Search Results (169)

This research presents a comparative analysis of numerical methods for solving first-kind Fredholm integral equations using the Bubnov–Galerkin method with various wavelet and orthogonal polynomial bases. The bases considered are constructed from Legendre, Laguerre, Chebyshev, and Hermite wavelets, as well as Alpert multiwavelets and CAS wavelets. The effectiveness of these bases is evaluated by measuring errors relative to known analytical solutions at different discretization levels. Results show that global orthogonal systems—particularly the Chebyshev and Hermite—achieve the lowest error norms for smooth target functions. CAS wavelets, due to their localized and oscillatory nature, produce higher errors, though their accuracy improves with finer discretization. The analysis has been extended to incorporate perturbations in the form of additive noise, enabling a rigorous assessment of the method’s stability with respect to different wavelet bases. This approach provides insight into the robustness of the numerical scheme under data uncertainty and highlights the sensitivity of each basis to noise-induced errors. Full article

Data from earth observation satellites provide unique and valuable information about water quality conditions in freshwater lakes but require significant processing before they can be used, even with the use of tools like Google Earth Engine. We use imagery from Sentinel 2 and MODIS and in situ data from the State of Utah Ambient Water Quality Management System (AQWMS) database to develop models and to generate a highly accessible, easy-to-use CSV file of chlorophyll-a (which is an indicator of algal biomass), turbidity, and water temperature measurements on Utah Lake. From a collection of 937 Sentinel 2 images spanning the period from January 2019 to May 2025, we generated 262,081 estimates each of chlorophyll-a and turbidity, with an additional 1,140,777 data points interpolated from those estimates to provide a dataset with a consistent time step. From a collection of 2333 MODIS images spanning the same time period, we extracted 1,390,800 measurements each of daytime water surface temperature and nighttime water surface temperature and interpolated or imputed an additional 12,058 data points from those estimates. We interpolated the data using piecewise cubic Hermite interpolation polynomials to preserve the original distribution of the data and provide the most accurate estimates of measurements between observations. We demonstrate the processing steps required to extract usable, accurate estimates of these three water quality parameters from satellite imagery and format them for analysis. We include summary statistics and charts for the resulting dataset, which show the usefulness of this data for informing Utah Lake management issues. We include the Jupyter Notebook with the implemented processing steps and the formatted CSV file of data as ^{supplemental materials}. The Jupyter Notebook can be used to update the Utah Lake data or can be easily modified to generate similar data for other waterbodies. We provide this method, tool set, and data to make remotely sensed water quality data more accessible to researchers, water managers, and others interested in Utah Lake and to facilitate the use of satellite data for those interested in applying remote sensing techniques to other waterbodies. Full article

The cislunar space navigation satellite system is essential infrastructure for lunar exploration in the next phase. It relies on high-precision orbit determination to provide the reference of time and space. This paper focuses on constructing a navigation constellation using special orbital locations such as Earth–Moon libration points and distant retrograde orbits (DRO), and it discusses the simplification of planetary perturbation models for their autonomous orbit determination on board. The gravitational perturbations exerted by major solar system bodies on spacecraft are first analyzed. The minimum perturbation required to maintain a precision of 10 m during a 30-day orbit extrapolation is calculated, followed by a simulation analysis. The results indicate that considering only gravitational perturbations from the Moon, Sun, Venus, Saturn, and Jupiter is sufficient to maintain orbital prediction accuracy within 10 m over 30 days. Based on these findings, a method for simplifying the ephemeris is proposed, which employs Hermite interpolation for the positions of the Sun and Moon at fixed time intervals, replacing the traditional Chebyshev polynomial fitting used in the JPL DE ephemeris. Several simplified schemes with varying time intervals and orders are designed. The simulation results of the inter-satellite links show that, with a 6-day orbit arc length, a 1-day lunar interpolation interval, and a 5-day solar interpolation interval, the accuracy loss for cislunar space navigation satellites remains within the meter level, while memory usage is reduced by approximately 60%. Full article

Pupillometry is commonly used to evaluate cognitive effort, attention, and facial expression response, offering valuable insights into human performance. The combination of eye tracking and facial expression data under the iMotions platform provides great opportunities for multimodal research. However, there is a lack of standardized pipelines for managing pupillometry data on a multimodal platform. Preprocessing pupil data in multimodal platforms poses challenges like timestamp misalignment, missing data, and inconsistencies across multiple data sources. To address these challenges, the authors introduced a systematic preprocessing pipeline for pupil diameter measurements collected using iMotions 10 (version 10.1.38911.4) during an endoscopy simulation task. The pipeline involves artifact removal, outlier detection using advanced methods such as the Median Absolute Deviation (MAD) and Moving Average (MA) algorithm filtering, interpolation of missing data using the Piecewise Cubic Hermite Interpolating Polynomial (PCHIP), and mean pupil diameter calculation through linear regression, as well as normalization of mean pupil diameter and integration of the pupil diameter dataset with facial expression data. By following these steps, the pipeline enhances data quality, reduces noise, and facilitates the seamless integration of pupillometry other multimodal datasets. In conclusion, this pipeline provides a detailed and organized preprocessing method that improves data reliability while preserving important information for further analysis. Full article

Some classical texts on the Sturm–Liouville equation ${(p(x)y^{\prime })}^{\prime }-q(x)y+\lambda \rho (x)y=0$ are revised to highlight further properties of its solutions. Often, in the treatment of the ensuing integral equations, $\rho =const$ is assumed (and, further, $\rho =1$). Instead, here we preserve $\rho (x)$ and make a simple change only of the independent variable that reduces the Sturm–Liouville equation to $y^{″}-q(x)y+\lambda \rho (x)y=0$. We show that many results are identical with those with $\lambda \rho -q=const$. This is true in particular for the mean value of the oscillations and for the analog of the Riemann–Lebesgue Theorem. From a mechanical point of view, what is now the total energy is not a constant of the motion, and nevertheless, the equipartition of the energy is still verified and, at least approximately, it does so also for a class of complex $\lambda$. We provide here many detailed properties of the solutions of the above equation, with $\rho =\rho (x)$. The conclusion, as we may easily infer, is that, for large enough $\lambda$, locally, the solutions are trigonometric functions. We give the proof for the closure of the set of solutions through the Phragmén–Lindelöf Theorem, and show the separate dependence of the solutions from the real and imaginary components of $\lambda$. The particular case of $q(x)=\alpha \rho (x)$ is also considered. A direct proof of the uniform convergence of the Fourier series is given, with a statement identical to the classical theorem. Finally, the proof of J. von Neumann of the completeness of the Laguerre and Hermite polynomials in non-compact sets is revisited, without referring to generating functions and to the Weierstrass Theorem for compact sets. The possibility of the existence of a general integral transform is then investigated. Full article

This paper presents a novel generalization of the mth-order Laguerre and Laguerre-based Appell polynomials and examines their fundamental properties. By establishing quasi-monomiality, we derive key results, including recurrence relations, multiplicative and derivative operators, and the associated differential equation. Additionally, both series and determinant representations are provided for this new class of polynomials. Within this framework, several subpolynomial families are introduced and analyzed including the generalized mth-order Laguerre–Hermite Appell polynomials. Furthermore, the generalized mth-order Laguerre–Gould–Hopper-based Appell polynomials are defined using fractional operators and we investigate their structural characteristics. New families are also constructed, such as the mth-order Laguerre–Gould–Hopper–based Bernoulli, Laguerre–Gould–Hopper–based Euler, and Laguerre–Gould–Hopper–based Genocchi polynomials, exploring their operational and algebraic properties. The results contribute to the broader theory of special functions and have potential applications in mathematical physics and the theory of differential equations. Full article

Multivariate normal moments are foundational for statistical methods. The derivation and simplification of these moments are critical for the accuracy of various statistical estimates and analyses. Normal moments are the building blocks of the Hermite polynomials, which in turn are the building blocks of the Edgeworth expansions for the distribution of parameter estimates. Isserlis (1918) gave the bivariate normal moments and two special cases of trivariate moments. Beyond that, convenient expressions for multivariate variate normal moments are still not available. We compare three methods for obtaining them, the most powerful being the differential method. We give simpler formulas for the bivariate moment than that of Isserlis, and explicit expressions for the general moments of dimensions 3 and 4. Full article

Transfinite interpolation, originally proposed in the early 1970s as a global interpolation method, was first implemented using Lagrange polynomials and cubic Hermite splines. While initially developed for computer-aided geometric design (CAGD), the method also found application in global finite element analysis. With the advent of isogeometric analysis (IGA), Bernstein–Bézier polynomials have increasingly replaced Lagrange polynomials, particularly in conjunction with tensor product B-splines and non-uniform rational B-splines (NURBSs). Despite its early promise, transfinite interpolation has seen limited adoption in modern CAD/CAE workflows, primarily due to its mathematical complexity—especially when blending polynomials of different degrees. In this context, the present study revisits transfinite interpolation and demonstrates that, in four broad classes, Lagrange polynomials can be systematically replaced by Bernstein polynomials in a one-to-one manner, thus giving the same accuracy. In a fifth class, this replacement yields a robust dual set of basis functions with improved numerical properties. A key advantage of Bernstein polynomials lies in their natural compatibility with weighted formulations, enabling the accurate representation of conic sections and quadrics—scenarios where IGA methods are particularly effective. The proposed methodology is validated through its application to a boundary-value problem governed by the Laplace equation, as well as to the eigenvalue analysis of an acoustic cavity, thereby confirming its feasibility and accuracy. Full article

High-quality data are foundational to reliable environmental monitoring and urban planning in smart cities, yet challenges like missing values and outliers in air pollution and meteorological time series data are critical barriers. This study developed and validated a dual-phase framework to improve data quality using a 60-month gas and weather dataset from Jubail Industrial City, Saudi Arabia, an industrial region. First, outliers were identified via statistical methods like Interquartile Range and Z-Score. Machine learning algorithms like Isolation Forest and Local Outlier Factor were also used, chosen for their robustness to non-normal data distributions, significantly improving subsequent imputation accuracy. Second, missing values in both single and sequential gaps were imputed using linear interpolation, Piecewise Cubic Hermite Interpolating Polynomial (PCHIP), and Akima interpolation. Linear interpolation excelled for short gaps (R² up to 0.97), and PCHIP and Akima minimized errors in sequential gaps (R² up to 0.95, lowest MSE). By aligning methods with gap characteristics, the framework handles real-world data complexities, significantly improving time series consistency and reliability. This work demonstrates a significant improvement in data reliability, offering a replicable model for smart cities worldwide. Full article

This study investigates the computation of fractional and higher integer-order moments for a stochastic process governed by a one-dimensional, non-homogeneous linear stochastic differential equation (SDE) driven by fractional Brownian motion (fBm). Unlike conventional approaches relying on moment-generating functions or Fokker–Planck equations, which often yield intractable expressions, we derive explicit closed-form formulas for these moments. Our methodology leverages the Wick–Itô calculus (fractional Itô formula) and the properties of Hermite polynomials to express moments efficiently. Additionally, we establish a recurrence relation for moment computation and propose an alternative approach based on generalized binomial expansions. To validate our findings, Monte Carlo simulations are performed, demonstrating a high degree of accuracy between theoretical and empirical results. The proposed framework provides novel insights into stochastic processes with long-memory properties, with potential applications in statistical inference, mathematical finance, and physical modeling of anomalous diffusion. Full article

For an extended lattice Boltzmann model based on a product-form equilibrium distribution function, an improved regularization model with enhanced numerical stability is proposed. In this paper’s regularized collision model, coefficients are calculated using two distinct methods during the reconstruction of the non-equilibrium distribution. The first method stems from the direct projection of the non-equilibrium distribution, while the second method relies on the regularization step, which is refined through the recursive calculation of the coefficients of non-equilibrium Hermite polynomials. Compared to the original lattice Boltzmann model, the recursive regularization method significantly enhances the stability of the numerical scheme by appropriately filtering out second-order and/or higher-order non-hydrodynamic contributions. Initially, under isothermal conditions, the periodic double-shear layer simulations are conducted at Reynolds numbers ranging from 10⁴ to 10⁶, testing the enhanced effect of the regularized model in broadening its available speed range. Subsequently, with a fixed Reynolds number, simulations are performed at various temperature values to assess the model’s performance when deviating from the lattice reference temperature. The results demonstrate that, compared to the original model, the recursive regularization model exhibits improved stability and widens the model’s usable speed and temperature ranges. Full article

In this paper, a new and general form of truncated-exponential-based general-Appell polynomials is introduced using the two-variable general-Appell polynomials. For this new polynomial family, we present an explicit representation, recurrence relation, shift operators, differential equation, determinant representation, and some other properties. Finally, two special cases of this family, truncated-exponential-based Hermite-type and truncated-exponential-based Laguerre–Frobenius Euler polynomials, are introduced and their corresponding properties are obtained. Full article

An experiment is conducted to investigate the turbulent flow field close to a wall-fastened horizontal cylinder. The evolution of the flow field is analyzed by evaluating turbulent flow characteristics and fluid dynamics along the lengthwise direction. The approach flow velocity retards in the immediate upstream area of the cylinder. At the crest level of the cylindrical pipe, the turbulence characteristics such as Reynolds stresses and turbulence intensities are attaining their peaks. Gram–Charlier (GC) series-based Hermite polynomials yield probability density functions that better match experimental data than those from Gram–Charlier (GC) series-based exponential distributions, demonstrating the superiority of the Hermite polynomial method. Quadrant analysis reveals that sweeps (Q4) dominate intermediate and free-surface zones, while ejections (Q2) prevail near the bed, both being primary contributors to Reynolds shear stress (RSS). The stress component remains minimal or zero for all events when $\mathrm{hole} \mathrm{size} \left(\mathrm{H}\right)\ge \mathrm{six}$. Larger hole sizes (≥five) drastically reduced the stress fraction, approaching zero. The stress fraction was highest near the cylinder, decreasing with distance and eventually plateauing. The study enhances the understanding of flow hydraulics around cylindrical objects in rough-bed natural streams. Full article

In this paper, we define a new generalization of three-variable q-Laguerre polynomials and derive some properties. By using these polynomials, we introduce a new generalization of three-variable q-Laguerre-based Appell polynomials (3VqLbAP) through a generating function approach involving zeroth-order q-Bessel–Tricomi functions. These polynomials are studied by means of generating function, series expansion, and determinant representation. Also, these polynomials are further examined within the framework of q-quasi-monomiality, leading to the establishment of essential operational identities. We then derive operational representations, as well as q-differential equations for the three-variable q-Laguerre-based Appell polynomials. Some examples are constructed in terms of q-Laguerre–Hermite-based Bernoulli, Euler, and Genocchi polynomials in order to illustrate the main results. Full article

In this paper, by using the zeroth-order q-Tricomi functions, the theory of three-variable q-Legendre-based Appell polynomials is introduced. These polynomials are studied by means of generating functions, series expansions, and determinant representation. Further, by utilizing the concepts of q-quasi-monomiality, these polynomials are examined as several q-quasi-monomial and operational representations; the q-differential equations for the three-variable q-Legendre-based Appell polynomials were obtained. In addition, we established a new generalization of three-variable q-Legendre-Hermite-Appell polynomials, and we derive series expansion, determinant representation, and q-quasi-monomial and q-differential equations. Some examples are framed to better illustrate the theory of three-variable q-Legendre-based Appell polynomials, and this is characterized by the above properties. Full article