Set Of Orthogonal Polynomials Research Articles

Eighth-Kind Chebyshev Polynomials Collocation Algorithm for the Nonlinear Time-Fractional Generalized Kawahara Equation

In this study, we present an innovative approach involving a spectral collocation algorithm to effectively obtain numerical solutions of the nonlinear time-fractional generalized Kawahara equation (NTFGKE). We introduce a new set of orthogonal polynomials (OPs) referred to as “Eighth-kind Chebyshev polynomials (CPs)”. These polynomials are special kinds of generalized Gegenbauer polynomials. To achieve the proposed numerical approximations, we first derive some new theoretical results for eighth-kind CPs, and after that, we employ the spectral collocation technique and incorporate the shifted eighth-kind CPs as fundamental functions. This method facilitates the transformation of the equation and its inherent conditions into a set of nonlinear algebraic equations. By harnessing Newton’s method, we obtain the necessary semi-analytical solutions. Rigorous analysis is dedicated to evaluating convergence and errors. The effectiveness and reliability of our approach are validated through a series of numerical experiments accompanied by comparative assessments. By undertaking these steps, we seek to communicate our findings comprehensively while ensuring the method’s applicability and precision are demonstrated.

Recurrence and Eigenfunction Methods for Non-Trivial Models of Discrete Binary Choice.

Understanding how systems relax to equilibrium is a core theme of statistical physics, especially in economics, where systems are known to be subject to extrinsic noise not included in simple agent-based models. In models of binary choice-ones not much more complicated than Kirman's model of ant recruitment-such relaxation dynamics become difficult to determine analytically and require solving a three-term recurrence relation in the eigendecomposition of the stochastic process. In this paper, we derive a concise closed-form solution to this linear three-term recurrence relation. Its solution has traditionally relied on cumbersome continued fractions, and we instead employ a linear algebraic approach that leverages the properties of lower-triangular and tridiagonal matrices to express the terms in the recurrence relation using a finite set of orthogonal polynomials. We pay special attention to the power series coefficients of Heun functions, which are also important in fields such as quantum mechanics and general relativity, as well as the binary choice models studied here. We then apply the solution to find equations describing the relaxation to steady-state behavior in social choice models through eigendecomposition. This application showcases the potential of our solution as an off-the-shelf solution to the recurrence that has not previously been reported, allowing for the easy identification of the eigenspectra of one-dimensional, one-step, continuous-time Markov processes.

Vibration analysis of thick-section sandwich structures in thermal environments

An ultra-light sandwich structure with a thin ceramic-based composite face sheet bonded to a thick thermal insulator is developed as an integrated thermal protection system (ITPS) structure to protect the substructure of a hypersonic flight vehicle. For vibration fatigue assessment and failure prediction, the global dynamic response of the three-dimensional ITPS structure when subjected to a combination of thermal and harmonic excitation is desired. In this study, a novel unified model for vibration analysis of a thick-section sandwich structure is proposed based on the variational asymptotic method. The original 3D sandwich structure problem is reduced to one-dimensional through-the-thickness analysis and two-dimensional (2D) reference plane analysis. The stiffness matrices, including an equivalent transverse shear matrix, are obtained; hence, a shear correction factor is not required. The mechanical response of the 2D reduced-order plate under general boundary conditions in thermal environments is investigated. The static thermal deformation and forced vibration of the sandwich structure with the temperature gradient in the thickness direction are determined using a set of algebraic orthogonal polynomials through the Gram–Schmidt procedure. The thermal deflection is first introduced into the kinematic relationship to modify the total strain energy for the sandwich plate with temperature-independent material properties. To demonstrate the validity and applicability of the model, some numerical results relating to displacements and fundamental frequencies of simply supported sandwich plates are presented and compared to those obtained by other models. The effects of temperature gradients in the thickness direction, boundary conditions, core thickness, and temperature-dependent material properties on the dynamic characteristics and forced vibration of the sandwich structure are discussed by a detailed parametric study. The global dynamic strain response is first obtained using the plate model, which is desirable to determine the 3D local stress and strain field and predict vibration fatigue failure for the ITPS structure.

Uncertainty Analysis and Sensitivity Estimation on an Artillery External Ballistic System

Abstract In the design of artillery external ballistics, sensitivity analysis can effectively quantify the influence of multi-source uncertain parameters on the dispersion of projectile landing points to improve the precise attack ability of artillery. However, for a complicated artillery external ballistic system containing multiple inputs and outputs, its mapping relationships are not definite under uncertainty and it is difficult to estimate a comprehensive sensitivity index due to involving the calculation of high dimensional integral. Therefore, a sensitivity analysis method based on the combination of variance and covariance decomposition with the approximate high dimensional model representation (AHDMR) is proposed to measure the influence of muzzle state parameters, projectile characteristic parameters, etc. on projectile landing points under uncertainty in this paper. First, we establish the numerical simulation model of artillery external ballistics by combing the external ballistic theory and Runge–Kutta algorithm to acquire the mapping relationships between the uncertain input parameters and the dispersion of projectile landing points and implement uncertainty analysis under different uncertainty levels (UL) and distributions. Then, with the use of a set of orthogonal polynomials for uniform and Gaussian distribution, respectively, the high dimensional model representation of the mapping relationship is approximately expressed and the compressive sensitivity indices can be effectively estimated based on the Monte Carlo simulation. Moreover, the comparison results of two numerical examples indicate the proposed sensitivity analysis method is accurate and practical. Finally, through the method, the importance rankings of multi-uncertain parameters on projectile landing points for two distributions are effectively quantified under the UL = [0.01, 0.02, 0.03, 0.04, 0.05].

An analytical method for vibration analysis of arbitrarily shaped non-homogeneous orthotropic plates of variable thickness resting on Winkler-Pasternak foundation

A highly applicable analytical approach is proposed for the dynamic analysis of composite plate structures with complex variation of geometric characteristics (shape/boundary and thickness) and material characteristics (Young's modulus, shear modulus and density) resting on Winkler-Pasternak foundation with general elastic boundary conditions. Boundary conditions of the plates are treated by the penalty method, so that the admissible functions can use any complete set of orthogonal polynomials, which is obtained by Gram-Schmidt orthogonalization process. The interval where the in-plane coordinate variables of the plate domain are located can be selected as the orthogonalization intervals, so as to avoid the normalization procedure of coordinates and simplify the operation process. In order to calculate the energy integral of the arbitrarily shaped plate, the plate domain segmentation strategy is introduced. To verify the effectiveness and accuracy of the proposed method, convergence study and numerical verifications for different orthogonalization intervals, weight functions, penalty spring stiffness and the truncation number of orthogonal polynomials are carried out. Taking regular hexagonal plates as examples, the effects of the parameters of thickness, nonhomogeneity and foundations on the vibration characteristics of the plates are studied. These results can be used as reference data for future research.

Hahn polynomials for hypergeometric distribution

Orthogonal polynomials for the multivariate hypergeometric distribution are defined on lattices in polyhedral domains in Rd. Their structures are studied through a detailed analysis of classical Hahn polynomials with negative integer parameters. Factorization of the Hahn polynomials is explored and used to explain the relationship between the index set of orthogonal polynomials and the lattice set in the polyhedral domain. In the multivariate case, these constructions lead to nontrivial families of hypergeometric polynomials vanishing on lattice polyhedra. The generating functions and bispectral properties of the orthogonal polynomials are also discussed.

Pearson equations for discrete orthogonal polynomials: I. Generalized hypergeometric functions and Toda equations

AbstractThe Cholesky factorization of the moment matrix is applied to discrete orthogonal polynomials on the homogeneous lattice. In particular, semiclassical discrete orthogonal polynomials, which are built in terms of a discrete Pearson equation, are studied. The Laguerre–Freud structure semiinfinite matrix that models the shifts by ±1 in the independent variable of the set of orthogonal polynomials is introduced. In the semiclassical case it is proven that this Laguerre–Freud matrix is banded. From the well‐known fact that moments of the semiclassical weights are logarithmic derivatives of generalized hypergeometric functions, it is shown how the contiguous relations for these hypergeometric functions translate as symmetries for the corresponding moment matrix. It is found that the 3D Nijhoff–Capel discrete Toda lattice describes the corresponding contiguous shifts for the squared norms of the orthogonal polynomials. The continuous 1D Toda equation for these semiclassical discrete orthogonal polynomials is discussed and the compatibility equations are derived. It is also shown that the Kadomtesev–Petviashvilii equation is connected to an adequate deformed semiclassical discrete weight, but in this case, the deformation does not satisfy a Pearson equation.

Some limit relationships between some two-variable finite and infinite sequences of orthogonal polynomials

The aim of paper is to give some limit relationships between finite and infinite sequences of orthogonal polynomials in two variables and to obtain the well-known relations of some infinite sets of two-variable orthogonal polynomials by taking limit of the properties verified by finite classes of two-variable orthogonal polynomials. Furthermore, the fourth-order partial differential equations satisfied by the polynomials, which are the products of finite orthogonal polynomials, are presented and by taking limit of the derived fourth-order equations, partial differential equations for some infinite sequences of two-variable orthogonal polynomials are found.

Multiple orthogonal polynomials with respect to Gauss' hypergeometric function

AbstractA new set of multiple orthogonal polynomials of both type I and type II with respect to two weight functions involving Gauss' hypergeometric function on the interval (0,1) is studied. This type of polynomials has direct applications in the investigation of singular values of products of Ginibre random matrices and are connected with branched continued fractions and total‐positivity problems in combinatorics. The pair of orthogonality measures is shown to be a Nikishin system and to satisfy a matrix Pearson‐type differential equation. The focus is on the polynomials whose indices lie on the step‐line, for which it is shown that the differentiation gives a shift in the parameters, therefore satisfying Hahn's property. We obtain Rodrigues‐type formulas for type I polynomials and functions, while a more detailed characterization is given for the type II polynomials (aka 2‐orthogonal polynomials) that include an explicit expression as a terminating hypergeometric series, a third‐order differential equation, and a third‐order recurrence relation. The asymptotic behavior of their recurrence coefficients mimics those of Jacobi–Piñeiro polynomials, based on which their asymptotic zero distribution and a Mehler–Heine asymptotic formula near the origin are given. Particular choices of the parameters and confluence relations give some known systems such as special cases of the Jacobi–Piñeiro polynomials, Jacobi‐type 2‐orthogonal polynomials, components of the cubic decomposition of threefold symmetric Hahn‐classical polynomials, and multiple orthogonal polynomials with respect to confluent hypergeometric functions of the second kind.

Dynamic modeling and orbit maneuvering response analysis for a three-axis attitude stabilized large scale flexible spacecraft installed with hinged solar arrays

An analytical dynamic model for a three-axis attitude stabilized large scale flexible spacecraft (LSFS) installed with hinged solar arrays is established. The natural property of the LSFS is investigated by simplifying it as a rigid central body hinged with two sets of multi-panel structures in this paper. The Gram-Schmidt process is used to construct a set of characteristic orthogonal polynomials as the displacement field of the solar panel. Lagrange multipliers are introduced to describe the constraints at hinges. For the model of LSFS, the Rayleigh-Ritz method is employed to obtain analytical global modes which are explicitly functions of space coordinates and effective for the modeling of lower-order discrete dynamic equations and design of active vibration controllers for the system. Moreover, the method is applied to investigate the parametric influence on natural characteristics for the LSFS. The paper develops a novel method to obtain rigid-flexible coupling global modes and construct the low-dimensional dynamical model for LSFS and study the rigid-flexible coupling dynamic response of in-orbit LSFS with complicated external excitation.

Pseudo-Lucas Functions of Fractional Degree and Applications

In a recent article, the first and second kinds of multivariate Chebyshev polynomials of fractional degree, and the relevant integral repesentations, have been studied. In this article, we introduce the first and second kinds of pseudo-Lucas functions of fractional degree, and we show possible applications of these new functions. For the first kind, we compute the fractional Newton sum rules of any orthogonal polynomial set starting from the entries of the Jacobi matrix. For the second kind, the representation formulas for the fractional powers of a r×r matrix, already introduced by using the pseudo-Chebyshev functions, are extended to the Lucas case.

Arbitrary polynomial chaos expansion and its application to power flow analysis-Fast approximation of probability distribution by arbitrary polynomial expansion

This paper introduces an arbitrary polynomial chaos expansion method for performing probabilistic power flow analysis in power systems. The proposed method is used for uncertainty analysis, expressing the uncertainty of a system as random variables with an arbitrary output distribution based on orthogonal polynomial expansion. This method is advantageous because of its calculation speed and accuracy. This study expresses probabilistic power flow in a power system with many uncertain power sources using linear combination polynomial expansion. The orthogonal polynomial system employed is generated by moment analysis from renewable energy output data, with the polynomial coefficients derived from a collocation method. Simulation of probabilistic power flow using the proposed method is applied to a 29-bus transmission network model including three renewable energies, and the calculation speed and accuracy are evaluated by changing the expansion order of the polynomial. In addition, the influence on the polynomial coefficient is assessed when the system topology is changed due to a line fault. Therefore, since the arbitrary polynomial chaos expansion method can represent complex networks by linear combination of orthogonal polynomial sets, calculation based on it is several hundred times faster than the conventional Monte Carlo method. The results demonstrate that the proposed method is very useful for analyzing the probabilistic power distribution and that third-order expansion is practically appropriate.

Modal analysis of a rotating pre-twisted beam axially loaded by an internally guided tendon

A guided tendon concept in the context of rotating helicopter blades is proposed to modify their dynamic characteristics, so that the rotorcraft can operate under a wide range of working conditions. In this paper, a rotating pre-twisted beam with bending-bending-torsion coupling axially loaded by a tendon is used to model the proposed system. An internally guided tendon is attached to the beam at several spanwise locations to avoid the resonant vibrations due to insufficient separation between the rotor harmonics and natural frequencies of the beam. A set of orthogonal polynomials that satisfy the geometric boundary conditions is used to represent the deflections of the beam and tendon. The equations of motion are derived by means of the Lagrange's equation of the second kind combined with the appropriate energy expressions. In vacuo modal analysis is then carried out after the modal properties of the rotating pre-twisted beam without the tendon are validated against the reference. The effect of the attachments and rotation on the free vibration of the beam-tendon system is studied. It is found that the attachments and rotation can improve the structural stability of the system significantly, so as to allow a significantly larger axial force to be applied. The natural frequencies of all studied modes increase with rotation due to the centrifugal stiffening effect. The tendon-guiding attachments cause an increase in the natural frequencies of the tendon-dominated modes because of their reduced effective length but decrease the frequency reduction rate of the beam-dominated modes with the applied axial force. The beam part of the mode shapes of the beam-dominated modes is not significantly sensitive to the varying axial force, rotor speed or number of attachments expect when the studied modes enter the modal veering regions with other neighbouring modes.

New set of adapted Gegenbauer–Chebyshev invariant moments for image recognition and classification

Orthogonal moments and their invariants to geometric transformations for gray-scale images are widely used in many pattern recognition and image processing applications. In this paper, we propose a new set of orthogonal polynomials called adapted Gegenbauer–Chebyshev polynomials (AGC). This new set is used as a basic function to define the orthogonal adapted Gegenbauer–Chebyshev moments (AGCMs). The rotation, scaling, and translation invariant property of (AGCMs) is derived and analyzed. We provide a novel series of feature vectors of images based on the adapted Gegenbauer–Chebyshev orthogonal moments invariants (AGCMIs). We practice a novel image classification system using the proposed feature vectors and the fuzzy k-means classifier. A series of experiments is performed to validate this new set of orthogonal moments and compare its performance with the existing orthogonal moments as Legendre invariants moments, the Gegenbauer and Tchebichef invariant moments using three different image databases: the MPEG7-CE Shape database, the Columbia Object Image Library (COIL-20) database and the ORL-faces database. The obtained results ensure the superiority of the proposed AGCMs over all existing moments in representation and recognition of the images.

Image Classification by Mixed Finite Element Method and Orthogonal Legendre Moments

In this paper, we propose a new classification system for image databases, particularly valid for noisy and geometrically distorted images. This system consists of three steps. In the first step, we apply a new image denoising technique based on the resolution of the Perona and Malik model using the finite element method (FEM). In the second step, we use the orthogonal invariant moments, applied to the obtained denoised images, to extract the feature vectors of images. In this step, we use a new set of orthogonal polynomials derived from the orthogonal Legendre polynomials, we call them orthogonal adapted-Legendre polynomials. These polynomials are used to define a series of orthogonal moments, which are invariant to translation, rotation, and scale. In the third steps, we use the radial basis function neural network (RBF), where the calculated feature vectors are the inputs of the input layer. To show the effectiveness of the proposed approach, we perform experimental tests and a comparative study with other well-known classification systems. The results obtained show the superiority and efficiency of our system.

Dynamic stochastic finite element method using time‐dependent generalized polynomial chaos

SummaryWe present an intrusive formulation for the dynamic stochastic finite‐element method to propagate the epistemic uncertainty in material properties into a finite‐element system over time. The stochastic finite‐element method, originally developed for the static case, uses generalized polynomial chaos (gPC) expansions to represent the uncertainty in both material/load fields and displacement fields and solves for the unknown PC coefficients of displacement at each degree of freedom of the finite‐element system. In this case, the gPC basis used to represent the solution is optimal and can be kept the same throughout the static simulation; however, when integrating a stochastic system over time, it is proven that using gPC tends to break down for integrations over long times. The reason is that the solution's complexity increases in time, and the set of polynomials used in the gPC expansion to represent the solution, therefore, does not stay optimal. In this formulation, new stochastic variables and orthogonal polynomials are constructed as time progresses. These variables are obtained as the Kahrunen‐Loeve expansion of the finite‐element solution at the times of update, thus, optimally representing the distribution of the solution using a minimal set of orthogonal polynomials. The result from the method is a time‐domain polynomial chaos representation of the entire finite‐element solution. A fast post‐processing phase can be used to (a) obtain the probability distribution of the solution at each degree of freedom and (b) generate any number of time series realizations of the solution, which correspond to the same time series that would be obtained from a set of simulations based on direct Monte‐Carlo simulations of the uncertain material properties.

Object recognition and classification by mixed finite element method and invariants of orthogonal adapted-Legendre moments

In this paper, we present a new technique of extraction of descriptor vectors of the images and a new 2D image classification system especially valid for the image databases which contain noisy or geometrically distorted objects. This system consists of three steps. In the first steps, we use an image denoising technique by solving the Perona–Malik equations using the mixed finite element method based on Taylor–Hood finite element. In the second, we extract the descriptor vectors from the images by using the orthogonal moments applied to the obtained denoised images. In this context we present a new set of orthogonal polynomials based on orthogonal Legendre polynomials, we call them orthogonal adapted-Legendre polynomials. This set of orthogonal polynomials is used to define a new type of orthogonal moments. This helps to build a set of orthogonal moments that are invariant to translation, rotation and scale. These invariant moments are derived algebraically from the invariant geometric moments. In the third steps, we use the multi layer perceptron neural network where the calculated descriptor vectors are the inputs of the input layer. To show the effectiveness of our approach we perform experimental tests on two databases and we present a comparative study with other well-known classification systems. The experimental results obtained show the superiority of our system.

Doppler Compensation of Orthogonal Frequency Division Multiplexing for Ocean Intelligent Multimodal Information Technology

The era of intelligence has arrived. Intelligent multimodal information is essential for underwater engineering. Sound is an indispensable medium in underwater multi-modal information technology. Orthogonal frequency division multiplexing underwater acoustic communication technology can achieve high bit rate communication. However, the time-varying Doppler of the underwater acoustic channel will cause inter carrier interference in the OFDM system, destroy the orthogonality between subcarriers, and affect system performance. In this paper, we propose a method based on wavelet function to compensate for the Doppler distortions of the underwater acoustic channel for OFDM data. This method uses a set of orthogonal polynomials to perform channel estimation in both the time and frequency domains and uses the matched filtering for channel equalization. Compared with the traditional polynomials, the smoothly varying polynomials in Wavelet FFT greatly reduces the correlation of the processed data, enable more accurate channel estimation and equalization. Finally, this paper verifies the feasibility of the algorithm through the Songhua Lake underwater experiment.

Investigation on global analytic modes for a three-axis attitude stabilized spacecraft with jointed panels

The natural property of a three-axis attitude stabilized spacecraft is investigated by simplifying it as a rigid central body jointed with 4 solar panels in this paper. For the spacecraft composed of a rigid central body and flexible panels, the Rayleigh-Ritz method is employed to obtain global modes which are explicitly functions of space coordinates and are useful for the modeling of lower-order discrete dynamic equations and design of active vibration controllers for the system. The Gram-Schmidt process is used to construct a set of characteristic orthogonal polynomials as the displacement field of the solar panel. Lagrange multipliers are introduced to describe the constraint at joints. The characteristic equation of the whole system is derived through the Rayleigh-Ritz procedure. Then, natural frequencies and corresponding global modes of the spacecraft with jointed panels are obtained. Taking the natural frequency obtained from the finite element method as a reference value, the process of getting natural frequencies is validated by comparing results obtained with those from the finite element method. Moreover, an excellent convergence and high accuracy of the present method is demonstrated by a very good agreement between results from ANSYS and theoretical method proposed here. Finally, parametric studies on characteristics of the spacecraft are conducted. The interesting mode shift phenomenon is observed when parameters of the spacecraft are changed.

Fractional Charlier moments for image reconstruction and image watermarking

In this paper, we propose a new set of discrete orthogonal polynomials called fractional Charlier polynomials (FrCPs). This new set will be used as a basic function to define the fractional discrete orthogonal Charlier moments (FrCMs). The proposed FrCPs are derived algebraically using the spectral decomposition of Charlier polynomials (CPs), then the Lagrange interpolation formula is used to derive the spectral projectors. Then, each spectral projector matrix is decomposed by the singular value decomposition (SVD) technique in order to build a basic set of orthonormal eigenvectors which help to develop FrCPs. FrCMs are deduced in matrix form from the proposed FrCPs and are applied for image reconstruction and watermarking. The experimental results show the capacity of the FrCMs proposed for image reconstruction and image watermarking against different attacks such as noise and geometric distortions.