A coupled Legendre-Laguerre polynomial method with analytical integration for the Rayleigh waves in a quasicrystal layered half-space with an imperfect interface

The Laguerre polynomial method has been successfully used to investigate the dynamic responses of a half-space.However,it fails to obtain the correct stress at the interfaces in a layered half-space,especially when there are significant differences in material properties.Therefore,a coupled Legendre-Laguerre polynomial method with analytical integration is proposed.The Rayleigh waves in a one-dimensional(1D)hexagonal quasicrystal(QC)layered half-space with an imperfect interface are investigated.The correctness is validated by comparison with available results.Its computation efficiency is analyzed.The dispersion curves of the phase velocity,displacement distributions,and stress distributions are illustrated.The effects of the phonon-phason coupling and imperfect interface coefficients on the wave characteristics are investigated.Some novel findings reveal that the proposed method is highly efficient for addressing the Rayleigh waves in a QC layered half-space.It can save over 99%of the computation time.This method can be expanded to investigate waves in various layered half-spaces,including earth-layered media and surface acoustic wave(SAW)devices.

Sensitivity Analysis of Structural Dynamic Behavior Based on the Sparse Polynomial Chaos Expansion and Material Point Method

This paper presents a framework for constructing surrogate models for sensitivity analysis of structural dynamics behavior.Physical models involving deformation,such as collisions,vibrations,and penetration,are devel-oped using the material point method.To reduce the computational cost of Monte Carlo simulations,response surface models are created as surrogate models for the material point system to approximate its dynamic behavior.An adaptive randomized greedy algorithm is employed to construct a sparse polynomial chaos expansion model with a fixed order,effectively balancing the accuracy and computational efficiency of the surrogate model.Based on the sparse polynomial chaos expansion,sensitivity analysis is conducted using the global finite difference and Sobol methods.Several examples of structural dynamics are provided to demonstrate the effectiveness of the proposed method in addressing structural dynamics problems.

The Interpolation Polynomial Based on the Zeros of the Legendre Polynomial

Let P_n(z) be the Legendre polynomial satisfing P_n(1)=1. For a function f(z) we denote by L_n(f;z) the Lagrange interpolation polynomial based on the zeros of the P_n(z), we denote by G_d the elliptic region with foci at—1, 1, where d is sum of its semi-axes. Theorem 1. If f(z) is analytic in G_d and continuous on G_d then where a, b are semimajor and semiminor axis of the ellipse G_d respectively. By P_n(z) denote the Legendre polynomial whose highest coefficient is 1. Let A be any fixed posititve number, By L_n^((A))(f;z) and L_n^((A))(f;z) denote interpolation polynomial based on the zeros of the P_n(z)-A and the P_n(z)-A respectively. Theorem2. If f(z) is analytic in G_d, then when d>2, where d^2/2 cannot be improved. If f(z) is analytic in G_d and continuous on G_d(d>2), then If f(z) is analytic in G_d and continuous on G_d, with modulus of continuity ω(f;δ)=o(δ^(1/2), then

New Form of Legendre Polynomials Obtained by Virtue of Excited Squeezed State and IWOP Technique in Quantum Optics

Using the technique of integration within an ordered product of operators and the intermediate coordinatemomentum representation in quantum optics, as well as the excited squeezed state we derive a new form of Legendre polynomials.

缔合Legendre函数的快速插值计算及其在六边形网格模型重力异常计算中的应用

鉴于局部区域非等纬度分布点的超高阶次球谐综合计算效率较低,本文深入研究了缔合Legendre函数的插值算法,结合模型重力异常求解的展开式,提出了超高阶模型空间重力异常插值快速计算方法,为了验证本文方法在东西狭长分布的点集更具有应用优势,采用球谐旋转变换技术进一步提升了计算效率。试验结果表明,利用2160阶次EGM2008模型计算日本地区同一高度的30303个非等纬度分布的六边形网格点处模型重力异常,插值快速计算方法相比逐点计算,在误差不超±0.005 mGal水平下,计算耗时从3669.41 s缩减到98.05 s,同时经过球谐旋转变换,将南北狭长的日本区域点集旋转为东西狭长分布,使得上述相应计算内容的耗时进一步由98.05 s缩减到19.06 s,计算效率相比最初方法提速比达到近200倍,有效解决了非等纬度分布点的超高阶模型重力异常快速解算的效率难题,且验证了该方法在东西狭长分布的情况下具有更高的提速比。

New Implementation of Legendre Polynomials for Solving Partial Differential Equations

In this paper we present a proposal using Legendre polynomials approximation for the solution of the second order linear partial differential equations. Our approach consists of reducing the problem to a set of linear equations by expanding the approximate solution in terms of shifted Legendre polynomials with unknown coefficients. The performance of presented method has been compared with other methods, namely Sinc-Galerkin, quadratic spline collocation and LiuLin method. Numerical examples show better accuracy of the proposed method. Moreover, the computation cost decreases at least by a factor of 6 in this method.

Numerical Solutions of Fractional Variable Order Differential Equations via Using Shifted Legendre Polynomials

In this manuscript,an algorithm for the computation of numerical solutions to some variable order fractional differential equations(FDEs)subject to the boundary and initial conditions is developed.We use shifted Legendre polynomials for the required numerical algorithm to develop some operational matrices.Further,operational matrices are constructed using variable order differentiation and integration.We are finding the operationalmatrices of variable order differentiation and integration by omitting the discretization of data.With the help of aforesaid matrices,considered FDEs are converted to algebraic equations of Sylvester type.Finally,the algebraic equations we get are solved with the help of mathematical software like Matlab or Mathematica to compute numerical solutions.Some examples are given to check the proposed method’s accuracy and graphical representations.Exact and numerical solutions are also compared in the paper for some examples.The efficiency of the method can be enhanced further by increasing the scale level.

时变动力学Legendre级数解的Duhamel积分改进算法

时变系统受迫振动的响应计算理论仍不完善,数值解法不能准确反映振动响应的解析性。利用Legendre多项式逼近法(Legendre polynomial approximation method, LPAM)可以得到时变系统响应的连续近似函数解,但该方法的计算效率较低。将时不变系统的时域响应求解理论移植到时变系统,基于线性系统的叠加原理提出基于Duhamel积分的改进Legendre级数算法,利用Dirac函数的Legendre多项式近似和Duhamel积分求解时变系统的响应。通过一个一阶非齐次变系数常微分方程组说明改进算法的有效性。设计刚度阻尼均线性变化和非线性变化的单自由度系统仿真算例,对比利用该方法得到的瞬态激励和简谐激励下系统位移响应和四阶Runge-Kutta数值法计算结果,说明了改进算法敛散性问题和计算速度的提升。

A Block Procedure with Linear Multi-Step Methods Using Legendre Polynomials for Solving ODEs

In this article, we derive a block procedure for some K-step linear multi-step methods (for K = 1, 2 and 3), using Legendre polynomials as the basis functions. We give discrete methods used in block and implement it for solving the non-stiff initial value problems, being the continuous interpolant derived and collocated at grid and off-grid points. Numerical examples of ordinary differential equations (ODEs) are solved using the proposed methods to show the validity and the accuracy of the introduced algorithms. A comparison with fourth-order Runge-Kutta method is given. The ob-tained numerical results reveal that the proposed method is efficient.

Legendre Polynomial and Nonlinear Oscillating Point-Like Charged Particle

In this article we explore the kinematics of a point-like charged particle placed within the interior plane of a charged ring. Analytically we formulate the electric field of the ring along a representative diagonal. Graph of the field as a function of the distance from the center of the ring assists foreseeing oscillating movement of the charged particle. We formulate the equation of motion;this is a nonlinear differential equation. Applying Computer Algebra System (CAS), specifically Mathematica [1] we solve the equation numerically. Utilizing the solution we quantify the kinematic quantities of interest including oscillations period. Although the equation of motion is nonlinear its period is regulated. For better understanding we take an advantage of Mathematica animation features animating the nonlinear oscillations.

Legendre Polynomial Kernel: Application in SVM

In machines learning problems, Support Vector Machine is a method of classification. For non-linearly separable data, kernel functions are a basic ingredient in the SVM technic. In this paper, we briefly recall some useful results on decomposition of RKHS. Based on orthogonal polynomial theory and Mercer theorem, we construct the high power Legendre polynomial kernel on the cube [-1,1]d. Following presentation of the theoretical background of SVM, we evaluate the performance of this kernel on some illustrative examples in comparison with Rbf, linear and polynomial kernels.

Characterization of Optical Aberrations Induced by Thermal Gradients and Vibrations via Zernike and Legendre Polynomials

For every astronomical instrument, the operating conditions are undoubtedly different from those defined in a setup experiment. Besides environmental conditions, the drives, the electronic cabinets containing heaters and fans introduce disturbances that must be taken into account already in the preliminary design phase. Such disturbances can be identified as being mostly of two types: heat sources/sinks or cooling systems responsible for heat transfer via conduction, radiation, free and forced convection on one side and random and periodic vibrations on the other. For this reason, a key role already from the very beginning of the design process is played by integrated model merging the outcomes based on a Finite Element Model from thermo-structural and modal analysis into the optical model to estimate the aberrations. The current paper presents the status of such model, capable of analyzing the deformed surfaces deriving from both thermo-structural and vibrational analyses and measuring their effect in terms of optical aberrations by fitting them by Zernike and Legendre polynomial fitting respectively for circular and rectangular apertures. The independent contribution of each aberration is satisfied by the orthogonality of the polynomials and mesh uniformity.

Probabilistic analysis of tunnel face seismic stability in layered rock masses using Polynomial Chaos Kriging metamodel

Face stability is an essential issue in tunnel design and construction.Layered rock masses are typical and ubiquitous;uncertainties in rock properties always exist.In view of this,a comprehensive method,which combines the Upper bound Limit analysis of Tunnel face stability,the Polynomial Chaos Kriging,the Monte-Carlo Simulation and Analysis of Covariance method(ULT-PCK-MA),is proposed to investigate the seismic stability of tunnel faces.A two-dimensional analytical model of ULT is developed to evaluate the virtual support force based on the upper bound limit analysis.An efficient probabilistic analysis method PCK-MA based on the adaptive Polynomial Chaos Kriging metamodel is then implemented to investigate the parameter uncertainty effects.Ten input parameters,including geological strength indices,uniaxial compressive strengths and constants for three rock formations,and the horizontal seismic coefficients,are treated as random variables.The effects of these parameter uncertainties on the failure probability and sensitivity indices are discussed.In addition,the effects of weak layer position,the middle layer thickness and quality,the tunnel diameter,the parameters correlation,and the seismic loadings are investigated,respectively.The results show that the layer distributions significantly influence the tunnel face probabilistic stability,particularly when the weak rock is present in the bottom layer.The efficiency of the proposed ULT-PCK-MA is validated,which is expected to facilitate the engineering design and construction.

变系数Volterra型积分微分方程的2种Legendre谱Galerkin数值积分方法

为了进一步提高求解Volterra型积分微分的数值精度,针对一种变系数Volterra型积分微分方程,提出了2种Legendre谱Galerkin数值积分法。采用Galerkin Legendre数值积分对Volterra型积分微分方程的积分项进行预处理,对其构造Legendre tau格式,同时用Chebyshev-Gauss-Lobatto配置点对变系数和积分项部分进行计算,并通过对方程的定义区间进行分解,提出了一种多区间Legendre谱Galerkin数值积分法。该方法的格式对于奇数阶模型具有对称结构。此外,通过引入Volterra型积分微分方程的最小二乘函数,构造了Legendre谱Galerkin最小二乘数值积分法。该方法对应的代数方程系数矩阵是对称正定的。数值算例验证了这2种Legendre谱Galerkin数值积分方法的高阶精度和有效性。

Diophantine equations and Fermat's last theorem for multivariate(skew-)polynomials

Fermat’s Last Theorem is a famous theorem in number theory which is difficult to prove.However,it is known that the version of polynomials with one variable of Fermat’s Last Theorem over C can be proved very concisely.The aim of this paper is to study the similar problems about Fermat’s Last Theorem for multivariate(skew)-polynomials with any characteristic.

Legendre-Weighted Residual Methods for System of Fractional Order Differential Equations

The numerical approach for finding the solution of fractional order systems of boundary value problems (BPVs) is derived in this paper. The implementation of the weighted residuals such as Galerkin, Least Square, and Collocation methods are included for solving fractional order differential equations, which is broadened to acquire the approximate solutions of fractional order systems with differentiable polynomials, namely Legendre polynomials, as basis functions. The algorithm of the residual formulations of matrix form can be coded efficiently. The interpretation of Caputo fractional derivatives is employed here. We have demonstrated these methods numerically through a few examples of linear and nonlinear BVPs. The results in absolute errors show that the present method efficiently finds the numerical solutions of fractional order systems of differential equations.

Invariant measure for cubic Fibonacci-like polynomials

A special class of cubic polynomials possessing decay of geometry property is studied.This class of cubic bimodal maps has generalized Fibonacci combinatorics.For maps with bounded combinatorics,we show that they have an absolutely continuous invariant probability measure.

An Extended Numerical Method by Stancu Polynomials for Solution of Integro-Differential Equations Arising in Oscillating Magnetic Fields

In this study, the Bernstein collocation method has been expanded to Stancu collocation method for numerical solution of the charged particle motion for certain configurations of oscillating magnetic fields modelled by a class of linear integro-differential equations. As the method has been improved, the Stancu polynomials that are generalization of the Bernstein polynomials have been used. The method has been tested on a physical problem how the method can be applied. Moreover, numerical results of the method have been compared with the numerical results of the other methods to indicate the efficiency of the method.

Improving Video Watermarking through Galois Field GF(2^(4)) Multiplication Tables with Diverse Irreducible Polynomials and Adaptive Techniques

Video watermarking plays a crucial role in protecting intellectual property rights and ensuring content authenticity.This study delves into the integration of Galois Field(GF)multiplication tables,especially GF(2^(4)),and their interaction with distinct irreducible polynomials.The primary aim is to enhance watermarking techniques for achieving imperceptibility,robustness,and efficient execution time.The research employs scene selection and adaptive thresholding techniques to streamline the watermarking process.Scene selection is used strategically to embed watermarks in the most vital frames of the video,while adaptive thresholding methods ensure that the watermarking process adheres to imperceptibility criteria,maintaining the video's visual quality.Concurrently,careful consideration is given to execution time,crucial in real-world scenarios,to balance efficiency and efficacy.The Peak Signal-to-Noise Ratio(PSNR)serves as a pivotal metric to gauge the watermark's imperceptibility and video quality.The study explores various irreducible polynomials,navigating the trade-offs between computational efficiency and watermark imperceptibility.In parallel,the study pays careful attention to the execution time,a paramount consideration in real-world scenarios,to strike a balance between efficiency and efficacy.This comprehensive analysis provides valuable insights into the interplay of GF multiplication tables,diverse irreducible polynomials,scene selection,adaptive thresholding,imperceptibility,and execution time.The evaluation of the proposed algorithm's robustness was conducted using PSNR and NC metrics,and it was subjected to assessment under the impact of five distinct attack scenarios.These findings contribute to the development of watermarking strategies that balance imperceptibility,robustness,and processing efficiency,enhancing the field's practicality and effectiveness.

基于循环Legendre模糊神经网络的DFIG二阶滑模容错控制

针对双馈感应发电机(DFIG)易受外界干扰而对并网产生影响的问题,将循环Legendre模糊神经网络(RLFNN)与二阶滑模控制(SOSMC)应用于DFIG控制中,从而提高了DFIG在传感器故障和不确定条件下的功率跟踪能力。首先,SOSMC采用超螺旋算法进行推导,并使用Lyapunov第二定理证明了控制系统的渐近稳定性。其次,提出了使用RLFNN来估计不确定部分,RLFNN的控制律与参数可在线训练,以进一步确保系统鲁棒性。仿真结果表明,所提出的方法能够使DFIG在发生传感器故障、参数变化以及外部干扰情况下保持正常运行,实现了有效容错控制。