Filter integrals for orthogonal polynomials

Motivated by an expression by Persson and Strang on an integral involving Legendre polynomials, stating that the square of $P_{2n+1}(x)/x$ integrated over $[-1,1]$ is always $2$, we present analog results for Hermite, Chebyshev, Laguerre and Gegenbauer polynomials as well as the original Legendre polynomial with even index.

Solution of Multi-order Fractional Differential Equation Based on Conformable Derivative by Shifted Legendre Polynomial

The aim of this article is the way for finding approximation solution of multi-order fractional differential equation with conformable sense with use approximated function by shifted Legendre polynomial, the method is easy and powerful for get our results of the linear and non-linear equation, the background idea behind this method is finding system of algebra after achieving messing variable is that mean obtain approximate solution, a few examples illustrates for presented how much our method is capable.

On Weighted Pál Type (0,2)-interpolation on the Unit Circle

In this paper, we study the explicit representation of weighted Pál-type (0,2)-interpolation on two pairwise disjoint sets of nodes on the unit circle, which are obtained by projecting vertically the zeros of (1−x2)Pn(x) and Pn′′(x) respectively, where Pn(x) stands for nth Legendre polynomial.AMS Classification (2000): 41A05, 30E10.

A Recursive Legendre Polynomial Analytical Integral Method for the Fast and Efficient Modelling Guided Wave Propagation in Rectangular Section Bars of Orthotropic Materials

Ultrasonic guided waves are widely used in non-destructive testing (NDT), and complete guided wave dispersion, including propagating and evanescent modes in a given waveguide, is essential for NDT. Compared with an infinite plate, the finite lateral width of a rectangular bar introduces a greater density of modes, and the dispersion solutions become more complicated. In this study, a recursive Legendre polynomial analytical integral (RLPAI) method is presented to calculate the dispersion behaviours of guided waves in rectangular bars of orthotropic materials. The existing polynomial method involves a large number of numerical integration steps, and it is often computationally costly to compute these integrals. The presented RLPAI method uses analytical integration instead of numerical integration, thus leading to a significant improvement in the computational speed. The results are compared with those published previously to validate our method, and the computational efficiency is discussed. The full three-dimensional dispersion curves are plotted. The dispersion characteristics of propagating and evanescent waves are investigated in various rectangular bars. The influences of different width-to-thickness ratios on the dispersion curves of four types of low-order modes for a rectangular bar of an orthotropic composite are illustrated.

Comparison and estimation of different linear and nonlinear lactation curve submodels in random regression analyses on dairy cattle

In the random regression model (RRM) for milk yield, by replacing empirical lactation curves with the five-order Legendre polynomial to fit fixed groups, the RRM can be transformed to a hierarchical model that consisted of a RRM in the first hierarchy with Legendre polynomials as individuals’ lactation curves resolved by restricted maximum likelihood (REML) software, and a multivariate animal model for phenotypic regression coefficients in the second hierarchy resolved by DMU software. Some empirical lactation functions can be embedded into the RRM at the first hierarchy to well fit phenotypic lactation curve of the average observations across all animals. The functional relationship between each parameter and time can be described by a Legendre polynomial or an empirical curve usually called submodel, and according to three commonly used criteria, the optimal submodels were picked from linear and nonlinear submodels except for polynomials. The so-called hierarchical estimation for the RRMs in dairy cattle indicated that more biologically meaningful models were available to fit the lactation curves; moreover, with the same number of parameters, the empirical lactation curves (MIL1, MIL5, and MK1 for 3, 4, and 5 parameters, respectively) performed higher goodness of fit than Legendre polynomial when modelling individuals’ phenotypic lactation curves.

Retrieval of Photometric Parameters of Minerals Using a Self-Made Multi-Angle Spectrometer Based on the Hapke Radiative Transfer Model

In this paper, a self-made, mineral, multi-angle, spectrum measurement device is employed to measure the multi-angle spectra of olivine and plagioclase; the multi-angle spectra of ilmenite in the Reflectance Experiment Laboratory (RELAB) Spectral Library are collected; and the optimized retrieval of the photometric parameters of the Hapke model is realized. Importantly, the derived result of the single-scattering albedo (SSA) is stable and has both mathematical meaning and physical meaning. The derived Legendre polynomial coefficients of the phase function can better simulate the variation in the mineral spectra with angle. This paper compares the effects of multi-angle and single-angle spectral data on the photometric parameter derived results. The setting of the Legendre polynomial coefficient of the scattering phase function mainly affects the simulation accuracy of the mineral spectra as a function of angle. Using this coefficient to optimize the retrieval, the simulation accuracy is moderately improved compared with the single-angle simulation. The estimation of photometric parameters based on multi-angle spectral data can provide a basis for setting the empirical values of the phase function parameters from single-angle spectral calculations, which can more truly reflect the law of reflectance spectra changing with angle than Lucey’s traditional empirical value of the phase function (b = −0.4 and c = 0.25). The results of multi-angle spectra retrieval in this paper show that the Legendre polynomial coefficients of the phase function vary with wavelength rather than being constant and that different minerals differ greatly.