Search Results (164)

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

In this paper, we introduce the Hermite wavelet method (HWM), a numerical method for the fractional-order Bagley–Torvik equation (BTE) solution. The recommended method is based on a polynomial called the Hermite polynomial. This method uses collocation points to turn the given differential equation into an algebraic equation system. We can find the values of the unknown constants after solving the system of equations using the Maple program. The required approximation of the answer was obtained by entering the numerical values of the unknown constants. The approximate solution for the given fractional-order differential equation is also shown graphically and numerically. The suggested method yields straightforward results that closely match the precise solution. The proposed methodology is computationally efficient and produces more accurate findings than earlier numerical approaches. Full article

The primary objective of this study is the computation of the matching polynomials of a number of symmetric, semisymmetric, double group graphs, and solids in third and higher dimensions. Such computations of matching polynomials are extremely challenging problems due to the computational and combinatorial complexity of the problem. We also consider a series of recursive graphs possessing symmetries such as D_2h-polyacenes, wheels, and fans. The double group graphs of the Möbius types, which find applications in chemically interesting topologies and stereochemistry, are considered for the matching polynomials. Hence, the present study features a number of vertex- or edge-transitive regular graphs, Archimedean solids, truncated polyhedra, prisms, and 4D and 5D polyhedra. Such polyhedral and Möbius graphs present stereochemically and topologically interesting applications, including in chirality, isomerization reactions, and dynamic stereochemistry. The matching polynomials of these systems are shown to contain interesting combinatorics, including Stirling numbers of both kinds, Lucas polynomials, toroidal tree-rooted map sequences, and Hermite, Laguerre, Chebychev, and other orthogonal polynomials. Full article

During switching in electrical systems, transient electromagnetic processes occur. The resulting dangerous current surges are best studied by computer simulation. However, the time required for computer simulation of such processes is significant for complex electromagnetic devices, which is undesirable. The use of spectral methods can significantly speed up the calculation of transient processes and ensure high accuracy. At present, we are not aware of publications showing the use of spectral methods for calculating transient processes in electromagnetic devices containing ferromagnetic cores. The purpose of the work: The objective of this work is to develop a highly effective method for calculating electromagnetic transient processes in a coil with a ferromagnetic magnetic core connected to a voltage source. The method involves the use of nonlinear magnetoelectric substitution circuits for electromagnetic devices and a spectral method for representing solution functions using orthogonal polynomials. Additionally, a schematic model for applying the spectral method is developed. Obtained Results: A method for calculating transients in magnetoelectric circuits based on approximating solution functions with algebraic orthogonal polynomial series is proposed and studied. This helps to transform integro-differential state equations into linear algebraic equations for the representations of the solution functions. The developed schematic model simplifies the use of the calculation method. Representations of true electric and magnetic current functions are interpreted as direct currents in the proposed substitution circuit. Based on these methods, a computer program is created to simulate transient processes in a magnetoelectric circuit. Comparing the application of various polynomials enables the selection of the optimal polynomial type. The proposed method has advantages over other known methods. These advantages include reducing the simulation time for electromagnetic transient processes (in the examples considered, by more than 12 times than calculations using the implicit Euler method) while ensuring the same level of accuracy. The simulation of processes over a long time interval demonstrate error reduction and stabilization. This indicates the potential of the proposed method for simulating processes in more complex electromagnetic devices, (for example, transformers). Full article

This manuscript introduces the family of Gould–Hopper–Lambda polynomials and establishes their quasi-monomial properties through the umbral method. This approach serves as a powerful mechanism to analyze the characteristic of multi-variable special polynomials. Several summation formulas for these polynomials are explored, and their operational identities are obtained using partial differential equations. The corresponding results for Hermite–Lambda polynomials are also obtained. In addition, a conclusion is given. Full article

This paper explores the operational principles and monomiality principles that significantly shape the development of various special polynomial families. We argue that applying the monomiality principle yields novel results while remaining consistent with established findings. The primary focus of this study is the introduction of degenerate multidimensional Hermite-based Appell polynomials (DMHAP), denoted as ${}_{\mathbb{H}}\mathbb{A}_n^{\left[r\right]}(l_1,l_2,l_3,\dots ,l_r;\vartheta )$. These DMHAP forms essential families of orthogonal polynomials, demonstrating strong connections with classical Hermite and Appell polynomials. Additionally, we derive symmetric identities and examine the fundamental properties of these polynomials. Finally, we establish an operational framework to investigate and develop these polynomials further. Full article