The Fourier series of Gegenbauer's function.
Fourier series of Gegenbauer function, examining convergence characteristics
CONVERGENCE↗
SEARCH · Engineering Papers
Results for “LEGENDRE FUNCTION SERIES”
Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.
Error curves for Lanczos' ''selected points'' method.
Maximum error curves for Lanczos selected point method of polynomial solution to ordinary differential equations, using Chebyshev and Legendre functions
ANALYTIC FUNCTION↗
A note of expansions of functions of velocity in the two-body problem
Fourier expansions based on both the true anomaly and the mean anomaly are obtained for the functions of velocity in the two-body problem; the series of coefficients is written from classical formulae involving associated Legendre polynomials, Gegenbauer polynomials, or Bessel functions. The Fourier expansions are compared with the expansions in powers of eccentricity developed by Broucke (1974) through use of computerized Poisson series manipulation.
Kinoshita, H.↗
The application of Legendre-tau approximation to parameter identification for delay and partial differential equations
Approximation schemes based on Legendre-tau approximation are developed for application to parameter identification problem for delay and partial differential equations. The tau method is based on representing the approximate solution as a truncated series of orthonormal functions. The characteristic feature of the Legendre-tau approach is that when the solution to a problem is infinitely differentiable, the rate of convergence is faster than any finite power of 1/N; higher accuracy is thus achieved, making the approach suitable for small N.
Ito, K.↗
An analytic study on bounds for the associated Legendre functions
The bounds for the normalized associated Legendre functions P sub nm were studied to provide a rational basis for the truncation of the geopotential series in spherical harmonics in various orbital analyses. The conjecture is made that the largest maximum of the normalized associated Legendre function lies in the interval which indicates the greatest integer function. A procedure is developed for verifying this conjecture. An on-line algebraic manipulator, IAM, is used to implement the procedure and the verification is carried out for all n equal to or less than 2m, for m = 1 through 6. A rigorous proof of the conjecture is not available.
Payne, M. H.↗
A simple method for obtaining a three-dimensional proton distribution function from Voyager plasma data
The main sensor of the Vogager plasma experiment consists of a cluster of three, modulated-grid Faraday cups whose normals are arranged symmetrically about the symmetry axis of the cluster at an angle of 20 degrees to that axis. In interplanetary space, each cup explores the positive ion distribution by accepting particles from contiguous slices in velocity space. The slices are narrow in the direction of the normal to the modulating grid but are broad in planes parallel to that grid. The resulting three sets of measurements can be combined to yield the three-dimensional distribution function in the following way: the distribution function is assumed to be gyrotropic. For each value of speed in a frame of reference moving with the bulk velocity of the solar wind, the variation of the distribution function with angle from the field direction is represented by a series of Legendre polynomials. Effects such as double-streaming and heat flow can be well represented by using only the first three terms of the series which are fully specified by the measurements. Examples of the use of this method in the analysis of Voyager data are shown.
Olbert, S.↗
Legendre-Tau approximations for functional differential equations
The numerical approximation of solutions to linear functional differential equations are considered using the so called Legendre tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time differentiation. The approximate solution is then represented as a truncated Legendre series with time varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximations is made.
Ito, K.↗
Legendre-tau approximations for functional differential equations
The numerical approximation of solutions to linear retarded functional differential equations are considered using the so-called Legendre-tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time-differentiation. The approximate solution is then represented as a truncated Legendre series with time-varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximation is made.
Ito, K.↗
Determination of volume fraction in multiphase systems using incomplete pole figures
The determination of the volume fraction of a second phase in a multiphase sample by X-ray diffraction becomes more difficult if the diffracting planes have a preferred orientation. Lopata and Kula have described a method of treating this problem using complete pole figures for each of the phases. With some samples, it is not always possible or convenient to obtain data over the full hemisphere. Equations and an example are given which require X-ray data over a limited range of approximately 0 to 75 deg. This can be obtained by reflection without a specially cut sample or transmission data. A series of Legendre polynomials are fitted to data collected while spinning the sample about its normal. An extrapolation is made possible by introducing two conditions on the end points which must be satisfied if the extrapolation functions are to be valid.
Houska, C. R.↗
Orthogonality of spherical harmonic coefficients
Orthogonality relations are obtained for the spherical harmonic coefficients of functions defined on the surface of a sphere. Following a brief discussion of the orthogonality of Fourier series coefficients, consideration is given to the values averaged over all orientations of the coordinate system of the spherical harmonic coefficients of a function defined on the surface of a sphere that can be expressed in terms of Legendre polynomials for the special case where the function is the sum of two delta functions located at two different points on the sphere, and for the case of an essentially arbitrary function. It is noted that the orthogonality relations derived have found applications in statistical studies of the geomagnetic field.
Mcleod, M. G.↗
Analysis of the vibratory excitation of gear systems. II - Tooth error representations, approximations, and application
The second part of a theory for predicting the vibratory excitation of gear systems from fundamental descriptions of gear tooth elastic properties and deviations of tooth faces from perfect involute surfaces is presented. The first part of the theory provides expressions for the Fourier-series coefficients of the vibratory excitation, and this paper gives expressions for these Fourier-series coefficients in terms of easily interpreted gear tooth metrics that are readily evaluated from tooth-face measurements. Results are given for rectangular tooth-face contact regions using two-dimensional Legendre polynomial expansions of local tooth-pair stiffnesses and stiffness-weighted deviations of tooth faces from perfect involute surfaces. A rigorous transfer function approach is developed that permits separation of the effects of gear tooth errors and gear design parameters; the theory is applicable to helical and spur gears and is illustrated with measurements of tooth-spacing errors and tooth profiles obtained from a pair of spur gears.
Mark, W. D.↗