Modified Legendre Polynomials Research Articles

Modal spectral element method with modified Legendre polynomials to analyze binary crossed gratings.

In a previous paper, a modal spectral element method (SEM), the originality of which comes from the use of a hierarchical basis built with modified Legendre polynomials, was shown to be very powerful for the analysis of lamellar gratings. In this work, keeping the same ingredients, the method has been extended to the general case of binary crossed gratings. The geometric versatility of the SEM is illustrated with gratings whose patterns are not aligned with the boundaries of the elementary cell. The method is validated by a comparison to the Fourier modal method (FMM) in the case of anisotropic crossed gratings and with the FMM with adaptive spatial resolution in the case of a square-hole array in a silver film.

Galerkin Method and Its Residual Correction with Modified Legendre Polynomials

Accuracy and error analysis is one of the significant factors in computational science. This study employs the Galerkin method to solve second order linear or nonlinear Boundary Value Problems (BVPs) of Ordinary Differential Equations (ODEs) with modified Legendre polynomials to seek numerical solutions. The residual function of a differential operator is used as non-homogeneous term information of an error differential equation. The Galerkin approximation is then improved or corrected by solving the error differential equation by the Galerkin method using the same polynomials. Thus we apply the double layer Galerkin method to a variety of instances. We compare approximate solutions with exact ones and results available in the literature, and in every case, we find better accuracy.

Spectral element method with modified Legendre polynomials for modal analysis of lamellar gratings.

We report on the derivation of a spectral element method whose originality comes from the use of a hierarchical basis built with modified Legendre polynomials. We restrict our work to TM polarization, which is the most challenging. The validation and convergence are carefully checked for metallic dielectric gratings. The method is shown to be highly efficient and remains stable for huge truncation numbers. All the necessary information is given so that non-specialists can implement the method.

Numerical Solutions of Linear Fractional Order BVP by Galerkin Residual Method with Differentiable Polynomials

The main objective of this paper is to solve numerically the differential equations of fractional order with homogeneous boundary conditions by the Galerkin weighted residual method. In this method, linear combinations of some types of functions are used to find the approximate solutions which must satisfy the homogeneous boundary conditions. Such type of functions should be differentiated and integrated easily, so the piecewise polynomials, namely, Bernstein, Bernoulli and Modified Legendre polynomials are used as basis functions in this paper. The fractional derivatives are used in the conjecture of Riemann-Liouville and Caputo sense. Thus, we develop the Galerkin weighted residual formulation, in matrix form, to the linear fractional order boundary value problems, in details, which is easy to understand. The accuracy and applicability of the present method are demonstrated through few numerical examples. We observe that the approximate results converge monotonically to the exact solutions. In addition, we compare the approximate results with the exact solutions, and also with the existing solutions which are available in the literature. The absolute errors are depicted in tabular form as well as graphical representations, a reliable accuracy is achieved. The proposed method may be applied to fractional order partial differential equations also.

Numerical Solutions of Black-Scholes Model by Du Fort-Frankel FDM and Galerkin WRM

The main objective of this paper is to find the approximate solutions of the Black-Scholes (BS) model by two numerical techniques, namely, Du Fort-Frankel finite difference method (DF3DM), and Galerkin weighted residual method (GWRM) for both (call and put) type of European options. Since both DF3DM and GWRM are the most familiar numerical techniques for solving partial differential equations (PDE) of parabolic type, we estimate options prices by using these techniques. For this, we first convert the Black-Scholes model into a modified parabolic PDE, more specifically, in DF3DM, the first temporal vector is calculated by the Crank-Nicolson method using the initial boundary conditions and then the option price is evaluated. On the other hand, in GWRM, we use piecewise modified Legendre polynomials as the basis functions of GWRM which satisfy the homogeneous form of the boundary conditions. We may observe that the results obtained by the present methods converge fast to the exact solutions. In some cases, the present methods give more accurate results than the earlier results obtained by the adomian decomposition method [14]. Finally, all approximate solutions are shown by the graphical and tabular representations.

Generating Functions for Legendre Polynomials by using Group Theoretic Method

In this paper we obtain generating functions for the Legendre polynomials \(P_n (x)\) in its modified form by using Weisner's group theoretic method. Whereas, we deployed it to determine the new generating relations between the generalized Legendre polynomials and with easy way. The ideas in consent with proofs are originated from the book of McBride [8] and is used to determine new generating relations which involve modified Legendre polynomials.

Milk mid-infrared spectra enable prediction of lactation-stage-dependent methane emissions of dairy cattle within routine population-scale milk recording schemes

Mitigating the proportion of energy intake lost as methane could improve the sustainability and profitability of dairy production. As widespread measurement of methane emissions is precluded by current in vivo methods, the development of an easily measured proxy is desirable. An equation has been developed to predict methane from the mid-infrared (MIR) spectra of milk within routine milk-recording programs. The main goals of this study were to improve the prediction equation for methane emissions from milk MIR spectra and to illustrate its already available usefulness as a high throughput phenotypic screening tool. A total of 532 methane measurements considered as reference data (430 ± 129 g of methane/day) linked with milk MIR spectra were obtained from 165 cows using the SF6 technique. A first derivative was applied to the MIR spectra. Constant (P0), linear (P1) and quadratic (P2) modified Legendre polynomials were computed from each cows stage of lactation (days in milk), at the day of SF6 methane measurement. The calibration model was developed using a modified partial least-squares regression on first derivative MIR data points × P0, first derivative MIR data points × P1, and first derivative MIR data points × P2 as variables. The MIR-predicted methane emissions (g/day) showed a calibration coefficient of determination of 0.74, a cross-validation coefficient of determination of 0.70 and a standard error of calibration of 66 g/day. When applied to milk MIR spectra recorded in the Walloon Region of Belgium (≈2 000 000 records), this equation was useful to study lactational, annual, seasonal, and regional methane emissions. We conclude that milk MIR spectra has potential to be used to conduct high throughput screening of lactating dairy cattle for methane emissions. The data generated enable monitoring of methane emissions and production characteristics across and within herds. Milk MIR spectra could now be used for widespread screening of dairy herds in order to develop management and genetic selection tools to reduce methane emissions.

Hot topic: Innovative lactation-stage-dependent prediction of methane emissions from milk mid-infrared spectra

The main goal of this study was to develop, apply, and validate a new method to predict an indicator for CH4 eructed by dairy cows using milk mid-infrared (MIR) spectra. A novel feature of this model was the consideration of lactation stage to reflect changes in the metabolic status of the cow. A total of 446 daily CH4 measurements were obtained using the SF6 method on 142 Jersey, Holstein, and Holstein-Jersey cows. The corresponding milk samples were collected during these CH4 measurements and were analyzed using MIR spectroscopy. A first derivative was applied to the milk MIR spectra. To validate the novel calibration equation incorporating days in milk (DIM), 2 calibration processes were developed: the first was based only on CH4 measurements and milk MIR spectra (independent of lactation stage; ILS); the second included milk MIR spectra and DIM information (dependent on lactation stage; DLS) by using linear and quadratic modified Legendre polynomials. The coefficients of determination of ILS and DLS equations were 0.77 and 0.75, respectively, with standard error of calibration of 63g/d of CH4 for both calibration equations. These equations were applied to 1,674,763 milk MIR spectra from Holstein cows in the first 3 parities and between 5 and 365 DIM. The average CH4 indicators were 428, 444, and 448g/d by ILS and 444, 467, and 471g/d by DLS for cows in first, second, and third lactation, respectively. Behavior of the DLS indicator throughout the lactations was in agreement with the literature with values increasing between 0 and 100 DIM and decreasing thereafter. Conversely, the ILS indicator of CH4 emission decreased at the beginning of the lactation and increased until the end of the lactation, which differs from the literature. Therefore, the DLS indicator seems to better reflect biological processes that drive CH4 emissions than the ILS indicator. The ILS and DLS equations were applied to an independent data set, which included 59 respiration chamber measurements of CH4 obtained from animals of a different breed across a different production system. Results indicated that the DLS equation was much more robust than the ILS equation allowing development of indicators of CH4 emissions by dairy cows. Integration of DIM information into the prediction equation was found to be a good strategy to obtain biologically meaningful CH4 values from lactating cows by accounting for biological changes that occur throughout the lactation.

Solving Duffing equation using an improved semi-analytical method

Since achieving the precise analytical solution of most nonlinear problems is not possible, numerical methods are meant to be used. In fact a very limited number of nonlinear problems have the exact analytical solution. Let us consider some other methods which are known as semi-analytical methods and applied to solve nonlinear problems. In these methods by using a primary approximation and an iterative process, the approximate solution can be provided, so that the iterations converge to the desired solution. Methods which have been employed and support researchers in this field are categorized as Adomian decomposition method(ADM), Homotopy perturbation method(HPM), Homotopy analysis method(HAM), and He variational iteration method(VIM). The aim of this paper is to suggest an improvement of reliability of Adomian decomposition method to obtain a solution to Duffing equation. In details, Adomian decomposition method using modified Legendre polynomials has been applied in the present work that allows researchers to provide approximation of the solution with high accuracy. This method is tested for example. Moreover the obtained numerical results will be presented.

Numerical Solutions of Sixth Order Linear and Nonlinear Boundary Value Problems

The aim of paper is to find the numerical solutions of sixth order linear and nonlinear differential equations with two point boundary conditions. The well known Galerkin method with Bernstein and modified Legendre polynomials as basis functions is exploited. In this method, the basis functions are transformed into a new set of basis functions, which satisfy the homogeneous form of Dirichlet boundary conditions. A rigorous matrix formulation is derived for solving the sixth order BVPs. Several numerical examples are considered to verify the efficiency and implementation of the proposed method. The numerical results are compared with both the exact solutions and the results of the other methods available in the literature. The comparison shows that the performance of the present method is more efficient and yields better results.

Image analysis by modified Legendre moments

In the paper, a new set of orthogonal moments based on the modified Legendre polynomials is introduced. Three properties of the modified Legendre polynomials, which are orthogonality, orthogonal invariance and the characteristic that an interval on the center of [ - 1 , 1 ] covers more zeros than do that on the edge of [ - 1 , 1 ] , are discussed detailedly. The orthogonality of the proposed moments ensures minimal information redundancy in a moment set. The orthogonal invariance of the proposed polynomials makes the proposed moments have the property of translation invariance. And, the third property ensures that the modified Legendre moments have superior feature representation capabilities over Legendre moments in analyzing small images. For small images, the description by the modified Legendre moments is better than that by the Legendre moments and the Chebyshev moments in terms of image-reconstruction errors. Theoretical and experimental measures of performance are carried out to investigate the image-representation capabilities of the proposed moments for images and noisy-images. The computational aspects of the moments using recurrence, integral, symmetry and translation invariance are also discussed. Experimental results are shown.

Approximation of Linear Age-Structured Population Models Using Legendre Polynomials

We develop a numerical algorithm for approximation of solutions for linear age-structured population models. The construction is based on approximation of age distributions by modified Legendre polynomials and uses the Trotter-Kato theorem of semigroup theory for the corresponding abstract Cauchy problem. Unbounded resp. nonintegrable mortality rates are admissible.

A numerical solution for axisymmetric contact problems involving rigid indenters on elastic layers

A simple numerical scheme is developed for the solution of axisymmetric frictionless contact problems involving rigid indenters on bonded and unbonded elastic layers. The integral equation relating the unknown pressure to the given displacement over the contact region is reduced, using modified Legendre polynomials, to a system of algebraic linear equations whose solution yields the unknown coefficients in the expansion for the pressure. Results for pressure distribution, total load, penetration depth over the contact region and normal surface displacement outside the contact region are presented for both spherical and flat-ended circular indenters for β (contact radius to layer thickness ratio) in the range 0 ⩽ β ⩽ 20 and Poisson's ratio 0 ⩽ ν ⩽ 0.5.

A Legendre Approximation Method for the Circular Microstrip Disk Problem

The quasi-static solution for the circular microstrip disk is studied using a GaIerkin solution to the Fredholm integral equation of the first kind derived by using the Green's function approach. The basis functions are modified Legendre polynomials combined with a reciprocal square root to provide the correct singularity in charge density at the edge of the disk. The integrals involving the singular part of the Green's function are evaluated exactly, the remainder by using Gaussian quadrature. The method is compared in computational efficiency with recent methods based either on a Galerkin approach in the spectral domain, or the use of dual integral equations. Numerical results are given for charge distribution and capacitance; they are compared to exact results and those obtained by others, and the limitations of those methods are discussed. Closed form expressions are given for the capacitance of a disk based on two simple charge distributions.

Solution thermodynamic quantities represented by modified legendre polynomials

Solution thermodynamic quantities represented by modified legendre polynomials

Low-Pass Filters Using Ultraspherical Polynomials

The problem of approximating the ideal normalized amplitude response of a low-pass filter by the use of a set of ultraspherical polynomials is considered. The amplitude response obtained is more general than the analogous response of the Chebyshev filter because of an additional parameter available with the ultraspherical polynomials. It is shown that this additional parameter may be used to obtain a response having either less ripple or sharper cutoff than the Chebyshev response. The ultraspherical polynomial filter is shown to include as special cases the Chebyshev filter, the Butterworth filter, and a filter recently developed utilizing modified Legendre polynomials.