Legendre Series Expansion Research Articles

The impedance of a center-fed strip dipole by the Poynting vector method

The impedance of a center-fed strip dipole of width w and thickness t, with t/spl Lt/w, is obtained by the Poynting vector method. With w also taken to be small with respect to the operating wavelength (w/spl Lt//spl lambda/) the surface current on the dipole is modeled by the classical sinusoidal standing wave along its length. There are, however, appropriate transverse singularities assumed at the edges of the strip. The Poynting vector method leads to a two-dimensional integral in the transverse coordinates involving the classical mutual impedance between filamentary dipoles located on the surface of the strip itself as well as the above mentioned transverse edge singularities. A Legendre series expansion of this mutual impedance (the coefficients obtained by numerical integration) leads to a rapidly convergent series solution (eight terms) for the overall impedance of the strip dipole. Numerical results are provided For a variety of strip widths, thicknesses and lengths. They are compared with the impedances of equivalent circular dipoles of radius a=(w+t)/4 and good agreement occurs except for quite wide strips (w=0.05/spl lambda/).< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Legendre series estimate of a distribution function

An estimation procedure for the cumulative distribution function of a random variable based on Legendre series expansion of the density is introduced. Given a sequence of independently and identically distributed random variables, an estimate of the density function of the underlying distribution is constructed using its Legendre expansion. The operational matrix of integration associated with Legendre polynomials is then utilized to derive the estimate of the corresponding distribution function. The limiting properties of the estimate are investigated and discussed. The results of some simulations are used to compare the estimate with the celebrated empirical distribution function and illustrate the applicability of the proposed method.

Error analysis of the Tau method: dependence of the approximation error on the choice of perturbation term

We consider a system of ordinary differential equations with constant coefficients and deduce asymptotic estimates for the Tau Method approximation error vector per step for different choices of the perturbation term H n ( x). The cases considered are Legendre polynomials, Chebyshev polynomials, powers of x and polynomials of the form ( x 2 − r 2) n, −r ⩽ x ⩽ r . The first two are standard choices for the Tau Method, for Chebyshev and Legendre series expansion techniques and also for collocation; the third one realizes the classical power series expansion techniques in the framework of the Tau Method and the last is related to the trial functions used in weighted residuals methods; we shall refer to it as the weighted residuals choice. We show that the resulting Tau Method implementations can be arranged into the following scale of increasing error estimates at the end point x = r : Legendre < Chebyshev ⪡ Power series < Weighted residuals. For the interesting case of Legendre Tau approximations, we offer upper and lower error bounds for the end point of the interval of approximation. In particular, this last estimates solve a conjecture on increased accuracy at the end point of the interval of approximation formulated by Lanczos in 1956. Such conjecture has equivalent forms for other polynomial methods for the numerical solution of differential equations. Although formulated in the convenient framework of the recursive Tau Method (see Ortiz [1]), the results given here apply, without essential modifications, to Chebyshev or Legendre series expansion techniques for differential equations, collocation and spectral methods. We give numerical examples which confirm the sharpness of the lemmas and theorems given in this paper. Finally, we discuss in an example the application of our results to the analysis of singularly perturbed differential equations.

霧粒の粒径分布239例を用いた可視および赤外域の位相関数の平均的特性

The mean characteristics of the phase function and of the coefficients of its Legendre series expansion were studied within a sample of 239 observed fog droplet size spectra, by computing their mean value, dispersion coefficient, skewness and kurtosis, at 86 wavelengths in the λ=0.35-90μm spectral interval.The results showed that the mean spectral behavior of the above quantities retains all the features pertaining to single spectra, and that the values of the optical quantities obtained for the different spectra are widely dispersed about their mean values. In particular, the dispersion for the phase function, at visible wavelengths, has a minimum close to zero (disp=l.3%) around 40°, surrounded by a wide interval (15-55°) of low dispersion; for the asymmetry parameter the dispersion increases with wavelength from 1.7% at λ=0.35μm up to 48% at λ=90μm. The data produced can serve as reference values for both experimental and theoretical needs.

霧粒の粒径分布239例を用いた霧の雲物理的性質と光学的特性の研究

Mean radius (r) and moments around r, total concentration of droplets, liquid water content (W), terminal fall speed and visual range (Vm), have been computed for a sample of 239 droplet size distri-butions of valley, advection and radiation fog. The scattering, absorption and extinction coefficients, the albedo for single scattering, the phase function P(cosθ) at several scattering angles θ, the coeffi-cients χn, χm connected with the Legendre series expansion and the δ-M approximation of the phase function have also been computed at seventy-four wavelengths from 0.35 to 90μm.Data elaboration has basically consisted in the determination of the standard statistical parameters (histogram, mean, value, dispersion, skewness and kurtosis) for each of the above quantities and each of the various fog groups considered.Results concerning the grouping of fog spectra according to the fog type show that all the quantities are widely dispersed within each fog group and, in addition, the respective histograms partially overlap to each other; similar results also emerge by grouping the spectra according to their values of W and Vm, so that it appears unreliable to consider mean values as characteristic of a given fog group. Finally, a description of the mean physical and optical properties of the whole sample is presented, with the spectra all grouped together.

Some methods for reducing propagation-induced phase errors in coherent imaging systems I Formalism

The random phase errors induced in an electromagnetic wave propagating through turbulence in the troposphere and/or the ionosphere can degrade the performance of a coherent imaging system such as a synthetic-aperture radar. Several methods for removing or reducing these phase errors are described. Some of these methods, such as cubic-spline approximations with map drift and a deterministic technique based on a Legendre-series expansion of the phase error, are not new but are included for comparison. At least three techniques are (to our knowledge) new; a binary multiplexing scheme, a method using the Wigner–Ville distribution, and an optimal estimator based on the covariance matrix of the Legendre-series expansion coefficients. The method based on the Wigner–Ville distribution promises to be the most effective in reducing the deleterious effects of the phase errors on image quality.

Some methods for reducing propagation-induced phase errors in coherent imaging systems II Numerical results

The phase-error mitigation methods are applied to computer-generated one-dimensional (the azimuthal dimension) synthetic-aperture radar images. These images represent a superposition of five point targets of different intensities and locations. Specific realizations of the turbulence-induced phase errors are constructed by the Karhunen–Loeve technique. A specific realization is added to the phase history of the point targets to obtain the phase history of the distorted image. Reasonable reconstructions from the phase histories of the distorted images, using cubic-spline fits based on either the map drift or the mean frequency difference, are obtained when low-frequency phase errors dominate the spectrum. Binary multiplexing and a deterministic method based on the Legendre-series expansion of the phase error also produce good results for the first eight terms of the expansion. The best reconstructions are obtained by using the Wigner–Ville distribution.

ON THE DEFORMATION OF DROPS AND BUBBLES WITH VARYING INTERFACIAL TENSION

In this paper the deformation of a translating fluid drop is examined in the limit of low Reynolds and capillary numbers. The general case of an arbitrarily varying interfacial tension is considered here. With the capillary number as a perturbation parameter, an expansion is carried out for small deformation to satisfy the normal stress condition. This leads to a rapidly converging Legendre series expansion for the deformation. This general result is applied to the special case of a translating drop with a stagnant cap of surfactant. An interesting feature of this special case is that although the shear stress at the interface has a square root singularity, the total normal stress is non-singular.

A direct approach using the shifted Legendre series expansion for near optimum control of linear time-varying systems with multiple state and control delays

A computationally efficient method, which uses finite orthogonal expansions to approximate state and control variables, is developed to obtain suboptimal control for time-varying systems with multiple state and control delays and with quadratic cost. Shifted Legendre polynomials form the basis of expansions as a consequence of their useful properties. The Galerkin method is used to reduce the problem to one of minimizing a quadratic form with linear algebraic constraints. The solution of a coupled two-point boundary-value problem with both delayed and advanced terms, which is always required in applying the Pontryagin's maximum principle to the optimization of delay systems, is thus avoided.

Optimization of batch crystallization processes via legendre polynomials

Abstract The characteristics of crystallization in a batch Couette flow crystallizer are modeled by population balance, mass and energy equations. The effective and powerful method of Legendre polynomials is employed to transform the partial differential equation of population balance into a series of ordinary differential equations of Legendre series expansion coefficients. Optimization theory is applied to find the minimum coefficient of variations, which accounts for the uniformity of crystal‐size distribution within the crystallizer. The results are obtained by the Runge‐Kutta‐Gill computational method and are explained by the physical significance of the growth and nucleation phenomena.

Relaxation and bremsstrahlung of thick-target electron streams - A simple application of the Legendre expansion method

view Abstract Citations (6) References (17) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Relaxation and bremsstrahlung of thick-target electron streams: a simple application of the Legendre expansion method. Hoyng, P. ; Melrose, D. B. ; Adams, J. C. Abstract The Legendre series expansion of the three-dimensional equations for electron velocity and Langmuir wave distributions provides a numerical treatment for the relaxation of a stationary, axisymmetric electron stream in a homogeneous and fully ionized hydrogen plasma. Both quasilinear and Coulomb interactions figure in the treatment. The electrons are represented by a Fokker-Planck system of equations, which allows elimination of the distinction between background and stream electrons. A computation of the bremsstrahlung associated with the isotropic component of the electron distribution yields a simple model for radiation from thick-target electron streams, for example in impulsive hard X-ray emission from solar flares. Results of the study suggest power law-like bremsstrahlung spectra as they are often observed can be produced quite readily. Publication: The Astrophysical Journal Pub Date: June 1979 DOI: 10.1086/157152 Bibcode: 1979ApJ...230..950H Keywords: Bremsstrahlung; Electron Diffusion; Hydrogen Plasma; Solar Flares; Solar Spectra; Solar X-Rays; Accuracy; Boundary Value Problems; Coulomb Collisions; Electron Distribution; Electron Radiation; Electrostatic Probes; Legendre Functions; Linear Equations; Maxwell Equation; Solar Electrons; Velocity Distribution; X Ray Spectra; Solar Physics; Plasma; Solar Flares:X Rays full text sources ADS |

Photoneutron Angular Distributions for 4He

Differential cross sections for the reaction 4He(γ, n)3He have been measured at laboratory angles of 53°, 68°, 83°, 98°, 128°, and 143° for excitation energies between 22 and 32 MeV by the photoneutron time of flight technique. The photoneutron spectra were used to obtain the angular distribution coefficients in the Legendre series expansion for the differential photoneutron cross sections. Analysis of these coefficients indicates that the photodisintegration process in this nucleus, while dominated by E1 photon absorption followed by channel spin zero (no spin flip) p wave neutron emission, is affected by E1 channel spin one and E2 contributions. In the higher energy region (E &gt; 27 MeV), the amount of E2 absorption observed was ~6% of the total and is dominated by the channel spin one E2 matrix element. The total cross section obtained from the measured differential cross sections yields a value of σ(γ, p)/σ(γ, n) near unity, in agreement with a calculation of this ratio which requires no charge symmetry breaking component of the nuclear force.

Algorithm 473: Computation of Legendre series coefficients [C6

LEGSER approximates the first N + 1 coefficients B n of the Legendre series expansion of a function ƒ( x ) having known Chebyshev series coefficients A n . Several algorithms are available for the computation of coefficients A n of the truncated Chebyshev series expansion on [-1, 1] ƒ( x ) ≃ ∑′ N n =0 A n T n ( x ), (1) where ∑′ donotes a sum whose first term is halved. The commonly used algorithms are based on the orthogonal property of summation of the Chebyshev polynomials [1]. The application of the analogous property of the Legendre polynomials for the calculation of the coefficients B n of the expansion ƒ( x ) ≃ ∑ N n =0 B n P n ( x ) (2) is less suitable for practical use since it requires the abscissas and weights of the Gauss-Legendre quadrature formulas [2].

High-field galvanomagnetic effects in two-valley n-type semiconductors

With the carrier distribution function approximated by the first two terms in its Legendre series expansion and the scattering effect by a momentum relaxation time, current-voltage characteristics have been derived with and without considering the effects of applied transverse magnetic field in two-valley n-type semiconductors. The analysis indicates that the applied transverse magnetic field tends to change the electron mobility and to lower the electron population in the upper valley relative to that in the lower valley. The computed results for n-GaAs are in reasonable agreement with presently available experimental ones.

Phenomenology of [formula omitted] below 1 GeV

The angular distribution for the production reaction πN → πN ∗ at energies below 1 GeV. is investigated in terms of the familiar Legendre series expansion. By incorporating results from pion-nucleon phase-shift analysis we show that the most probable dominant inelastic decays of three pion-nucleon resonances are D 13(1520)→N ∗(1238)+Π, D 15(1650)→D 13(1520)+Π, F 15(1670)→D 13(1520)+Π,

Differential cross sections for the T(p, n)He 3 reaction

The zero-degree differential cross section of the T(p, n)He 3 reaction was measured with a recoil-proton counter telescope at about 0.5-MeV intervals from 5 to 13.5 MeV. Relative angular distributions of the neutrons from the reaction were also obtained at 0.5-MeV intervals from 5 to 13 MeV proton energy. Cross sections were obtained by normalization to the zero-degree telescope measurements. The distributions, when transformed to the centre-of-mass system, are strongly peaked in the backward hemisphere. The centre-of-mass distributions were fitted with a Legendre series expansion and the total reaction cross section obtained from the coefficients.