Regularized numerical method for the space-fractional logarithmic Klein-Gordon equation

Projects integrating Science, Technology, Engineering, and Mathematics (STEM) are essential to interdisciplinary research. This study presents a STEM (Science, Technology, Engineering, and Mathematics) methodology with the primary objective of designing, constructing, and validating a functional cannabinoid extraction device. To inform the device’s drying parameters, the dehydration kinetics of female hemp buds or flowering buds (FHB) were first analyzed using infrared drying at 100 °C for different durations. The plants were cultivated and harvested in accordance with good agricultural practices using Dinamed CBD Autoflowering seeds. The FHB were harvested and prepared by manually separating them from the stems and leaves. Six 5 g samples were prepared, each with a slab geometry of varying surface area and thickness. Two of these samples were ground: one into a fine powder and the other into a coarse powder. Mathematical fits were obtained for each resulting curve using either an exponential decay model or the logarithmic equation yt=Ae−kt+y0 calculate the equilibrium moisture (mE). The Moisture Rate (MR) was calculated, and by modelling with the logarithmic equation, the constant k and the effective diffusivity (Deff) were determined with the analytical solution of Fick’s second law. The Deff values (ranging from 10−7 to 10−5) were higher than previously reported. The coarsely ground powder sample yielded the highest k and Deff values and was selected for oil extraction. The device was then designed using Quality Function Deployment (QFD), specifically the House of Quality (HoQ) matrix, to systematically translate user requirements into technical specifications. A 200 g sample of coarsely ground, dehydrated FHB was prepared for ethanol extraction. Chemical results obtained by Liquid Chromatography coupled with Photodiode Array Detection (LC-PDA) revealed the presence of THC, CBN, CBC, and CBG. The extraction device design was validated using previous results showing the presence of CBD and CBDA. The constructed device successfully extracted cannabinoids, including Δ9-THC, CBG, CBC, and CBN, from coarsely ground FHB, validating the integrated STEM approach. This work demonstrates a practical framework for developing accessible agro-technical devices through interdisciplinary collaboration.

Experimental results of a study of the effect of ionizing radiation with different dose rates on fiber-optic systems using fibers with a germanosilicate core (GeO 2 ) and an undoped pure silica core (SiO 2 ) are presented. A mathematical approximation of the experimental curves for the growth of radiation-induced optical losses in the fiber is performed using a modified power-law equation that includes the contribution of the ionizing radiation dose rate. A correlation between the values of empirical coefficients and the dose rate of ionizing radiation is experimentally established. A natural logarithmic function equation is proposed for describing the dependence of the empirical coefficients defining the shape of the radiation-induced loss growth curve on the dose rate. An approach to reconstructing the radiation-induced loss growth curve using equations for the dependence of the empirical approximation coefficients on the dose rate is proposed; this technique enables predicting the radiation response of the optical fiber under new conditions without experimental setup. The method demonstrates applicability for single-mode fibers with silica and germanosilicate fiber cores of different designs. Based on the described approach, a methodology is developed for determining correlation equations for empirical coefficients.

I investigate a dynamical dark energy model based on a generalized logarithmic equation of state, derived from a canonical scalar field Lagrangian and extending the Ma–Zhang parametrization via two parameters [Formula: see text]. Fixing [Formula: see text], I compute the cosmic time–redshift relation [Formula: see text] by numerically evaluating the integral [Formula: see text], where [Formula: see text] follows from the Friedmann equation and a redshift-dependent [Formula: see text]. I reconstruct the scalar field dynamics [Formula: see text] and potential [Formula: see text] corresponding to this phenomenological input, and calculate the evolution of key cosmological quantities including [Formula: see text], [Formula: see text] and [Formula: see text]. Using Markov Chain Monte Carlo (MCMC) sampling, I constrain the model against observational data for [Formula: see text], recovering posterior distributions for [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. I further analyze the onset of future singularities and the coincidence problem, noting that the current age of the universe may lie near the midpoint to a big rip. Perturbative stability is confirmed within the parameter space considered. The framework provides a flexible and physically consistent approach to modeling evolving dark energy within a scalar field theory.

It is known that the logarithmic Volterra integral equations are so hard to find its exact analytic solution, so we are obliged to find a new numerical method strong and reliable.. In this work, we develop a technical approximation to obtain a numerical solution of the logarithmic Volterra and Fredholm integral equations, This new approximation is based on the Shifted second Chebyshev polynomials the unknown function will be approximated by a truncated sum of shifted Chebyshev Polynomials. our spectral method sends the given equation into an algebraic system of equations with the unknown expansion coefficients. The convegence, the efficiency and the high accuracy of this method are realized by many examples.

Bamboo is an environmentally friendly construction material because it is renewable, lightweight, and possesses good mechanical strength. However, due to the lack of basic knowledge about the properties of bamboo culms, only a few species are commonly utililized. This study aims to investigate the longitudinal variation of physical and mechanical properties of Dendrocalamus asper and Gigantochloa apus that grow naturally on Lombok Island. Sampling was carried out by selecting 10 bamboo culms, aged 3-4 years old, from different clumps at each location. The tests conducted include moisture content, basic density, tangential and radial shrinkage, modulus of elasticity (MOE), and modulus of rupture (MOR). Statistical analysis was performed using R software, incorporating linear and non-linear mixed-effects models to evaluate longitudinal variations and the influence of individual and location on the distribution of bamboo properties. The results showed that the values for green moisture content, basic density, tangential and radial shrinkage at 1% moisture content change, MOE, and MOR for G. apus were 99.97%, 0.60 g/cm³, 0.29%, 0.35%, 8.27 GPa, and 108.80 MPa, respectively. Meanwhile, the values for D. apser were 108.13%, 0.58 g/cm³, 0.34%, 0.42%, 9.71 GPa, and 102.47 MPa, respectively. The longitudinal variation of moisture content in G. apus followed a linear pattern, while a logarithmic equation best described the variation in moisture content of D. asper, basic density, MOE, and MOR for both bamboo species. Additionally, the longitudinal variation in tangential and radial shrinkage for both species remained constant, following a linear equation with a y-intercept.

The ocean is a priority for governments and international organizations. Large-scale, in situ ocean water quality monitoring programs are not very feasible due to the high costs associated with their implementation and operation. In this work, we present a tool for assessing ocean conditions, the SIMAR Integrated Marine Water Quality Index (ICAM-SIMAR-Integral), composed of two satellite parameters and three numerical models. We evaluated its spatiotemporal variability at 10 sites in the Gulf of Mexico, which have dissimilar environmental conditions. We validated its use by comparing it with the TRIX trophic index at 41 sites. To construct the index, the five parameters were standardized using a logarithmic equation and then summed, weighted according to their relationship with eutrophication. An index with a scale of 1 to 100 was obtained, divided into five classification intervals: oligotrophic, mesotrophic, eutrophic, supertrophic, and hypertrophic. The median values of the index and its parameters exhibited significant spatial and temporal variability, consistent with the literature’s criteria regarding their values and eutrophication thresholds. Comparison with TRIX showed no significant differences, validating the implementation of ICAM-SIMAR-Integral as an easily interpreted early warning system for managers and decision-makers in conservation matters. This index will allow for continuous, large-scale monitoring of the ocean, thereby contributing synoptically to its sustainable use.

Abstract Some relevant aspects of a new form of generalized entropic cosmology, recently introduced by Nojiri, Odintsov and Faraoni, are considered. The setup is a logarithmic equation of state for a viscous dark fluid coupled with dark matter, in the ordinary Friedmann–Lemaître–Robertson–Walker flat universe. The influence of thermal effects, caused by Hawking radiation, near the singularity, are carefully investigated. In particular, their role on the formation and specific type of the Big Rip expected to occur within a finite time. It is shown that a scenario arises, where a qualitative change towards the good direction, in the type of the singularity formed, does occur. On top of that, another very interesting scenario is obtained, where the singularity vanishes completely.

Abstract This paper is concerned with the study of the following double phase equation with logarithmic nonlinearity $$\begin{aligned}&-\operatorname {div}\left( |\nabla u|^{p-2}\nabla u+\mu (x) |\nabla u|^{q-2}\nabla u\right) + |u|^{p-2}u+\mu (x) |u|^{q-2}u \\&=K_{1}(x)|u|^{p^*-2}u+\lambda K_{2}(x)|u|^{r-2}u\log (|u|)\quad \text {in } \mathbb {R}^N, \end{aligned}$$ - div | ∇ u | p - 2 ∇ u + μ ( x ) | ∇ u | q - 2 ∇ u + | u | p - 2 u + μ ( x ) | u | q - 2 u = K 1 ( x ) | u | p ∗ - 2 u + λ K 2 ( x ) | u | r - 2 u log ( | u | ) in R N , with dimension $$N\ge 2$$ N ≥ 2 , parameter $$\lambda >0$$ λ > 0 , $$1<p<q<N$$ 1 < p < q < N , $$\mu :\mathbb {R}^{N}\rightarrow [0,\infty )$$ μ : R N → [ 0 , ∞ ) is a Lipschitz continuous function and $$\max \{p,N(p-1)/(N-p)\}<r<p^*=Np/(N-p)$$ max { p , N ( p - 1 ) / ( N - p ) } < r < p ∗ = N p / ( N - p ) . Here, the weight function $$K_1$$ K 1 is positive, while $$K_2$$ K 2 may change sign on $$\mathbb {R}^{N}$$ R N . By a different variational approach, we prove an existence result which in some aspects improves our contribution in [A. Bahrouni, A. Fiscella, P. Winkert, J. Math. Anal. Appl. 547 (2025), no. 2, Paper No. 129311, 24 pp.]. For this, we need some restrictive assumptions on the weights $$\mu (\cdot )$$ μ ( · ) , $$K_1$$ K 1 and $$K_2$$ K 2 .

This study characterizes sphere-packed beds with diameter gradient for optimizing gas diffusion layer (GDL) in fuel cells. The key indices of the porous media are permeability and corresponding homogeneity of velocity field. The former guarantees high accessibility to reaction sites from fuel and oxygen sources (e.g., flow channels) and the latter promotes effective utilization of reaction sites’ surface area leading to large electrochemical active area (ECSA). For this objective, five different porous media were prepared: 1. fast increase to upper limitation, 2. linear increase, 3. Slow increase, 4. fixed upper limitation, and 5. fixed lower limitation. Each type includes 8 binarized models, and accordingly, 40 porous media were generated with stable porosity of 0.5. The type of 1. fast increase exhibited high portion of large spheres compared to the types of 2. linear and 3. slow increases. The sphere diameter of them (types 1, 2, and 3) on the outlet is the same with the lower limitation. For arrangement of spheres, the fast and slow increases employed logarithmic equations with the coordinate of z-direction. To assess the key performances, permeability and homogeneity of velocity distribution were figured out by single-phase lattice Boltzmann method (LBM) under a steady pressure difference. The index representing homogeneity is Geary constant showing degree of clustering on the outlet plane that consists of the finest spheres. In addition, mass diffusion simulations governed by the explicit linear diffusion equation (the Fick’s laws) were implemented under steady fluxes on inlet and outlet planes. The dimensionless diffusion coefficients were derived to assess the pore network with pore centers and throats for smooth mass diffusion. As boundary conditions of permeability and diffusion simulations, periodic boundary conditions on x- and y- directions (orthognathic to main flow) were applied. As a result, the porosity of porous media exhibits from 0.48 to 0.51, indicating stable values with a negligible standard deviation of 0.01. The mean pore size distribution on z-direction planes according to gradient direction shows the same tendency with the sphere diameter. The mean pore sizes across the whole pore network are 3.95, 2.79, 1.95, 4.17, and 1.50 lattices, in order of the types from 1 to 5, respectively. The permeability of each model exhibits figures of 1.51, 0.15, 0.36, 2.58, and 0.18 in the LBM unit, with the same order, respectively. The porous media with fixed upper and lower diameters show the maximum and minimum permeabilities, showing predominant influence of the fraction of higher grains on permeability with a similar tendency to that of mean pore size. In terms of uniform supply of fuel and oxygen to reaction sites, the Geary constants on outlet’s x-velocity field of 0.043, 0.069, 0.069, 0.011, and 0.073 were observed, in order of the types from 1 to 5, respectively. A higher Geary constant signifies more severe clustering, ununiform supply to reaction sites. The linear increasing diameter demonstrates a comparable even distribution to the slow increase and fixed lower limitation, while outperforming permeability to those. Regarding mass diffusion coefficient, the difference between the models seems to be insignificant from 0.32 to 0.40. Since this study does not account for diffusion regime such as Knudsen and molecular diffusions, it is affected by porosity, tortuosity, and constructivity, not the length scale of the pore network. Due to the most tortuous and highest deviation of pore size of the model with the fixed lower diameter, it shows the lowest diffusion coefficient of 0.323. These results imply that linear gradient of length scale could be optimized structure to maximize ECSA while minimizing degradation of fuel and oxygen accessibility. Since the conventional structure of the GDL with the gas diffusion substrate and microporous layer in polymer electrolyte fuel cells exhibits a similar structure to the slow increase in the present study, there may be room for the design to be further optimized. Figure 1

From the point of view of physics, the main task in the analysis of technical systems is to determine the real state of the system from the full set of permissible ones, by conducting an appropriate set of studies. In this case, it becomes possible to use both full-scale objects and their imitations, that is, models. When using a full-scale experiment, the means of experimental research interact directly with the object (sample) being studied. In a model experiment, tests are carried out not with the sample itself, but with its substitute − a model. On the other hand, from the point of view of applied materials science, an example of a “black box” system can be the dependence of parameters that characterize the structural state of rolled metal (dimensions of structural components, their percentage content, etc.) on its geometric properties (thickness of rolled metal). Therefore, the meaning of any modeling is the ability, based on the results of experiments conducted on models, to obtain qualitative and quantitative relationships between physical quantities that determine the behavior of the system in full-scale conditions. At the same time, the use of their models instead of experimental samples allows to significantly reduce the cost of full-scale experimental research. Thus, based on the principles of building models of this type, it is relevant to develop methods for their application to solve a number of applied problems of modern materials science. Purpose of the article. Obtaining a mathematical model of the relationship between the parameters of the structural state and the thickness of rolled metal from low-carbon low-alloy steels. Conclusion. Mathematical models of the relationship between the parameters of the structural state and the thickess of rolled metal from low-carbon low-alloy steels were obtained. Microstructural analysis showed that the microstructural components of all the studied systems are ferrite and pearlite. With increasing thickness, the percentage content of the ferrite component increases with a simultaneous decrease in the amount of pearlite. At the same time, an increase in the size of the structural components is observed. Based on the obtained quantitative data on the dependence of the parameters of the structural state on the thickness of the rolled metal, the corresponding regression models were constructed. Analysis of the models shows that the analyzed dependencies are nonlinear and are described by logarithmic equations of the type Y = b0 + b1 × log10(x). The adequacy of the obtained regression models was checked using the quasi-Newton method of residues (graphical method). The analysis showed that the regression models describe the dependencies that were investigated with a sufficient degree of adequacy.

Measuring soil moisture has a big impact on resource efficiency and decision-making in geotechnical engineering, agriculture, and environmental sustainability. This study presents a revolutionary Internet of Things-based method for predicting soil type and moisture in real time. Water content was verified using the traditional oven-dry method for accuracy evaluation, and capacitance measurements were gathered using a sensor to generate a custom dataset in the lab. The method outperformed linear regression in water content prediction, achieving 96.49% accuracy using an Excel logarithmic regression equation. Furthermore, a machine learning model that employed polynomial regression was able to measure the water content of the soil and predict values for fine, medium-coarse, and coarse sand types. With an R2 score of 0.79, the model can account for almost 79% of the variation in water content and produce 1.71% of MAE(Mean Absolute of Error), indicating a strong relationship between capacitance and water content. On the expanded dataset, Random Forest classifier was chosen for classification, which correctly identified the intended soil types with an accuracy of roughly 97.77%. By combining sensor data with sophisticated algorithms, the suggested methodology makes it possible to analyze soil qualities effectively and non-destructively. This scalable method offers substantial potential for environmental management, soil monitoring, and precision agriculture and is flexible enough for the creation of mobile applications. Predictive modeling and real-time data processing combined improve resource management effectiveness while lowering the need for human intervention. In order to improve forecast accuracy and application and support more environmentally friendly farming methods and environmental monitoring, further research will surely give efforts to enrich the dataset, incorporate a variety of soil types, and take environmental elements like temperature and salinity into consideration.

What degree of dilution is needed to reduce matrix effects in SERS: An example of the Cu(OH)2-Ag/CN-CDots substrate for the detection of malachite green?

ABSTRACT This paper addresses the global existence and asymptotic behavior of solutions to a logarithmic wave equation posed in a bounded domain and incorporating strong damping, a fractional time derivative, and an infinite‐memory term. The model includes a nonlinear logarithmic source, which arises in various physical contexts such as structural vibrations, fluid dynamics, and quantum mechanics. The combined effects of strong damping and fractional dissipation are essential for ensuring well‐posedness and system stabilization, while the infinite‐memory term introduces a complex, history‐dependent dynamic. This study extends the recent work of the first two authors ( Nonlinear logarithmic wave equations: Blow‐up phenomena and the influence of fractional damping, infinite memory, and strong dissipation, Evol. Equ. Control Theory, 13 (2024), 1423–1435) by establishing new results on global dynamics. Numerical simulations are also provided to illustrate the long‐term behavior of solutions and support the theoretical findings.

Instability of Equilibrium Solutions for a Logarithmic Heat Equation in Commutative Banach Algebras

Background and Aims:Monitoring oxygen saturation (SpO2) during general anaesthesia using pulse oximetry is mandatory, though it may not always reflect the actual oxygenation status. Arterial blood gas (ABG) analysis remains the gold standard for measuring the partial pressure of oxygen (PaO2), but it is invasive and limited by complications. This study aims to validate five existing equations for predicting PaO2 from SpO2 and, if applicable, to derive and validate a new equation in adult patients undergoing surgeries under general anaesthesia.Methods:A prospective cohort study was conducted on adult patients undergoing general anaesthesia and requiring ABG measurements at several time points intraoperatively. Using SPSS software, a Bland-Altman analysis was performed to assess the agreement between derived and measured PaO2 values using five existing equations. Regression analysis was then performed to develop a new equation for predicting PaO2, which was validated in a second cohort.Results:During the initial phase, 150 ABG samples were collected from 83 patients. Bland-Altman analysis revealed weak agreements with all existing equations. A new logarithmic equation, El-Khatib’s equation (PaO2 = 10−25.6 × SpO213.9), was derived. The validation phase involved 150 ABG samples from 65 patients, demonstrating strong agreement with El-Khatib’s equation (systematic bias of 13 mmHg, limits of agreement: −189 to 214 mmHg).Conclusion:Five existing equations for predicting PaO2 from SpO2 revealed weak agreement in adult patients under general anaesthesia. El-Khatib’s equation demonstrated strong potential for predicting PaO2, providing a reliable non-invasive alternative for determining PaO2 in clinical practice.

Despite its potential danger, site amplification effects are often neglected in seismic hazard analysis. Appropriate amplification factors can be determined from shear wave velocity, but impracticality in in situ measurements leads to reliance on regional correlation with geotechnical parameters such as SPT N-value. Modified power law and logarithmic equations were derived from past correlation studies to determine Vs30 values for each borehole location in the City of Manila. Vs30 profiles were spatially interpolated using the inverse-distance weighted and thin-spline methods to approximate the variation in shear wave velocities and add more detail to the existing contour map for soil profile classification across Metro Manila. Statistical analysis of the interpolated models indicates percentage differences ranging from 0 to 10% with a normalized root mean square error of nearly 5%. Generated equations and geospatial models in the study may be used as a basis for a seismic microzonation model for Metro Manila, considering other geological and geophysical layers.

Octopus mimus is an octopod of interest to benthic fisheries in Peru and an important species in food webs. Its beaks or jaws can be found in the stomach contents of various species, making them a useful tool for species identification, as well as for estimating age and biomass. This study aimed to establish regression formulas between beak measurements and mantle length (ML) and body weight (W) of O. mimus specimens caught in the adjacent waters of the Lobos de Afuera Islands, Peru. Two hundred fifty beaks were analyzed. Our results showed that the most useful beak measurements for estimating the ML were upper hood length (UHL) and the length of the baseline of the lower beak (LBL). These measures fit logarithmic and exponential equations, respectively. For W, the best measurements were upper crest length (UCL) and LBL, both of which fit exponential equations. This study could be useful in analyzing the diet and trophic ecology of its predator species. It is the first study to develop a regression formula from O. mimus beak measurements to estimate ML and W.

Abstract Transforming forest inventory data into useful information about a stand requires equations to estimate tree volume and merchantable volume, or green weight. Taper equations that predict diameters at any point on the stem are essential for merchantable volume estimations. Here, we measured taper, total and merchantable volume, and green weight by destructively sampling 324 longleaf pine (Pinus palustris Mill.) trees grown on sites with differing site histories including 11 sites corresponding to former agricultural fields and 9 clearcut areas (formerly stocked with forest). We tested whether there is an effect of stand origin on taper and volume. We fit the measured data to available taper, total and merchantable volume, and green weight models published for longleaf pine plantations and developed updated models fit to the newly collected data. We assessed the updated models fit here to a pre-existing longleaf pine dataset. The best model for stem taper used the Max and Burkhart (Segmented polynomial regression applied to taper equations. For Sci 1976;22:283–9.) form [root mean square deviation (RMSD) = 0.7 cm], the best model for total volume used the generalized logarithmic equation (Abs. Bias = 4.4%), and the best total green weight equation used the Bullock and Burkhart (Equations for predicting green weight of loblolly pine trees in the south. South J Appl For 2003;27:153–9.) form (RMSD = 0.01 kg). Notably, stem taper and volume did not vary with stand origin. The models presented here for individual tree taper, volume, and green weight perform exceedingly well for planted longleaf pine stems grown on old-field and cut-over sites in Georgia.

Abstract Infrared thermography assessment using cold stress test (CST) is often used to diagnose pathologies that affect vascularization. However, the diagnostic capacity of the CST can still be questioned because of the high intersubject variability of hand skin temperature response. The objective of this study was to assess which is the main between-subject factors that explain hand skin temperature response after CST. A total of 78 participants (44 females and 34 males) did a CST based in a hand immersion during 30 s at 6.4 °C of water temperature. Skin temperature was measured using infrared thermography before and during the 3 min after the CST. To obtain coefficients that explain the thermal response after CST, a logarithmic equation for each region of interest of the hand was calculated, and intercept and slope coefficients were obtained. Stepwise multiple regression analysis was performed to explain each one of the logarithmic coefficients (β0 and β1) with the demographic data as inputs. Our results showed that the main between-subject factors that explain hand skin temperature response after CST were the body surface area, the physical activity volume, and the sex/menstrual cycle, explaining 20% of the variance of the response after CST. In conclusion, between-subject factors assessed in our study explained the 20% of the variance of the response after CST suggesting that there is still a large percentage of the variance that could be explained by pathology when the test is used to diagnose diseases that affect vascularization.