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In this paper, a kinetic equation of Euler–Bernoulli beam is established with variable order fractional viscoelastic model. An effective numerical algorithm is proposed. This method uses a combination of shifted Bernstein polynomial and Legendre polynomial to approximate the numerical solution. The effectiveness of the algorithm is tested and verified by mathematical examples. The dynamic behavior of viscoelastic beams made of two materials under various loading conditions is studied. [ABSTRACT FROM AUTHOR]

In this work, we have investigated the problem of assessing stability and designing an appropriate feedback control law for T-S fuzzy systems with time-varying delay. By way of designing a new Lyapunov-Krasovskii functional based on Legendre polynomials and membership functions, we have developed conditions for stability assessment and feedback gain synthesis. The resulting algebraic conditions form a set of hierarchical LMIs which, by increasing the order of the Bessel-Legendre polynomial, compete with the sum of squares in both conservatism and complexity. Finally, two examples are provided to demonstrate the effectiveness of the findings. [ABSTRACT FROM AUTHOR]

An effective numerical algorithm is presented to analyze the fractional viscoelastic plate in the time domain for the first time in this paper. The viscoelastic behavior of the plate is described with fractional Kelvin–Voigt (FKV) constitutive model in three-dimensional space. A governing equation with three independent variables is established. Ternary unknown function in the governing equation is solved by deriving integer and fractional order differential operational matrices of the shifted Legendre polynomials. Error analysis and mathematical example are presented to verify the effectiveness and accuracy of proposed algorithm. Finally, numerical analysis of the plate under different loading conditions is carried out. Effects of the damping coefficient on vibration amplitude of the viscoelastic plate are studied. The results obtained are consistent with the current reference and actual situation. It shows that shifted Legendre polynomials algorithm is suitable for numerical analysis of fractional viscoelastic plates. • The fractional order governing equation of a viscoelastic plate is established. • Shifted Legendre polynomials algorithm is used to solve the governing equation. • The feasibility and efficiency of the proposed algorithm are verified. • Transverse displacements of viscoelastic plate are calculated directly in the time domain. [ABSTRACT FROM AUTHOR]

We propose an algorithm for the orthogonal fast discrete spherical Bessel transform on a uniform grid. Our approach is based upon the spherical Bessel transform factorization into the two subsequent orthogonal transforms, namely the fast Fourier transform and the orthogonal transform founded on the derivatives of the discrete Legendre orthogonal polynomials. The method utility is illustrated by its implementation for the problem of a two-atomic molecule in a time-dependent external field simulating the one utilized in the attosecond streaking technique. [ABSTRACT FROM AUTHOR]

The frequency spectrum of a partially metallized piezoelectric disc resonator was studied using Legendre polynomials. The formulation, based on three-dimensional equations of linear elasticity, takes into account the high contrast between the electroded and non-electroded regions. The mechanical displacement components and the electrical potential were expanded in a double series of orthonormal functions and were introduced into the equations governing wave propagation in piezoelectric media. The boundary and continuity conditions were automatically incorporated into the equations of motion by assuming position-dependent physical material constants or delta-functions. The incorporation of electrical sources is illustrated. Structure symmetry was used to reduce the number of unknowns. The vibration characteristics of the piezoelectric discs were analyzed using a three-dimensional modelling approach with modal and harmonic analyses. The numerical results are presented as resonance and anti-resonance frequencies, electric input admittance, electromechanical coupling coefficient and field profiles of fully and partially metallized PIC151 and PZT5A resonator discs. In order to validate our model, the results obtained were compared with those published previously and those obtained using an analytical approach. [ABSTRACT FROM AUTHOR]

We propose a novel, highly-efficient approach for the evaluation of bond-orientational order parameters (BOPs). Our approach exploits the properties of spherical harmonics and Wigner 3 j -symbols to reduce the number of terms in the expressions for BOPs, and employs simultaneous interpolation of normalised associated Legendre polynomials and trigonometric functions to dramatically reduce the total number of arithmetic operations. Using realistic test cases, we show how the above, combined with a CPU-cache-friendly data structure, leads to a 10 to 50-fold performance increase over approaches currently in use, depending on the size of the interpolation grids and the machine used. As the proposed approach is an approximation, we demonstrate that the errors it introduces are well-behaved, controllable and essentially negligible for practical grid sizes. We benchmark our approach against other structure identification methods (centro-symmetry parameter (CSP), common neighbour analysis (CNA), common neighbourhood parameter (CNP) and Voronoi analysis), generally regarded as much faster than BOPs, and demonstrate that our formulation is able to outperform them for all studied systems. [ABSTRACT FROM AUTHOR]

The exact Green’s function of the diffusion equation (GFDE) is often considered to be the gold standard for the simulation of partially diffusion-controlled reactions. As the GFDE with angular dependency is quite complex, the radial GFDE is more often used. Indeed, the exact GFDE is expressed as a Legendre expansion, the coefficients of which are given in terms of an integral comprising Bessel functions. This integral does not seem to have been evaluated analytically in existing literature. While the integral can be evaluated numerically, the Bessel functions make the integral oscillate and convergence is difficult to obtain. Therefore it would be of great interest to evaluate the integral analytically. The first term was evaluated previously, and was found to be equal to the radial GFDE. In this work, the second term of this expansion was evaluated. As this work has shown that the first two terms of the Legendre polynomial expansion can be calculated analytically, it raises the question of the possibility that an analytical solution exists for the other terms. [ABSTRACT FROM AUTHOR]

Abstract: The method of direct collocation with nonlinear programming (DCNLP) is a powerful tool to solve optimal control problems (OCP). In this method the solution time history is approximated with piecewise polynomials, which are constructed using interpolation points deriving from the Jacobi polynomials. Among the Jacobi polynomials family, Legendre and Chebyshev polynomials are the most used, but there is no evidence that they offer the best performance with respect to other family members. By solving different OCPs with interpolation points not only taken within the Jacoby family, the behavior of the Jacobi polynomials in the optimization problems is discussed. This paper focuses on spacecraft trajectories optimization problems. In particular orbit transfers, interplanetary transfers and station keepings are considered. [Copyright &y& Elsevier]

Abstract: By applying hybrid functions of general block-pulse functions and Legendre polynomials, linear Volterra integrodifferential systems are converted into a system of algebraic equations. The approximate solutions of linear Volterra integrodifferential systems are derived. Using the results we obtain the optimal control and state as well as the optimal value of the objective functional. The numerical examples illustrate that the algorithms are valid. [Copyright &y& Elsevier]

Abstract: In the present paper, a new Legendre wavelet operational matrix of derivative is presented. Shifted Legendre polynomials and their properties are employed for deriving a general procedure for forming this matrix. The application of the proposed operational matrix for solving initial and boundary value problems is explained. Then the scheme is tested for linear and nonlinear singular examples. The obtained results demonstrate efficiency and capability of the proposed method. [Copyright &y& Elsevier]

Abstract: The problem of normal contact of two identical elastic spheres is considered. The corresponding mixed boundary-value problem is formulated at the surface of the sphere, thereby relaxing the assumption of the Hertz theory about small size of the contact area compared to the sizes of the contacting bodies. A general solution in the form of Legendre series expansions is used to reduce the problem to dual series equations, and, subsequently, to a Fredholm integral equation of the second kind. Analysis of the contact stress and surface displacements of the sphere is carried out. Comparison of the results with the Hertz theory shows that the latter predicts contact stresses with high accuracy even for relatively large contact areas. [Copyright &y& Elsevier]

Abstract: It is a well-known fact that Haar wavelet can exactly represent any piecewise constant function. Beam and Warming proved later, in 2000, that the supercompact wavelets can exactly represent any piecewise polynomial function in one variable. Higher level of accuracy is attained by higher order polynomials of supercompact wavelets. The initial approach of Beam and Warming, which is based on multiwavelets (family of wavelets) constructed in a one dimensional context, was later extended to the case of multidimensional multiwavelets (3D). The orthogonal basis used by these authors was defined as separable functions given by the product of three Legendre polynomials. In this paper we propose an extension of these previous works to the case of surfaces by using non separable orthogonal functions. Our construction keeps the same advantages attained by the just referenced articles in relation with orthogonality, short support, approximation of surfaces with no border effects, detection of discontinuities, higher degree of accuracy and compressibility, as it is shown in the presented graphical and numerical examples. In this sense, our work may be regarded as a new contribution to supercompact multiwavelets’ theory with great applicability to the approximation of surfaces. [Copyright &y& Elsevier]

Abstract: The solution of the problem of the stress–strain state of an elastic body of spherical shape when it enters the Earth''s atmosphere at a superorbital velocity is obtained in the form of a series in Legendre polynomials, in the case of a viscous gas at hypersonic velocity, in the quasi-stationary formulation. Using the Hubert–von Mises–Hencky criterion the limit stresses corresponding to the start of disintegration are obtained. This enables the nature of the disintegration of the body to be judged and also enables the heights at which destruction of known meteoroids begins to be estimated. [Copyright &y& Elsevier]

Abstract: In this paper, we use Legendre wavelet method for solving quadratic Riccati differential equations and perform a comparative study between the proposed method and other existing methods. Our results show that in comparison with other existing methods, the Legendre wavelet method provides a fast convergent series of easily computable components. The present study is illustrated by exploring two kinds of nonlinear Riccati differential equations that shows the pertinent features of the Legendre wavelet method. [Copyright &y& Elsevier]

Abstract: Using block-pulse functions (BPFs)/shifted Legendre polynomials (SLPs) a unified approach for computing optimal control law of linear time-varying time-delay systems with reverse time terms and quadratic performance index is discussed in this paper. The governing delay-differential equations of dynamical systems are converted into linear algebraic equations by using operational matrices of orthogonal functions (BPFs and SLPs). The problem of finding optimal control law is thus reduced to the problem of solving algebraic equations. One example is included to demonstrate the applicability of the proposed approach. [ABSTRACT FROM AUTHOR]

The next to leading order chiral corrections to the Gell-Mann-Oakes-Renner (GMOR) relation are obtained using the light-quark pseudoscalar correlator to five-loop order in perturbative QCD, together with new finite energy sum rules (FESR) incorporating polynomial, Legendre type, integration kernels. These kernels are designed to suppress hadronic contributions in the region where they are least known. We obtain for the corrections to the GMOR relation, , the value . As a byproduct, the chiral perturbation theory (unphysical) low energy constant is determined to be . [ABSTRACT FROM AUTHOR]

Abstract: This paper introduces a new approach to obtain the integration matrices using Legendre power expansion . This method generates approximations to the lower order derivatives of the function through successive integrations of the Legendre polynomials to the highest order derivatives. This method is used to solve integral and integro-differential equations. The advantages of the suggested integration matrices emerged through comparisons with other ones. [Copyright &y& Elsevier]

Abstract: This paper studies round jet with large eddy simulation (LES) method, in which spectral element technique is used as spacial discritization for the large eddy simulation Navier–Stokes equations. A local spectral discretization associated with Legendre polynomials is employed on each element of the structured mesh, which allows for high accurate simulations of turbulent flows. Discontinuities across the interfaces of the elements are resolved using a Riemann solver. An isoparametric representation of the geometry is implemented, with boundaries of the domain discretized to the same order of accuracy as the solution, and explicit low-storage Runge–Kutta methods are used for time integration. LES results of round jet are presented, in which the instantaneous and statistical turbulence structures of the round jet have been captured. The probability density function, and the spectral density function of the round jet that can reflect properties of turbulence have also been estimated. The work serves the purpose of allowing fast, convenient computations and comparisons with theoretical results and the ultimate goal is to develop it into an LES code featuring spectral accuracy with minimum dissipation and dispersion, a valuable tool for round jet computations. [Copyright &y& Elsevier]

Abstract: A new numerical method, based on hybrid of Block-pulse and Legendre polynomials for numerical evaluation of Hankel transform is proposed in this paper. Hybrid of Block-pulse and Legendre polynomials are used as a basis to expand a part of the integrand, , appearing in the Hankel transform integral. Thus transforming the integral into a Fourier–Bessel series. Truncating the series, an efficient algorithm is obtained for the numerical evaluations of the Hankel transforms of order . The method is quite accurate and stable, as illustrated by given numerical examples with varying degree of random noise terms added to the data function , where is a uniform random variable with values in . Finally, an application of the proposed method is given in solving the heat equation in an infinite cylinder with a radiation condition. [Copyright &y& Elsevier]

Abstract: In this paper, an efficient modification of the Adomian decomposition method by using Legendre polynomials is presented. Both linear and non-linear models are suited for the proposed method. Some examples here in are solved by using this method and this paper will demonstrate that the results are more reliable and efficient. [Copyright &y& Elsevier]

This paper presents a numerical technique for solving the variable-order fractional extended Fisher–Kolmogorov equation. The method suggested to solve this problem is based on the orthonormal shifted discrete Legendre polynomials and the collocation method. First, we expand the unknown solution of the problem using the these polynomialss. Also, we approximate the second- and fourth-order classical derivatives, as well as the variable-order fractional derivatives by these basis functions. Then, we substitute these approximations in the equation. Next, we utilize the classical and fractional derivative matrices together with the collocation method to convert the main equation into a system containing nonlinear algebraic equations. We show the correctness of the proposed scheme by providing several numerical examples. [ABSTRACT FROM AUTHOR]

A method for finding the solution of time-delay systems using a hybrid function is proposed. The properties of the hybrid functions which consists of block-pulse functions plus Legendre polynomials are presented. The method is based upon expanding various time functions in the system as their truncated hybrid functions. The operational matrices of product and delay are introduced. These matrices together with the operational matrix of integration are utilized to reduce the solution of time-delay systems to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. [Copyright &y& Elsevier]

This paper examines whether or not a discrete-time econometric test for nonlinearity in mean may be used in cases where the data are believed to be generated in continuous time. It is demonstrated that appropriate bootstrapping techniques are required to yield a test statistic with sensible statistical properties. The technique is demonstrated by using it to examine 7-day Eurodollar rates for nonlinearity in mean. [Copyright &y& Elsevier]

This paper presents a formula expressing Legendre polynomials in terms of their derivatives and a formula expressing a Legendre polynomial integrated k-times in terms of Legendre polynomials. In view of these formulae, the second- and fourth-order elliptic equation were solved. Moreover, the suggested method is applicable for a wide area of differential equations. The present results are in satisfactory agreement with the exact solutions. [Copyright &y& Elsevier]

The article reports on the cyclic radial shearing interferometer (CRSI) that a group of laser-fusion researchers at Sichuan University and the Research Center of Laser Fusion at the China Academy of Engineering Physics want to use. It aims to spot-check the beam wavefront in their laser system, and have developed a wavefront-reconstruction algorithm for the interferometer that is based on Legendre polynomials. The system allows diagnosis of transient pulses with high speed and accuracy.

In this article, a step-by-step collocation approach based on the shifted Legendre polynomials is presented to solve a fractional order system of nonlinear stochastic differential equations involving a constant delay. The problem is considered with suitable initial condition and the fractional derivative is in the Caputo sense. With a step-by-step process, first, the considered problem is converted into a non-delay fractional order system of nonlinear stochastic differential equations in each step and then, a shifted Legendre collocation scheme is introduced to solve this system. By collocating the obtained residual at the shifted Legendre points, we get a nonlinear system of equations in each step. The convergence analysis and rate of convergence of the proposed method are investigated. Finally, three test examples are provided to affirm the accuracy of this technique in the presence of different noise measures. [ABSTRACT FROM AUTHOR]

Conventional estimators of the anisotropic power spectrum and two-point correlation function (2PCF) adopt the "Yamamoto approximation," fixing the line of sight of a pair of galaxies to that of just one of its members. While this is accurate only to first-order in the characteristic opening angle θmax, it allows for efficient implementation via fast Fourier transforms (FFTs). This work presents practical algorithms for computing the power spectrum and 2PCF multipoles using pairwise lines of sight, adopting either the galaxy midpoint or angle bisector definitions. Using newly derived infinite series expansions for spherical harmonics and Legendre polynomials, we construct estimators accurate to arbitrary order in θmax, though note that the midpoint and bisector formalisms themselves differ at fourth-order. Each estimator can be straightforwardly implemented using FFTs, requiring only modest additional computational cost relative to the Yamamoto approximation. We demonstrate the algorithms by applying them to a set of realistic mock galaxy catalogs, and find that both procedures produce comparable results for the 2PCF, with a slight preference for the bisector power spectrum algorithm, albeit at the cost of greater memory usage. Such estimators provide a useful method to reduce wide-angle systematics for future surveys. [ABSTRACT FROM AUTHOR]

This paper is associated to investigate a stochastic SEIAQHR model for transmission of Coronavirus disease 2019 that is a recent great crisis in numerous societies. This stochastic pandemic model is established due to several safety protocols, for instance social-distancing, mask and quarantine. Three white noises are added to three of the main parameters of the system to represent the impact of randomness in the environment on the considered model. Also, the unique solvability of the presented stochastic model is proved. Moreover, a collocation approach based on the Legendre polynomials is presented to obtain the numerical solution of this system. Finally, some simulations are provided to survey the obtained results of this pandemic model and to identify the theoretical findings. [ABSTRACT FROM AUTHOR]

Propagation of acoustic waves is considered in a system consisting of two stiff quarter-spaces connected by a planar soft layer. The two quarter-spaces and the layer form a half-space with a planar surface. In a numerical study, surface waves have been found and analyzed in this system with displacements that are localized not only at the surface, but also in the soft layer. In addition to the semi-analytical finite element method, an alternative approach based on an expansion of the displacement field in a double series of Laguerre functions and Legendre polynomials has been applied. It is shown that a number of branches of the mode spectrum can be interpreted and remarkably well described by perturbation theory, where the zero-order modes are the wedge waves guided at a rectangular edge of the stiff quarter-spaces or waves guided at the edge of a soft plate with rigid surfaces. For elastic moduli and densities corresponding to the material combination PMMA–silicone–PMMA, at least one of the branches in the dispersion relation of surface waves trapped in the soft layer exhibits a zero-group velocity point. Potential applications of these 1D guided surface waves in non-destructive evaluation are discussed. • Surface waves were studied localized in a soft layer between two quarter-spaces. • Their dispersion relation and displacements were computed by FEM. • At small phase speeds they mostly behave like edge modes in plates with rigid walls. • Close to the cut-off speed they have the character of two interacting wedge waves. • They can be well described by a perturbation-theoretical approach. [ABSTRACT FROM AUTHOR]

• The study of COVID-19 mathematical model. • Use of non-singular Mittag-Leffler kernel fractional derivative. • Derivation of operational matrix of fractional differentiation for above fractional derivative. • Numerical solution of a system of fractional ODE with non-singular kernel with this newly developed operational matrix. • Effect of contact rate and transmissibility multiple on infected people and dynamics of all unknowns for different fractional order. We analyze a proposition which considers new mathematical model of COVID-19 based on fractional ordinary differential equation. A non-singular fractional derivative with Mittag-Leffler kernel has been used and the numerical approximation formula of fractional derivative of function (t − a) n is obtained. A new operational matrix of fractional differentiation on domain [0, a ], a ≥ 1, a ∈ N by using the extended Legendre polynomial on larger domain has been developed. It is shown that the new mathematical model of COVID-19 can be solved using Legendre collocation method. Also, the accuracy and validity of our developed operational matrix have been tested. Finally, we provide numerical evidence and theoretical arguments that our new model can estimate the output of the exposed, infected and asymptotic carrier with higher fidelity than the previous models, thereby motivating the use of the presented model as a standard tool for examining the effect of contact rate and transmissibility multiple on number of infected cases are depicted with graphs. [ABSTRACT FROM AUTHOR]