Hermite Polynomials Research Articles

New Summation and Integral Representations for 2-Variable (p,q)-Hermite Polynomials

In this paper, we introduce and study new features for 2-variable (p,q)-Hermite polynomials, such as the (p,q)-diffusion equation, (p,q)-differential formula and integral representations. In addition, we establish some summation models and their (p,q)-derivatives. Certain parting remarks and nontrivial examples are also provided.

Harmonic oscillator with a step and its isospectral properties

We investigate the one-dimensional Schrödinger equation for a harmonic oscillator with a finite jump a at the origin. The solution is constructed by employing the ordinary matching-of-wavefunctions technique. For the special choices of a, a = 4ℓ (ℓ = 1, 2, …), the wavefunctions can be expressed by the Hermite polynomials. Moreover, we explore isospectral deformations of the potential via the Darboux transformation. In this context, infinitely many isospectral Hamiltonians to the ordinary harmonic oscillator are obtained.

A q‐Erkuş–Srivastava polynomials operator

Chan et al. (Integral Transforms Spec. Funct. 12 (2), 139–148, (2001)) constructed a multivariable extension of the Lagrange polynomials, popularly known as the Chan–Chyan–Srivastava polynomials. Altin and Erkuş (Integral Transforms Spec. Funct. 17, 239–244, (2006)) proposed Lagrange–Hermite polynomials in several variables. Erkuş and Srivastava (Integral Transforms Spec. Funct., 17, 267–273, (2006)) presented an unification (and generalization) of the Chan–Chyan–Srivastava polynomials and the multivariable Lagrange–Hermite polynomials, called as Erkuş–Srivastava polynomials. Duman (Taiwanese J. Math. 12 (2), 539‐543, (2008)) defined a ‐analogue of these generalized multivariable polynomials. Inspired by these studies, we construct a linear positive operator by means of the ‐Erkuş–Srivastava multivariable polynomials and study the Korovkin‐type theorems and the rate of convergence of these operators by using summability techniques of weighted ‐statistical convergence and the power series method. We also define a th‐order Taylor generalization of the multivariable polynomials operator and investigate the approximation of th‐order continuously differentiable Lipschitz class elements. Finally, we define the bivariate case of ‐Erkuş–Srivastava multivariable polynomials operator and study its ‐statistical convergence by using four‐dimensional matrix transformation.

Role of catalytic reactions in a flow-induced due to outer stationary and inner stretched coaxial cylinders: An application of Probabilists’ Hermite collocation method

This paper investigates the impact of homogeneous-heterogeneous chemical reactions on the flow between the coaxial cylinders with heat generation/absorption. From an engineering perspective, in energy absorber design for the shrinking and expanding of a circular metal tube in the production of extruding materials by stretching cylinders designing with great care is necessary to regulate skin friction and heat transfer rate. Studying processes in confined geometries enhances the basic knowledge of chemical kinetics and fluid dynamics. Researchers may increase the chemical processes efficiency/heat transfer activities by designing systems with an optimized flow and analysing the effects of reactions on flow. Both cost reductions and higher-quality products may result from this optimization. This research may yield insights with far-reaching implications across multiple disciplines. The modelled equations are transformed into ordinary differential equations by employing apt similarity transformations and obtained equations are solved using the collocation method with Probabilists’ Hermite polynomial. Graphs illustrating the variation of velocity, thermal, and concentration profiles have been plotted using the influence of various parameters. The increase in curvature parameter causes the velocity and thermal profile upsurge. The rise in heat source/sink parameter improves the heat transport. As the heterogeneous and homogeneous parameters upsurge, the concentration profile decreases.

High-order thread-safe lattice Boltzmann model for high performance computing turbulent flow simulations

We present a highly optimized thread-safe lattice Boltzmann model in which the non-equilibrium part of the distribution function is locally reconstructed via recursivity of Hermite polynomials. Such a procedure allows the explicit incorporation of non-equilibrium moments of the distribution up to the order supported by the lattice. Thus, the proposed approach increases accuracy and stability at low viscosities without compromising performance and amenability to parallelization with respect to standard lattice Boltzmann models. The high-order thread-safe lattice Boltzmann is tested on two types of turbulent flows, namely, the turbulent channel flow at Reτ=180 and the axisymmetric turbulent jet at Re = 7000; it delivers results in excellent agreement with reference data [direct numerical simulations (DNS), theory, and experiments] and (a) achieves peak performance [∼5×1012 floating point operations (FLOP) per second and an arithmetic intensity of ∼7 FLOP/byte on a single graphic processing unit] by significantly reducing the memory footprint, (b) retains the algorithmic simplicity of standard lattice Boltzmann computing, and (c) allows to perform stable simulations at vanishingly low viscosities. Our findings open attractive prospects for high-performance simulations of realistic turbulent flows on GPU-based architectures. Such expectations are confirmed by excellent agreement among lattice Boltzmann, experimental, and DNS reference data.

Tool path generation with a uniform residual error distribution considering tool contour error for ultra-precision diamond turning

Tool path generation strategy plays a crucial role in improving the efficiency and surface quality of ultra-precision diamond turning mainly by influencing residual errors. However, current approaches in tool path generation have not achieved a uniform distribution of residual error due to the unexplored and significant impact of tool contour error on residual error. Therefore, this study introduces a novel tool path generation algorithm for ultra-precision diamond turning, aiming for a uniform residual error distribution with consideration of tool contour error. The tool contour is measured off-machine, fitted, and modeled to achieve its actual cutting-edge contour at a submicron scale. An estimation model is established to calculate residual error on the machined surface considering the tool contour error. A tool path generation approach is then proposed to achieve uniform residual error distribution by adaptively controlling stepovers. Then the proposed stepover is smoothed by cubic Hermite polynomial interpolation to avoid undesired tool path fluctuations. To further improve the machining accuracy, a spiral tool path is generated with proper discretization parameters to control chord error. Finally, a prototype software system is developed to optimize the tool path as well as predict the residual and machining error across the entire machined surface. Experiment results demonstrate a significant reduction of approximately 40 % in both residual error and machining error compared to unoptimized results.

Numerical analysis of the radiative heat transfer and fluid flow between two cylinders using a high-order efficient Probabilists’ Hermite polynomial based collocation method

Radiative effects are useful in physics and engineering, especially in high-temperature operations and space technology. The impact of thermal radiation is important in managing the heat transfer process in the polymer production sector. This work examines the impact of the Soret and Dufour effect on the mass and heat transfer in the fluid flow between two cylinders. The only thing that creates the flow is the inner cylinder extending beneath the motionless outer cylinder. Thermal radiation and nonuniform heat source/sink are also considered. Through the application of similarity transformations, the governing nonlinear differential equations for this flow dynamics are converted into nonlinear ordinary differential equations (ODEs). After that, these simplified equations are solved by applying the collocation method with Probabilists’ Hermite polynomial. The physical behavior of important parameters is examined in depth using graphical representations. The results of a comprehensive examination are presented in tabular data. Result reveals that, the heat, momentum, and mass transfer fields are all significantly impacted by the gap size. As the Dufour number and radiation parameter value grows, the thermal profile also rises. The concentration profile rises as the Soret number values increases, whereas the rise in the Schmidt number shows the reverse pattern.

Szász–Durrmeyer Operators Involving Confluent Appell Polynomials

This article is concerned with the Durrmeyer-type generalization of Szász operators, including confluent Appell polynomials and their approximation properties. Also, the rate of convergence of the confluent Durrmeyer operators is found by using the modulus of continuity and Peetre’s K-functional. Then, we show that, under special choices of A(t), the newly constructed operators reduce confluent Hermite polynomials and confluent Bernoulli polynomials, respectively. Finally, we present a comparison of newly constructed operators with the Durrmeyer-type Szász operators graphically.

Rationally extended harmonic oscillator potential, isospectral family and the uncertainty relations

We consider the rationally extended harmonic oscillator potential which is isospectral to the conventional one and whose solutions are associated with the exceptional, Xm- Hermite polynomials and discuss its various important properties for different even codimension of m. The uncertainty relations are obtained for different m and it is shown that for the ground state, the uncertainty increases as m increases. A one parameter (λ) family of exactly solvable isospectral potential corresponding to this extended harmonic oscillator potential is obtained. Special cases corresponding to the λ=0 and λ=−1, which give the Pursey and the Abraham-Moses potentials respectively, are discussed. The uncertainty relations for the entire isospectral family of potentials for different m and λ are also calculated.

Solving Neutral Delay Differential Equations Using Galerkin Weighted Residual Method Based on Successive Integration Technique and its Residual Error Correction

Objectives: The main objectives of this work are to solve Neutral Delay Differential Equations (NDDEs) using Galerkin weighted residual method based on successive integration technique and to obtain the Estimation of Error using Residual function. Methods: The Galerkin weighted residual method based on successive integration technique is proposed to obtain approximate solutions of the NDDEs. In this study, the most widely used classical orthogonal polynomials, namely, the Bernoulli polynomials, the Chebyshev polynomials, the Hermite polynomials, and the Fibonacci polynomials are considered. Findings: Numerical examples of linear and nonlinear NDDEs have been considered to demonstrate the efficiency and accuracy of the method. Approximate solutions obtained by the proposed method are well comparable with exact solutions. Novelty: From the results it is observed that the accuracy of the numerical solutions by the proposed method increases as N increases. The proposed method is very effective, simple, and suitable for solving the linear and nonlinear NDDEs in real-world problems. Keywords: Galerkin Weighted Residual method, Polynomials, Hermite, Bernoulli, Chebyshev, Fibonacci, Successive integration technique, Neutral Delay Differential Equations

Dynamics of pollutant dispersion and solid–fluid interfacial layer in Jeffrey nanofluid flow subjected to waste discharge concentration: Implementation of probabilists’ Hermite polynomial collocation method

The current study explores the flow of a Jeffrey nanofluid with the impact of a non-uniform heat source/sink and magnetic effects over a stretching sheet by accounting for a pollutant’s transient dispersion after being released from a source outside the system. The need to enhance the efficiency of prospective future technologies, including heat exchangers, solar collectors with nanoscale solid particles suspended in the base fluid, powered engines, pharmaceutical procedures, and hybrid microelectronics, has grown recently. Graphene is treated as a nanoparticle and NaA lg as a base liquid. To emphasize the thermal integrity of the flow that is now taken into consideration, the influence of the diameter of the nanoparticle and the liquid–solid interfacial layer is also shown at the molecular level. The modeled equations are transformed into ordinary differential equations by means of the appropriate similarity transformations. Probabilists’ Hermite polynomial collocation technique is employed to solve these nonlinear dimensionless ODEs, and the graphical representations are created for certain significant values of the underlying physical parameters in the flow model. The result reveals that as Deborah number increases, velocity upsurges but temperature declines. A rise in concentration is observed when the local pollutant external source variation parameter rises. The temperature is enhanced with the advanced values of non-uniform heat source/sink parameters.

A gradient method for high-dimensional BSDEs

Abstract We develop a Monte Carlo method to solve backward stochastic differential equations (BSDEs) in high dimensions. The proposed algorithm is based on the regression-later approach using multivariate Hermite polynomials and their gradients. We propose numerical experiments to illustrate its performance.

Hermite, Higher order Hermite, Laguerre type polynomials and Burgers like equations

The multivariable version of ordinary and generalized Hermite polynomials are the natural solutions of the classical heat equation and of its higher order versions. We derive the associated Burgers equations and show that analogous non-linear partial differential equations can be derived for Laguerre polynomials. The starting point of this extension is the Laguerre diffusive equation, whose nonlinear extension reveals interesting implications involving families of mixed polynomials. In this way we have a general scheme to obtain new exact explicit solutions for nonlinear PDEs by using Laguerre, Hermite and other families of polynomials of Appèl and non-Appèl type.

SEMIPARAMETRIC ESTIMATION AND VARIABLE SELECTION FOR SPARSE SINGLE INDEX MODELS IN INCREASING DIMENSION

This paper considers semiparametric sieve estimation in high-dimensional single index models. The use of Hermite polynomials in approximating the unknown link function provides a convenient framework to conduct both estimation and variable selection. The estimation of the index parameter is formulated from solutions obtained by the routine penalized weighted linear regression procedure, where the weights are used in order to tackle the unbounded support of the regressors. The resulting index parameter estimator is shown to be consistent and sparse, and the asymptotic normality for the estimators of both the index parameter and the link function is established. To perform variable selection in the ultra-high dimension case, we further suggest a forward regression screening method, which is shown to enjoy the sure independence screening property. This screening procedure can be used before the penalized variable selection to reduce the burden of dimensionality. Numerical results show that both the variable selection procedures and the associated estimators perform well in finite samples.

Transient dispersion of a reactive solute in an oscillatory Couette flow through an anisotropic porous medium

This research discusses the significance of reactive solute dispersion relevant to ecological, biological, and geological contexts. It examines solute movement under oscillatory Couette flow through anisotropic porous media between parallel plates under the effect of heterogeneous boundary reactions. The flow is driven by the combined effect of upper plate oscillation in its plane and time-dependent pressure gradient. The lower plate is assumed rough, which introduces slip velocity. A semi-analytical approach, with the method of moments and finite difference scheme, is utilized to explore the transient dispersion in steady and oscillatory flows with or without a non-zero mean. Using the Hermite polynomial, the mean concentration for purely oscillatory and combined flow is obtained, highlighting notable variations based on flow factors. The results suggest that increment in anisotropic angle ϕ reduces dispersion and enhances mean concentration for permeability ratio K < 1, but this reverses for K > 1. Three dispersion phases emerge: diffusive, anomalous, and Taylor's regimes. Gaussian cloud distribution occurs at small and large times, with intermediate stages displaying anomalous dispersion and asymmetric longitudinal distribution. Effects of boundary absorption stabilize over time. The research focuses on the practical significance of different permeabilities of porous media, emphasizing applications of anisotropic porous media in fields such as chemical engineering and industrial processes.

Multivariate Hermite polynomials and information matrix tests

The information matrix test for a normal random vector is shown to coincide with the sum of the moment tests for all third- and fourth-order multivariate Hermite polynomials. The statistic is decomposed as the sum of the marginal information matrix test for a subvector, the conditional information matrix test for the complementary subvector, and a third leftover component. It is also shown that exact finite sample distributions can be obtained by drawing spherical Gaussian vectors and orthogonalising them using sample moments. These tests are applied to assess the implications of Gibrat’s law for US city sizes using the three most recent censuses.

A New Extension of Combination of Linear and Bilinear Generating Relations in Function Spaces Associated with Hypergeometric Polynomials; Some New Applications of Two Signals in Special Functions

For a certain class of generalized hypergeometric polynomials, the first derive some special cases on linear and bilinear generating functions and then apply these generating functions in order to reduce the corresponding results for the classical Jacobi, Hermite, Laguerre and Gegenbauer Polynomials, hypergeometric functions of Gauss and functions of Bessel and Kelvin. They also consider several linear generating functions for these polynomials as well as for some multivariable Jacobi and multivariable Laguerre polynomials which were investigated in recent years. Some of the linear and bilinear generating functions, presented in this paper, are associated with the hypergeometric polynomials.

An efficient methodology for simulating multivariate non-Gaussian stochastic processes

Simulation of multivariate non-Gaussian stochastic processes is still a challenging task, where the efficiency and accuracy may not be balanced. In this paper, an efficient methodology is presented to address this challenge. This methodology makes use of a probability weighted moments-based Hermite polynomial model and a wavenumber–frequency spectrum density-based spectral representation approach. First, a probability weighted moments-based Hermite polynomial model is employed to establish a transformation relationship between Gaussian and non-Gaussian samples. Importantly, this model only requires the solution of a solvable system of equations. Second, the target non-Gaussian wavenumber–frequency spectrum density can be promptly transformed into the Gaussian one, where a fast root matching strategy and fast Fourier transform are utilized. Finally, the underlying Gaussian samples are generated by the spectral representation method, and the non-Gaussian samples are obtained via the proposed transformation model. The computational results from numerical examples demonstrate that the proposed methodology is considered to be an accurate, applicable, and efficient approach for simulating multivariate non-Gaussian stochastic processes.

Spectral properties of weighted Cauchy singular integral transform on S-poly-Barmgann spaces

We investigate some spectral properties of the weighted quaternionic Cauchy transform when acting on the right quaternionic Hilbert space of Gaussian integrable functions. We study its boundedness, compactness, and memberships to the k-Schatten class, and we identify its range. This is done by means of its restriction to the n-th S-polyregular Bargmann space of the second kind, for which we provide an explicit closed expression for its action on the quaternionic Itô–Hermite polynomials constituting an orthogonal basis. We also exhibit an orthogonal basis of eigenfunctions of its n-Bergman projection leading to the explicit determination of its singular values. The provided results extend to the quaternionic setting those obtained by A. Intissar and A. Intissar in (2006) [32] for the weighted Cauchy transform on the complex plane.

Post-quantum complex Itô–Hermite polynomials

Post-quantum complex Itô–Hermite polynomials