mechanism of drug transport from the matrices IPEC EPO/PecC, IPEC EPO/PecA_3, and EPO/PecA_4 can be characterized as Super Case II. Anomalous release (non-Fickian release) is typical for IPEC EPO/PecA_1 and EPO/PecA_2. Thus, the resulting systems can be further used for the effective delivery of the drugs to the colon. Full article

The Jeffreys-type heat conduction equation with flux precedence describes the temperature of diffusive hot electrons during the electron–phonon interaction process in metals. In this paper, the deformation resulting from ultrafast surface heating on a “nanoscale” plate is considered. The focus is on the anomalous heat transfer mechanisms that result from anomalous diffusion of hot electrons and are characterized by retarded thermal conduction, accelerated thermal conduction, or transition from super-thermal conductivity in the short-time response to sub-thermal conductivity in the long-time response and described by the fractional Jeffreys equation with three fractional parameters. The recent double-strip problem, Awad et al., Eur. Phy. J. Plus 2022, allowing the overlap between two propagating thermal waves, is generalized from the semi-infinite heat conductor case to thermoelastic case in the finite domain. The elastic response in the material is not simultaneous (i.e., not Hookean), rather it is assumed to be of the Kelvin–Voigt type, i.e., $\sigma =E\left(\epsilon +{\tau }_{\epsilon }\dot{\epsilon }\right)$, where $\sigma$ refers to the stress, $\epsilon$ is the strain, $E$ is the Young modulus, and ${\tau }_{\epsilon }$ refers to the strain relaxation time. The delayed strain response of the Kelvin–Voigt model eliminates the discontinuity of stresses, a hallmark of the Hookean solid. The immobilization of thermal conduction described by the ordinary Jeffreys equation of heat conduction is salient in metals when the heat flux precedence is considered. The absence of the finite speed thermal waves in the Kelvin–Voigt model results in a smooth stress surface during the heating process. The temperature contours and the displacement vector chart show that the anomalous heat transfer characterized by retardation or crossover from super- to sub-thermal conduction may disrupt the ultrafast laser heating of metals. Full article

The development of a lipid nano-delivery system was attempted for three specific poly (ADP-ribose) polymerase 1 (PARP1) inhibitors: Veliparib, Rucaparib, and Niraparib. Simple lipid and dual lipid formulations with 1,2-dipalmitoyl-sn-glycero-3-phospho-rac-(1′-glycerol) sodium salt (DPPG) and 1,2-dipalmitoyl-sn-glycero-3-phosphocoline (DPPC) were developed and tested following the thin-film method. DPPG-encapsulating inhibitors presented the best fit in terms of encapsulation efficiency (>40%, translates into concentrations as high as 100 µM), zeta potential values (below −30 mV), and population distribution (single population profile). The particle size of the main population of interest was ~130 nm in diameter. Kinetic release studies showed that DPPG-encapsulating PARP1 inhibitors present slower drug release rates than liposome control samples, and complex drug release mechanisms were identified. DPPG + Veliparib/Niraparib presented a combination of diffusion-controlled and non-Fickian diffusion, while anomalous and super case II transport was verified for DPPG + Rucaparib. Spectroscopic analysis revealed that PARP1 inhibitors interact with the DPPG lipid membrane, promoting membrane water displacement from hydration centers. A preferential membrane interaction with lipid carbonyl groups was observed through hydrogen bonding, where the inhibitors’ protonated amine groups may be the major players in the PARP1 inhibitor encapsulation mode. Full article

The application of fractional calculus in the field of kinetic theory begins with questions raised by Bernoulli, Clausius, and Maxwell about the motion of molecules in gases and liquids. Causality, locality, and determinism underly the early work, which led to the development of statistical mechanics by Boltzmann, Gibbs, Enskog, and Chapman. However, memory and nonlocality influence the future course of molecular interactions (e.g., persistence of velocity and inelastic collisions); hence, modifications to the thermodynamic equations of state, the non-equilibrium transport equations, and the dynamics of phase transitions are needed to explain experimental measurements. In these situations, the inclusion of space- and time-fractional derivatives within the context of the continuous time random walk (CTRW) model of diffusion encodes particle jumps and trapping. Thus, we anticipate using fractional calculus to extend the classical equations of diffusion. The solutions obtained illuminate the structure and dynamics of the materials (gases and liquids) at the molecular, mesoscopic, and macroscopic time/length scales. The development of these models requires building connections between kinetic theory, physical chemistry, and applied mathematics. In this paper, we focus on the kinetic theory of gases and liquids, with particular emphasis on descriptions of phase transitions, inter-phase mixing, and the transport of mass, momentum, and energy. As an example, we combine the pressure–temperature phase diagrams of simple molecules with the corresponding anomalous diffusion phase diagram of fractional calculus. The overlap suggests links between sub- and super-diffusion and molecular motion in the liquid and the vapor phases. Full article

In this paper, we study the phenomena of collapse and anomalous diffusion in shared mobility systems. In particular, we focus on a fleet of vehicles moving through a stations network and analyse the effect of self-journeys in system stability, using a mathematical simplex under stochastic flows. With a birth-death process approach, we find analytical upper bounds for random walk and we monitor how the system collapses by super diffusing under different randomization conditions. Using the multi-scale entropy metric, we show that real data from a bike-sharing fleet in the city of Salamanca (Spain) present a complex behaviour with more of a $1/f$ signal than a disorganized system with a white noise signal. Full article

Even though applications of direct numerical simulations are on the rise, today the most usual method to solve turbulence problems is still to apply a closure scheme of a defined order. It is not the case that a rising order of a turbulence model is always related to a quality improvement. Even more, a conceptual advantage of applying a lowest order turbulence model is that it represents the analogous method to the procedure of introducing a constitutive equation which has brought success to many other areas of physics. First order turbulence models were developed in the 1920s and today seem to be outdated by newer and more sophisticated mathematical-physical closure schemes. However, with the new knowledge of fractal geometry and fractional dynamics, it is worthwhile to step back and reinvestigate these lowest order models. As a result of this and simultaneously introducing generalizations by multiscale analysis, the first order, nonlinear, nonlocal, and fractional Difference-Quotient Turbulence Model (DQTM) was developed. In this partial review article of work performed by the authors, by theoretical considerations and its applications to turbulent flow problems, evidence is given that the DQTM is the missing (apparent) constitutive equation of turbulent shear flows. Full article

Fractional-order time and space derivatives are one way to augment the classical diffusion equation so that it accounts for the non-Gaussian processes often observed in heterogeneous materials. Two-dimensional phase diagrams—plots whose axes represent the fractional derivative order—typically display: (i) points corresponding to distinct diffusion propagators (Gaussian, Cauchy), (ii) lines along which specific stochastic models apply (Lévy process, subordinated Brownian motion), and (iii) regions of super- and sub-diffusion where the mean squared displacement grows faster or slower than a linear function of diffusion time (i.e., anomalous diffusion). Three-dimensional phase cubes are a convenient way to classify models of anomalous diffusion (continuous time random walk, fractional motion, fractal derivative). Specifically, each type of fractional derivative when combined with an assumed power law behavior in the diffusion coefficient renders a characteristic picture of the underlying particle motion. The corresponding phase diagrams, like pages in a sketch book, provide a portfolio of representations of anomalous diffusion. The anomalous diffusion phase cube employs lines of super-diffusion (Lévy process), sub-diffusion (subordinated Brownian motion), and quasi-Gaussian behavior to stitch together equivalent regions. Full article

Significant deviations from standard Boltzmann scaling, which corresponds to normal or Fickian diffusion, have been observed in the literature for water movement in porous media. However, as demonstrated by various researchers, the widely used conventional Richards equation cannot mimic anomalous diffusion and ignores the features of natural soils which are heterogeneous. Within this framework, governing equations of transient water flow in porous media in fractional time and multi-dimensional fractional soil space in anisotropic media were recently introduced by the authors by coupling Brooks–Corey constitutive relationships with the fractional continuity and motion equations. In this study, instead of utilizing Brooks–Corey relationships, empirical expressions, obtained by least square fits through hydraulic measurements, were utilized to show the suitability of the proposed fractional approach with other constitutive hydraulic relations in the literature. Next, a finite difference numerical method was proposed to solve the fractional governing equations. The applicability of the proposed fractional governing equations was investigated numerically in comparison to their conventional counterparts. In practice, cumulative infiltration values are observed to deviate from conventional infiltration approximation, or the wetting front through time may not be consistent with the traditional estimates of Richards equation. In such cases, fractional governing equations may be a better alternative for mimicking the physical process as they can capture sub-, super-, and normal-diffusive soil water flow processes during infiltration. Full article

In this work, we investigate a series of mathematical aspects for the fractional diffusion equation with stochastic resetting. The stochastic resetting process in Evans–Majumdar sense has several applications in science, with a particular emphasis on non-equilibrium physics and biological systems. We propose a version of the stochastic resetting theory for systems in which the reset point is in motion, so the walker does not return to the initial position as in the standard model, but returns to a point that moves in space. In addition, we investigate the proposed stochastic resetting model for diffusion with the fractional operator of Prabhakar. The derivative of Prabhakar consists of an integro-differential operator that has a Mittag–Leffler function with three parameters in the integration kernel, so it generalizes a series of fractional operators such as Riemann–Liouville–Caputo. We present how the generalized model of stochastic resetting for fractional diffusion implies a rich class of anomalous diffusive processes, i.e., $⟨{(\Delta x)}^2⟩\propto t^{\alpha }$ , which includes sub-super-hyper-diffusive regimes. In the sequence, we generalize these ideas to the fractional Fokker–Planck equation for quadratic potential $U\left(x\right)=a{x}^2+bx+c$ . This work aims to present the generalized model of Evans–Majumdar’s theory for stochastic resetting under a new perspective of non-static restart points. Full article

The microscopic and semi-macroscopic mechanisms responsible for photoinduced mass transport in functionalized azo-polymers are far from deeply understood. To get some insight into those mechanisms on “microscopic” scale, we studied the directed photoinduced motion of single functionalized polymer chains under various types of polarized light illumination using Monte Carlo bond fluctuation model and our kinetic Monte Carlo model for photoinduced mass transport. We found sub-diffusive, diffusive and super-diffusive regimes of the dynamics of single chains at constant illumination and mostly super-diffusive regime for directed motion in the presence of the gradient of light intensity. This regime is more enhanced for long than for short chains and it approaches the ballistic limit for very long chains. We propose a physical picture of light-driven inscription of Surface Relief Gratings (SRG) as corresponding to a dynamical coexistence of normal and anomalous diffusion in various parts of the system. A simple continuous time random walk model of SRG inscription based on this physical picture reproduced the light-driven mass transport found in experiments as well as the fine structure of SRG. Full article