Hermite Polynomials Research Articles

Family of Hermite Polynomials Adapted to Mixture Experiments With Random Perturbations in the Directional Effects of the Components

ABSTRACTIn the modelling of mixture designs, the use of the trace plot graph stands out, enabling the researcher to analyse the directional effects of the mixture in relation to a specific reference point. By fitting the Scheffé linear model, these directional effects are represented linearly, devoid of random oscillations, except for the experimental error imposed by the model. Consequently, the estimations of the directional effects remain unaffected by complex interactions that could indicate disturbances. Based on this rationale, the objective of this study is to propose a family of parameterizations capable of capturing the random oscillations disregarded by the Scheffé model. These parameterizations rely on orthogonal Hermite polynomials, employed in constructing an additional multiplicative term for the Scheffé model, referred to as the ‘perturbation term’. The application of this model for the graphical study of directional effects has proven to be a qualitative technique of straightforward implementation and efficacy in assessing the reliability of experimental data.

Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials

Abstract This study presents a novel family of bivariate 2D- q q Hermite polynomials. We derive explicit forms and q q -partial differential equations and investigate numerical aspects associated with these polynomials. Furthermore, we deduce the monomiality principle and other key properties of the bivariate 2D- q q Hermite polynomials, including q q -recurrence relations and q q -difference equations. In addition, the graphical representations for the bivariate 2D- q q Hermite polynomials and q q -Hermite polynomials are explored for different values of indices.

Theoretical and numerical study of a Burgers viscous equation type with moving boundary

In this article, we investigate the existence, uniqueness, and numerical aspects of a one‐ and two‐dimensional nonlinear viscous type Burgers problem defined in a noncylindrical domain. In order to obtain the existence and uniqueness of the solution, the problem with a moving ends is transformed into an equivalent problem in a cylindrical through a diffeomorphism between the domains. The numerical simulation for the one‐ and two‐dimensional cases is performed using Lagrange with degrees 1–3 and cubic Hermite polynomials as base functions for applying the linearized Crank–Nicolson–Galerkin method to obtain an approximate numerical solution. Graphs prove the efficiency of the numerical method along with the order of numerical convergence consistent with the degree of the base polynomial.

An immersed boundary-regularized lattice Boltzmann method for modeling fluid–structure–acoustics interactions involving large deformation

This work presents a numerical method for modeling fluid–structure–acoustics interaction (FSAI) problems involving large deformation. The method incorporates an immersed boundary method and a regularized lattice Boltzmann method (LBM) where a multi-block technique and a nonreflecting boundary condition are implemented. The von Neumann analysis is conducted to investigate the stability of the regularized LBM. It is found that the accuracy and stability of the regularized LBM can be improved when the collision operator is computed from the Hermite polynomials up to the fourth order instead of the second order. To validate the present method, four benchmark cases are conducted: the propagation of an acoustic monopole point source, the sound generated by a stationary cylinder in a uniform flow, the sound generation of a two-dimensional insect model in hovering flight, and the sound generation of a three-dimensional flapping wing. Predictions given by the current method show a good agreement with numerical simulations and analytical solutions reported in the literature, demonstrating its capability of solving FSAI problems involving complex geometries and large deformation. Finally, the method is applied in modeling sound generation in vortex-induced vibrations of a rigid cylinder and a sphere. It is found that vortex-induced vibration can enhance the acoustic intensity by approximately four times compared to that of the stationary case for a cylinder. In contrast, both vibrating and stationary spheres exhibited relatively less intense noise, primarily within the wake. Notably, the spanwise noise propagation is only observed when the sphere is vibrating.

Bivariate generalized Poisson distribution and its relation with 2D–Hermite polynomials

Bivariate generalized Poisson distribution and its relation with 2D–Hermite polynomials

Physics-informed neural networks for weakly compressible flows using Galerkin–Boltzmann formulation

In this work, we study the Galerkin–Boltzmann formulation within a physics-informed neural network (PINN) framework to solve flow problems in weakly compressible regimes. The Galerkin–Boltzmann equations are discretized with second-order Hermite polynomials in microscopic velocity space, which leads to a first-order conservation law with six equations. Reducing the output dimension makes this equation system particularly well suited for PINNs compared with the widely used D2Q9 lattice Boltzmann velocity space discretizations. We created two distinct neural networks to overcome the scale disparity between the equilibrium and non-equilibrium states in collision terms of the equations. We test the accuracy and performance of the formulation with benchmark problems and solutions for forward and inverse problems with limited data. We compared our approach with the incompressible Navier–Stokes equation and the D2Q9 formulation. We show that the Galerkin–Boltzmann formulation results in similar L2 errors in velocity predictions in a comparable training time with the Navier–Stokes equation and lower training time than the D2Q9 formulation. We also solve forward and inverse problems for a flow over a square, try to capture an accurate boundary layer, and infer the relaxation time parameter using available data from a high-fidelity solver. Our findings show the potential of utilizing the Galerkin–Boltzmann formulation in PINN for weakly compressible flow problems.

Least Squares Approximation Method for Estimation of Volterra Fractional Integro-differential equation using Hermite Polynomial as the basis functions

This research work utilizes the least squares approximation method to estimate approximate solutions for fractional-order integro-differential equations, with Hermite polynomials as basis functions. The process begins by assuming an approximate solution of degree N, which is then substituted into the fractional-order integro-differential equation under investigation. After evaluating the integral, the equation is rearranged to isolate one side, allowing the application of the least squares method. Three examples were solved using this approach. In Example 1, the numerical results for α=0.9 and α=0.8 were compared to the exact solution for α=1. In Examples 2 and 3, the results for α=1.9 and α=1.8 were compared to the exact solution for α=2. These comparisons showed favorable alignment with the exact solutions. The numerical results and graphical illustrations demonstrate the validity, competence, and accuracy of the proposed method.

On a Class of Generalized Multivariate Hermite–Humbert Polynomials via Generalized Fibonacci Polynomials

This paper offers a thorough examination of a unified class of Humbert’s polynomials in two variables, extending beyond well-known polynomial families such as Gegenbauer, Humbert, Legendre, Chebyshev, Pincherle, Horadam, Kinnsy, Horadam–Pethe, Djordjević, Gould, Milovanović, Djordjević, Pathan, and Khan polynomials. This study’s motivation stems from exploring polynomials that lack traditional nomenclature. This work presents various expansions for Humbert–Hermite polynomials, including those involving Hermite–Gegenbauer (or ultraspherical) polynomials and Hermite–Chebyshev polynomials. The proofs enhanced our understanding of the properties and interrelationships within this extended class of polynomials, offering valuable insights into their mathematical structure. This research consolidates existing knowledge while expanding the scope of Humbert’s polynomials, laying the groundwork for further investigation and applications in diverse mathematical fields.

An operational point of view for the theory of multi-variable/multi-index Hermite polynomials

The use of algebraic tools of operational and umbral nature is exploited to develop a new perspective and to extend the theory of Hermite polynomials, with more than one variable also of complex nature. The techniques we adopt includes multivariable/many index Hermite- Kampè-dè-Fèrièt polynomials of order two and higher. It will be shown that the treatment, foreseen here, simplifies the study of the relevant properties and the associated computational technicalities.

New control variates for pricing basket options

Abstract Accepted by: Giorgio Consigli Accurate pricing of basket options, which are financial derivatives on multiple underlying assets, is a challenging and practically important task for financial institutions. We propose several new control variates for accurate, fast and efficient pricing of basket options. The first approach to deriving new control variates is the use of Hermite polynomial approximation of appropriate function of the underlying asset prices, which leads to a Black–Scholes-like analytic solution. This approach is new in the option pricing context and opens up new possibilities in derivative pricing. Further control variates are analytically derived using Jensen’s inequality in one case, and distributional properties of multivariate Wiener processes in other cases. All the newly proposed control variates are shown to lead to excellent variance reduction in numerical experiments based on realistic data. The proposed methods are novel, computationally simple and have a strong potential to replace more conventional methods, such as the geometric lower bound in simulation-based pricing of basket options and similar products used in financial risk management.

Fourier transform of biorthogonal polynomials in one variable

In this paper, some new families of biorthogonal functions in one variable produced by using Fourier transforms of biorthogonal polynomials, which are suggested by Laguerre, Jacobi and Szegö–Hermite polynomials, and their biorthogonality relations obtained via Parseval identity are derived.

Fractional Wiener chaos: Part 1

In this paper, we introduce a fractional analogue of the Wiener polynomial chaos expansion. It is important to highlight that the fractional order relates to the order of chaos decomposition elements, and not to the process itself, which remains the standard Wiener process. The central instrument in our fractional analogue of the Wiener chaos expansion is the function denoted as Hα(x,y)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {H}}_\\alpha (x,y)$$\\end{document}, referred to herein as a power-normalised parabolic cylinder function. Through careful analysis of several fundamental deterministic and stochastic properties, we affirm that this function essentially serves as a fractional extension of the Hermite polynomial. In particular, the power-normalised parabolic cylinder function with the Wiener process and time as its arguments, Hα(Wt,t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {H}}_\\alpha (W_t,t)$$\\end{document}, demonstrates martingale properties and can be interpreted as a fractional Itô integral with 1 as the integrand, thereby drawing parallels with its non-fractional counterpart. To build a fractional analogue of polynomial Wiener chaos on the real line, we introduce a new function, which we call the extended Hermite function, by smoothly joining two power-normalized parabolic cylinder functions at zero. We form an orthogonal set of extended Hermite functions as a one-parameter family and use tensor products of the extended Hermite functions as building blocks in the fractional Wiener chaos expansion, in the same way that tensor products of Hermite polynomials are used as building blocks in the Wiener chaos polynomial expansion.

Lattice Boltzmann Simulation of Nonequilibrium Flows Using Spectral Multiple-Relaxation-Time Collision Model

Prediction of nonequilibrium flows is critical to space flight. The present work demonstrates that the recently developed spectral multiple-relaxation-time (SMRT) lattice Boltzmann (LB) model is theoretically equivalent to Grad’s eigensystem [“Principles of the Kinetic Theory of Gases,” Thermodynamik der Gase/Thermodynamics of Gases, Springer–Verlag, Berlin, 1958, pp. 205–294], where the eigenfunctions obtained by tensor decomposition of the Hermite polynomials are also those of the linearized Boltzmann equation. Numerical results of shock structure simulation using the Maxwell molecular model agree very well with those of a high-resolution fast spectral method code up to Mach 7, provided that the relaxation times of the irreducible tensor components match their theoretical values. If a reduced set of relaxation times is used, such as in the Shakhov model and lumped-sum relaxation of Hermite modes, non-negligible discrepancies start to occur as the Mach number is raised, indicating the necessity of the fine-grained relaxation model. Together with the proven advantages of LB, the LB-SMRT scheme offers a competitive alternative for nonequilibrium flow simulation.

A general solution for rotating lattices of identical point vortices

The rotating equilibrium solutions of N identical point vortices are the stationary energy states of a higher-dimensional object projected to the x − y plane, in this case, an ( N − 1 ) -dimensional regular simplex. Parameterizing the point vortex Hamiltonian with this fundamental geometrical object leads to a simple bivector condition, which contains the equilibrium states and resolves the origin of the asymmetric solutions. This novel approach uses the geometric product and the multivector derivative, ∂ ψ , of the Hamiltonian in the geometric algebra, C l ( N − 1 , 0 ) to find a geometric condition that determines the equilibrium states. The resulting bivector equation is then used as the input to an optimizer, which rotates a simplex until the equilibrium condition is met, leading to a wealth of new solutions. If the vertices of the oriented simplex are projected to the x -axis, the values form the roots of the Hermite polynomials, H N ( x ) , and obey the Stieltjes relations, capturing the collinear solutions. The vortex simplex exhibits a striking geometrical connection with the amplituhedron of quantum field theory and gives deep insight into the quantization of a classical system.

A novel hybrid function-based Ritz method for static and free vibration of axially loaded bi-directional functionally graded porous beams

This paper introduces a novel hybrid function, synthesizing trigonometric and Hermitian polynomials, to analyze the structural responses of bi-directional functionally graded porous (2D-FGP) beams. The displacement field of the beam is based on the higher-order shear deformation theory. A power-law relation governs the material variation along the x- and z-directions, with the porosity manifesting in three distinct distributions. The governing equations are deduced through the application of Lagrange’s equation. The proposed hybrid function is developed to delineate the displacement field. The deflections, stresses, buckling loads, and natural frequencies of 2D-FGP beams are computed for diverse gradation exponents in the x- and z-directions, boundary conditions, porosity type, porosity coefficient, and slenderness ratio. A comparative assessment with finite element method results indicates that the findings agree with available data. It substantiates the accuracy and efficiency of the proposed methodology.

Maximum correntropy polynomial chaos Kalman filter for underwater navigation

This paper develops an underwater navigation solution that utilizes a strapdown inertial navigation system (SINS) and fuses a set of auxiliary sensors such as an acoustic positioning system, Doppler velocity log, depth meter, and magnetometer to accurately estimate an underwater vessel's position and orientation. The conventional integrated navigation system assumes Gaussian measurement noise, while in reality, the noises are non-Gaussian, particularly contaminated by heavy-tailed impulsive noises. To address this issue, and to fuse the system model with the acquired sensor measurements efficiently, we develop a square root polynomial chaos Kalman filter based on maximum correntropy criteria. The proposed method uses Hermite polynomial chaos expansion to tackle the nonlinearity, and it has the potential to estimate the states in a more accurate way in presence of a non-Gaussian measurement noise. The filter is initialized using acoustic beaconing to accurately locate the initial position of the vehicle. The computational complexity of the proposed filter is calculated in terms of flops count. The proposed method is compared with the existing maximum correntropy sigma point filters in terms of estimation accuracy and computational complexity. It is found from the simulation results that the proposed method is more accurate compared to the conventional deterministic sample point filters and Huber's M-estimator.

Developments of the I-Function Several Complex Variables

The purpose of this study is to present the I-function of r-variables, a natural extension of thepopular, intriguing, and practical Fox H-function. The results are quite generalized when r=1, at which pointwe get the I-function. Second, we showed that an integral is formed by multiplying the I-function of a singlevariable by the Bessel function and the General polynomial class. A two-variable I-function may be obtainedby evaluating the Sine transform of the product of the Hermite polynomial and Fox's H-function, and itsspecial case, the Cosine transform of the product of the Legendre polynomial and Fox's H-function. Wehave also provided exceptional examples, basic features, and circumstances under which convergenceoccurs. Results for H-functions in r variables are generalized in this study.

The Extended Dunkl Oscillator and the Generalized Hermite Polynomials on the Radial Lines

Model of the extended Dunkl oscillator based on Milovanović generalized Hermite polynomials on radial rays is discussed. Simple explicit realization of creation and annihilation operators in terms of difference-differential operators, coherent states are investigated.

On the Nodal Set of Solutions to a Class of Nonlocal Parabolic Equations

We investigate the local properties, including the nodal set and the nodal properties of solutions to the following parabolic problem of Muckenhoupt-Neumann type: { ∂ t u ¯ − y − a ∇ ⋅ ( y a ∇ u ¯ ) = 0 a m p ; in B 1 + × ( − 1 , 0 ) − ∂ y a u ¯ = q ( x , t ) u a m p ; on B 1 × { 0 } × ( − 1 , 0 ) , \begin{equation*} \begin {cases} \partial _t \overline {u} - y^{-a} \nabla \cdot (y^a \nabla \overline {u}) = 0 \quad &\text { in } \mathbb {B}_1^+ \times (-1,0) \\ -\partial _y^a \overline {u} = q(x,t)u \quad &\text { on } B_1 \times \{0\} \times (-1,0), \end{cases} \end{equation*} where a ∈ ( − 1 , 1 ) a\in (-1,1) is a fixed parameter, B 1 + ⊂ R N + 1 \mathbb {B}_1^+\subset \mathbb {R}^{N+1} is the upper unit half ball and B 1 B_1 is the unit ball in R N \mathbb {R}^N . Our main motivation comes from its relation with a class of nonlocal parabolic equations involving the fractional power of the heat operator H s u ( x , t ) = 1 | Γ ( − s ) | ∫ − ∞ t ∫ R N [ u ( x , t ) − u ( z , τ ) ] G N ( x − z , t − τ ) ( t − τ ) 1 + s d z d τ . \begin{equation*} H^su(x,t) = \frac {1}{|\Gamma (-s)|} \int _{-\infty }^t \int _{\mathbb {R}^N} \left [u(x,t) - u(z,\tau )\right ] \frac {G_N(x-z,t-\tau )}{(t-\tau )^{1+s}} dzd\tau . \end{equation*} We characterise the possible blow-ups and we examine the structure of the nodal set of solutions vanishing with a finite order. More precisely, we prove that the nodal set has at least parabolic Hausdorff codimension one in R N × R \mathbb {R}^N\times \mathbb {R} , and can be written as the union of a locally smooth part and a singular part, which turns out to possess remarkable stratification properties. Moreover, the asymptotic behaviour of general solutions near their nodal points is classified in terms of a class of explicit polynomials of Hermite and Laguerre type, obtained as eigenfunctions to an Ornstein-Uhlenbeck type operator. Our main results are obtained through a fine blow-up analysis which relies on the monotonicity of an Almgren-Poon type quotient and some new Liouville type results for parabolic equations, combined with more classical results including Federer’s reduction principle and the parabolic Whitney’s extension.

Thermally induced oscillatory rarefied gas flow inside a rectangular cavity

Thermally induced oscillatory rarefied gas flow inside a two-dimensional rectangular cavity is investigated based on the hybrid macro-/mesoscopic scheme. The effects of the Knudsen (Kn) numbers and the oscillation frequency of lid temperature on the flow parameters are analyzed. The Shakhov model equation is solved numerically based on the mesoscopic approach in the near-wall region, and the macroscopic approach is adopted in the bulk flow region to reduce the computational cost. To close the numerical iteration procedure, the velocity distribution functions serving as the pseudo boundary between macroscopic and mesoscopic methods are reconstructed using the high-order Hermite polynomials. Numerical simulations demonstrate that the temperature profile at the central vertical of the cavity predicted by the hybrid method is in good agreement with results from the mesoscopic method, with a maximum error of 0.23%. In addition, the computational memory cost can be saved up to about 69.91%. The hybrid approach is able to capture the nonlinear phenomenon in the thermally induced oscillatory rarefied gas flow under high Kn numbers, where the horizontal velocity no longer obeys the law of periodic oscillating cosine function, and the rise time of the horizontal velocity is much longer than the fall time. The thickness of the viscous penetration layer and the disturbed region increases as the Kn number increases and decreases as the Strouhal number increases.