Hermite Series Research Articles

An approaching method based on integral for linear neutral delay differential equations by using Hermite polynomials

AbstractThis article presents a approximation method for linear neutral delay differential equations using Hermite polynomials. The method ensures the unknown function to be found by approximating the first derivative with the help of the finite Hermite series. This method reduces the problem to a linear algebraic system using the first derivative approach, matrix operations and collocation points. Also error of the solution is estimated by constructing an error problem. Numerical examples are solved to explain the method and error estimation technique. The results demonstrate the effectiveness and accuracy of the current study. Calculations have been made using MATLAB.

Multivariate form of Hermite sampling series

Multivariate form of Hermite sampling series

Wick–Fourier–Hermite Series in the Theory of Linear and Nonlinear Transformations of Gaussian Distributions

This article provides information on Hermite polynomials and its application to some problems in risk theory and site percolation.

Fourier–Hermite Dynamic Programming for Optimal Control

In this paper, we propose a novel computational method for solving non-linear optimal control problems. The method is based on the use of Fourier–Hermite series for approximating the action-value function arising in dynamic programming instead of the conventional Taylor-series expansion used in differential dynamic programming (DDP). The coefficients of the Fourier–Hermite series can be numerically computed by using sigma-point methods, which leads to a novel class of sigma-point based dynamic programming methods. We also prove the quadratic convergence of the method and experimentally test its performance against other methods.

Different higher order kinematics between star-forming and quiescent galaxies based on the SAMI, MAGPI, and LEGA-C surveys

ABSTRACT We present the first statistical study of spatially integrated non-Gaussian stellar kinematics spanning 7 Gyr in cosmic time. We use deep, rest-frame optical spectroscopy of massive galaxies (stellar mass $M_\star \gt 10^{10.5} \, \mathrm{M_\odot }$) at redshifts z = 0.05, 0.3, and 0.8 from the SAMI, MAGPI, and LEGA-C surveys, to measure the excess kurtosis h4 of the stellar velocity distribution, the latter parametrized as a Gauss–Hermite series. We find that at all redshifts where we have large enough samples, h4 anticorrelates with the ratio between rotation and dispersion, highlighting the physical connection between these two kinematic observables. In addition, and independently from the anticorrelation with rotation-to-dispersion ratio, we also find a correlation between h4 and M⋆, potentially connected to the assembly history of galaxies. In contrast, after controlling for mass, we find no evidence of independent correlation between h4 and aperture velocity dispersion or galaxy size. These results hold for both star-forming and quiescent galaxies. For quiescent galaxies, h4 also correlates with projected shape, even after controlling for the rotation-to-dispersion ratio. At any given redshift, star-forming galaxies have lower h4 compared to quiescent galaxies, highlighting the link between kinematic structure and star-forming activity.

Slip and jump coefficients for general gas–surface interactions according to the moment method

We develop a moment method based on the Hermite series of the arbitrary order to calculate viscous-slip, thermal-slip, and temperature-jump coefficients for general gas-surface scattering kernels. Under some usual assumptions of scattering kernels, the solvability is obtained by showing the positive definiteness of the symmetric coefficient matrix in the boundary conditions. For gas flows with the Cercignani–Lampis gas–surface interaction and inverse-power-law intermolecular potentials, the model can capture the slip and jump coefficients accurately with elegant analytic expressions. On the one hand, the proposed method can apply to the cases of arbitrary order moments with increasing accuracy. On the other hand, the explicit formulas for low-order situations are simpler and more accurate than some existing results in references. Therefore, one may apply these formulas in slip and jump conditions to improve the accuracy of macroscopic fluid dynamic models for gas flows.

Using parity to accelerate Hermite function computations: Zeros of truncated Hermite series, Gaussian quadrature and Clenshaw summation

Although Hermite functions have been studied for over a century and have been useful for analytical and numerical solutions in a myriad of areas, the theory of Hermite functions has gaps. This article is a unified treatment of all the operations—quadrature, summation, differentiation, and rootfinding—that can be accelerated by exploiting parity. Any function u(y) can be decomposed into its parts that are symmetric and antisymmetric with respect to the origin. Suppose that the quadrature on y∈[−∞,∞] is symmetric in the sense that if yj is a quadrature abscissa with weight wj, then −yj is also a quadrature point with weight wj. The number of multiplications is halved by evaluating the quadrature as ∫−∞∞u(y)dy≈w0u(0)+∑n=1Mwnu(yn)+u(−yn). Parity is equally useful in computing the zeros, maxima and minima of a truncated Hermite series of degree N for the important special cases that the series terms are all either symmetric or antisymmetric. The zeros and critical points are the eigenvalues of a companion matrix whose dimension is N/2 instead of N. In addition, for Hermite functions, we show that parity exploitation halves the dimension of the Jacobi matrix (a special case of the companion matrix) whose eigenvalues are the abcissas of Hermite–Gauss quadrature. The number of floating point operations for the recursion for the weights can likewise be halved. Lastly, the same is true for Chenshaw summation of a Hermite series.

Order Selection for Stochastic Bilinear Systems*

Identification of bilinear systems subject to white inputs is studied. Assuming bilinearity we use the crosscovariances between the output and the higher order Hermite polynomials of the input for estimating the coefficients of the Hermite series expansion of the output. The parameters of the bilinear model are obtained via a balanced factorization of the appropriately constructed Hankel matrix. The skewness of the singular values of the estimated Hankel matrix is applied for testing the order selection for the bilinear model.

Numerical Solution of Linear Volterra Integral Equations by using Hermite Polynomials

In this research, Hermite series method is used to find the convergent numerical solutions, this method is tested on linear Volterra integral equation of the second kind, the result was highly accurate. The algorithm and example is given to explain the solution procedures. The comparison of numerical solutions was compatible with exact solutions.

Collocation approaches to the mathematical model of an Euler–Bernoulli beam vibrations

Taylor-Matrix and Hermite-Matrix Collocation methods are presented to obtain the modal vibration behavior of an Euler–Bernoulli beam. The methods provide approximate solutions in the truncated Taylor series and the truncated Hermite series by using Chebyshev–Lobatto collocation points and operational matrices. The approaches are applied to the well-known transverse vibration model of a simply-supported Euler–Bernoulli type beam. The beam is assumed under the effect of external harmonic force with spatially varying amplitude. Firstly, the model problem is transformed into a system of linear algebraic equations. Then the approximate solution is computed by solving the obtained algebraic system. The proposed methods are compared with the exact solutions. Mode shapes of the first three fundamental frequencies are obtained for each approach. The convergence behaviors of transverse displacements and absolute errors are calculated for each mode. The numerical results are shown and compared. Based on the given figures and tables, one can state that the methods are remarkable, dependable, and accurate for approaching the mentioned model problems.

New Hermite series expansion for computing the matrix hyperbolic cosine

There are, currently, very few implementations to compute the hyperbolic cosine of a matrix. This work tries to fill this gap. To this end, we first introduce both a new rational-polynomial Hermite matrix expansion and a formula for the forward relative error of Hermite approximation in exact arithmetic with a sharp bound for the forward error. This matrix expansion allows obtaining a new accurate and efficient method for computing the hyperbolic matrix cosine. We present a MATLAB implementation, based on this method, which shows a superior efficiency and a better accuracy than other state-of-the-art methods. The algorithm developed on the basis of this method is also able to run on an NVIDIA GPU thanks to a MEX file that connects the MATLAB implementation to the CUDA code.

PROPERTIES OF THE EQUATION OF HEAT CONDUCTION WITH DISSIPATION SOLUTIONS

This paper investigates the properties of the solutions of the equation of heat conduction with dissipation, which is associated with a harmonic oscillator - the operator $-d^2/dx^2 + x^2$, $x\in \mathbb{R}$ (non-negative and self-adjoint in $L_2(\mathbb{R})$). An explicit form of the function is given, which is analogous to the fundamental solution of the Cauchy problem for the heat conduction equation. A formula that describes all infinitely differentiable (with respect to the variable $x$) solutions of such an equation was found, well-posedness of the Cauchy problem for the heat conduction equation with dissipation with the initial function, which is an element of the space of generalized functions $(S_{1/2}^{1/2})'$, is established. It is established that $(S_{1/2}^{1/2})'$ is the "maximum" space of initial data of the Cauchy problem, for which the solutions are infinite functions differentiable by spatial variable. The main means of research are formal Hermite series, which are identified with linear continuous functionals defined on $S_{1/2}^{1/2}$.

Motion of two-dimensional quantum particle under a linear potential in the presence of Rashba and Dresselhaus spin–orbit interactions

The quantum mechanical problem of a particle moving in the presence of electric field and Rashba and/or Dresselhaus spin–orbit interactions (SOIs) is solved exactly. Existence of the SOI removes the spin degeneracy, yielding two coupled Schrödinger equations for spin-up and spin-down spinor eigenfunctions. Fourier-transform of these equations provides two coupled first-order differential equations, which are shown to be reduced to two decoupled Hermite-like equations with complex coefficients. The convergent solutions of the equations are found to be described in the form of so called Hermite series. The recurrence relations and the integral representation of the Hermite series are provided. In the absence of Rashba and Dresselhaus spin–orbit coupling constants, these interconnected equations are decoupled and reduced to two Schrödinger equations for a quantum particle moving in a linear potential, solution of which is described by Airy functions.

Approximate solution of Lane-Emden problem via modified Hermite operation matrix method


 
 
 
 Lane-Emden equations are singular initial value problems and they are important in mathematical physics and astrophysics. The aim of this present paper is presenting a new numerical method for finding approximate solution to Lane-Emden type equations arising in astrophysics based on modified Hermite operational matrix of integration. The proposed technique is based on taking the truncated modified Hermite series of the solution function in the Lane-Emden equation and then transferred into a matrix equation together with the given conditions. The obtained result is system of linear algebraic equation using collection points. The suggested algorithm is applied on some relevant physical problems as Lane-Emden type equations. 
 
 
 

Sequential estimation of Spearman rank correlation using Hermite series estimators

In this article we describe a new Hermite series based sequential estimator for the Spearman rank correlation coefficient and provide algorithms applicable in both the stationary and non-stationary settings. To treat the non-stationary setting, we introduce a novel, exponentially weighted estimator for the Spearman rank correlation, which allows the local nonparametric correlation of a bivariate data stream to be tracked. To the best of our knowledge this is the first algorithm to be proposed for estimating a time varying Spearman rank correlation that does not rely on a moving window approach. We explore the practical effectiveness of the Hermite series based estimators through real data and simulation studies demonstrating good practical performance. The simulation studies in particular reveal competitive performance compared to an existing algorithm. The potential applications of this work are manifold. The Hermite series based Spearman rank correlation estimator can be applied to fast and robust online calculation of correlation which may vary over time. Possible machine learning applications include, amongst others, fast feature selection and hierarchical clustering on massive data sets.

Hermite wavelet based numerical method for the solution of two parameters singularly perturbed non-linear Benjamina-Bona-Mohany equation

In this paper, the Hermite wavelet based numerical method is developed for the solution of two parameters singularly perturbed non-linear Benjamina-Bona-Mohany partial differential equation.Present method is purely based on the time discretization of Hermite wavelets series approximations with collocation technique. In order to check the efficiency of the proposed method, it is applied on well known equations are called two parameters singularly perturbed non-linear Benjamina-Bona-Mohany partial differential equation with different cases. Hermite wavelet based numerical solutions are obtained by preparing matlab codes of the proposed method and in comparision with the exact and existing methods of solution, it is presented in the form of figures and tables at different time levels. Absolute error is also calculated at the different time levels and presented in figures and tables to exhibit the accuracy and applicability of the proposed method.

A compatible Hermite–Taylor matrix-collocation technique with convergence test for second-order partial integro-differential equations containing two independent variables with functional bounds

The aim of this study is to offer a compatible numerical technique to solve second-order linear partial integro-differential equations with variable (functional) bounds, including two independent variables, under the initial and/or boundary conditions by using hybrid Hermite and Taylor series. The method converts the presented integro-differential equation to a matrix equation including the unknown Hermite coefficients. Solving this matrix equation and applying the collocation method, the approximate solution of the problem is obtained in terms of the Hermite polynomials. Also, by means of an error estimation and convergence test related to residual functions, some examples to illustrate the accuracy and efficiency of the method are fulfilled; the obtained results are scrutinized and interpreted. All numerical computations have been performed on the computer programs.

Near-Gaussian distributions for modelling discrete stellar velocity data with heteroskedastic uncertainties

ABSTRACT The velocity distributions of stellar tracers in general exhibit weak non-Gaussianity encoding information on the orbital composition of a galaxy and the underlying potential. The standard solution for measuring non-Gaussianity involves constructing a series expansion (e.g. the Gauss–Hermite series) that can produce regions of negative probability density. This is a significant issue for the modelling of discrete data with heteroskedastic uncertainties. Here, we introduce a method to construct positive-definite probability distributions by the convolution of a given kernel with a Gaussian distribution. Further convolutions by observational uncertainties are trivial. The statistics (moments and cumulants) of the resulting distributions are governed by the kernel distribution. Two kernels (uniform and Laplace) offer simple drop-in replacements for a Gauss–Hermite series for negative and positive excess kurtosis distributions with the option of skewness. We demonstrate the power of our method by an application to real and mock line-of-sight velocity data sets on dwarf spheroidal galaxies, where kurtosis is indicative of orbital anisotropy and hence a route to breaking the mass–anisotropy degeneracy for the identification of cusped versus cored dark matter profiles. Data on the Fornax dwarf spheroidal galaxy indicate positive excess kurtosis and hence favour a cored dark matter profile. Although designed for discrete data, the analytic Fourier transforms of the new models also make them appropriate for spectral fitting, which could improve the fits of high-quality data by avoiding unphysical negative wings in the line-of-sight velocity distribution.

Spline Technique for Second Type of Fredholm Integral Equations

Integral equation plays very important tools in a developed science like numerous troubles in an engineering and mechanical sciences . a cubic spline method renowned to begin with one powerful role to solve many functional equations like integral equations ,ordinary and partial differential equation .In our paper, a spline method is used to solve the problem linear integral equation of Fredholm. Integral function is turned to polynomial by basis of cubic spline . efficiency test of the proposed method , the comparison is check together with method of power series (least square method) and quadrature(nystrom) method,Laguerre,Legendre and Hermite series method . In MATLAB, all programs are written.

On the properties of hermite series based distribution function estimators

Hermite series based distribution function estimators have recently been applied in the context of sequential quantile estimation. These distribution function estimators are particularly useful because they allow the online (sequential) estimation of the full cumulative distribution function. This is in contrast to the empirical distribution function estimator and smooth kernel distribution function estimator which only allow sequential cumulative probability estimation at particular values on the support of the associated density function. Hermite series based distribution function estimators are well suited to the settings of streaming data, one-pass analysis of massive data sets and decentralised estimation. In this article we study these estimators in a more general context, thereby redressing a gap in the literature. In particular, we derive new asymptotic consistency results in the mean squared error, mean integrated squared error and almost sure sense. We also present novel Bias-robustness results for these estimators. Finally, we study the finite sample performance of the Hermite series based estimators through a real data example and simulation study. Our results indicate that in the general (non-sequential) context, the Hermite series based distribution function estimators are inferior to smooth kernel distribution function estimators, but may remain compelling in the context of sequential estimation of the full distribution function.