Legendre Series Research Articles

A Classical Theorem on the Singularities of Legendre Series in C3 and Associated System of Hyperbolic Partial Differential Equations

A classical theorem of Nehari relates the singularities of Legendre series expansions in $C_z$ with those of associated Taylor's series in $C_t$. The generalization of Nehari's theorem is known for Legendre series in $C_{z_1 \times z_2}$. In this paper, function theoretic methods develop the analogous relationships between the singularities of series expanded as triple products of Legendre polynomials in $C_{z_1 \times z_2 \times z_3}$ and those of associated analytic functions in $C_t$. The singularities of these generalized Legendre series are determined by certain elliptic curves. Moreover, these series satisfy a system of hyperbolic partial differential equations (PDEs) in $C^3$ that are connected to Bochner's study of Poisson processes in $R^2$.

Diffusion fields associated with prolate spheroids in size and shape coarsening

The diffusion field or solute concentration distributed around a prolate spheroidal particle simulating a rod- or needle-shaped precipitate has been solved for varying aspect ratios of the precipitate and varying concentrations along the precipitate surface owing to the curvature effect. With the prolate spheroidal coordinates, the principal curvatures of the prolate spheroidal surface are derived as functions of the angular variable, and the Laplace field equation is separated into two Legendre equations either on the radial variable or on the angular variable. The analytical solution fitting the present boundary conditions is secured as the sum of a Legendre function of the second kind of order zero and a Legendre series. The Legendre function of the second kind gives the concentration distribution when the curvature effect can be ignored, whereas the Legendre series represents the concentration contributed from the curvature effect. Numerical results of normalized concentrations are presented as functions of the radial and angular variables for selected aspect ratios. The tangent component of the concentration gradients, contributed from the curvature effect, may cause complicated mass transfer, which is responsible for shape coarsening.

Diffusion fields associated with size and shape coarsening of oblate spheroids

The diffusion field or solute concentration distributed around an oblate spheroidal particle simulating a disc-shaped precipitate has been solved for varying particle aspect ratios and varying concentrations along the precipitate surface because of the curvature effect. With oblate spheroidal coordinates, the principal curvatures of the oblate spheroidal surface are derived as functions of the angular variable, and the Laplace field equation is separated into two Legendre equations on the angular variable and on the radial variable. The analytical solution to the Laplace equation, fitting the present boundary conditions, is secured as the sum of a Legendre function and a Legendre series composed of Legendre functions of the second kind with imaginary arguments. The Legendre function gives the concentration distribution with an ignored curvature effect, whereas the series shows the contribution from the curvature effect. Numerical results of normalized concentrations are presented as functions of the radial and angular variables for selected aspect ratios. The concentration distributions around both oblate and prolate spheroidal particles are shown to reduce to the concentration distributed around a spherical particle when the aspect ratio of the spheroids approaches unity.

Application of Legendre series to the control problems governed by linear parabolic equations

A method for finding the optimal control problems governed by linear parabolic equations using Legendre series is discussed. The optimal control of the one-dimensional diffusion equation is simplified into optimal control of linear system with a quadratic cost functional. The state variable, state rate and the control vector are expanded in the shifted Legendre series with unknown coefficients. The relation between the coefficients of the state rate and control vector with state variable is provided and the necessary condition of optimality is derived as a linear system of linear algebraic equations in terms of the unknown coefficient of the state vector. A numerical example is included to demonstrate the validity and the applicability of the technique.

On the Approximation of Continuous Functions by Fourier–Legendre Sums

Let {Xn}∞0be the orthonormal system of Legendre polynomials on [−1, 1]. Forf∈C[−1, 1] letSn(f) (n+1∈N) be thenth partial sum of the Fourier–Legendre series of the functionf. Some refinements of the classical inequality[formula]involving best approximation inLp-norms are discussed. For a class of examples we obtain better order estimates than those that can be derived from (1). Furthermore, we show that the results are best possible in a certain sense. It turns out that only in two particular cases (p=43andp=4) there is no proof of optimality of the results. In conclusion, we give without proof a generalization of the main theorem to the ultraspherical case.

Fourier transform summation of Legendre series and D-functions

The relation between D- and d-functions, spherical harmonic functions and Legendre functions is reviewed. Dmatrices and irreducible representations of the rotation group O(3) and SU(2) group are briefly reviewed. Two new recursive methods for calculations of D-matrices are presented. Legendre functions are evaluated as part of this scheme. Vector spherical harmonics in the form af generalized spherical harmonics are also included as well as derivatives of the spherical harmonics. The special dmatrices evaluated for argument equal toπ/2 offer a simple method of calculating the Fourier coefficients of Legendre functions, derivatives of Legendre functions and vector spherical harmonics. Summation of a Legendre series or a full synthesis on the unit sphere of a field can then be performed by transforming the spherical harmonic coefficients to Fourier coefficients and making the summation by an inverse FFT (Fast Fourier Transform). The procedure is general and can also be applied to evaluate derivatives of a field and components of vector and tensor fields.

AN OPTIMAL METHOD TO ESTIMATE THE SPHERICAL HARMONIC COMPONENTS OF THE SURFACE AIR TEMPERATURE

This paper describes a method that minimizes the mean squared error (MSE) in estimating the spherical harmonic components of the surface air temperature field. The ratio of the MSE to the variance of the spherical harmonic component is expressed in terms of the length scale λ0, and the positions and weights of the measurement stations. The weights are optimized by the condition of minimizing the sampling error. To present an analytical example, we assume the homogeneous statistics of the temperature anomaly field, and take the low frequency approximation (i.e. ignoring the time dependence). The spectra of the temperature anomaly are the coefficients of a Fourier–Legendre series of the covariance function, and they are analytically derived from a linear noise forced energy balance climate model. Consequently, the MSE, the percentage sampling error, and the signal–noise ratio are computed for a given network of stations. Our results show that: (i) the sampling errors computed from both optimal weights and uniform weights increase with respect to the order of the spherical harmonic component; (ii) the sampling errors computed from optimal weights are significantly smaller than those from uniform weights for sufficiently dense networks. With about 60 reasonably positioned stations for sampling the spherical harmonic components 00 T10 and T11, one can get the sampling error below 10 per cent when the optimal weights are applied. An experiment with 210 stations produces the sampling errors of less than 10 per cent for the spherical harmonic components from T00 up to T54

Fourier transform summation of Legendre series and D-functions

Fourier transform summation of Legendre series and D-functions

The oblique azimuthal projection of geodesic type for the biaxial ellipsoid: Riemann polar and normal coordinates

Riemann polar/normal coordinates are the constituents to generate the oblique azimuthal projection of geodesic type, here applied to the “reference” ellipsoid of revolution (biaxial ellipsoid).Firstly we constitute a minimal atlas of the biaxial ellipsoid built on {ellipsoidal longitude, ellipsoidal latitude} and {metalongitude, metalatitude}. TheDarboux equations of a 1-dimensional submanifold (curve) in a 2-dimensional manifold (biaxial ellipsoid) are reviewed, in particular to represent geodetic curvature, geodetic torsion and normal curvature in terms of elements of the first and second fundamental form as well as theChristoffel symbols. The notion of ageodesic anda geodesic circle is given and illustrated by two examples. The system of twosecond order ordinary differential equations of ageodesic (“Lagrange portrait”) is presented in contrast to the system of twothird order ordinary differential equations of ageodesic circle (Proofs are collected inAppendix A andB). A precise definition of theRiemann mapping/mapping of geodesics into the local tangent space/tangent plane has been found.Secondly we computeRiemann polar/normal coordinates for the biaxial ellipsoid, both in theLagrange portrait (“Legendre series”) and in theHamilton portrait (“Lie series”).Thirdly we have succeeded in a detailed deformation analysis/Tissot distortion analysis of theRiemann mapping. The eigenvalues — the eigenvectors of the Cauchy-Green deformation tensor by means of ageneral eigenvalue-eigenvector problem have been computed inTable 3.1 andTable 3.2 (Λ1, Λ2 = 1) illustrated inFigures 3.1, 3.2 and3.3. Table 3.3 contains the representation ofmaximum angular distortion of theRiemann mapping. Fourthly an elaborate global distortion analysis with respect toconformal Gauβ-Kruger, parallel Soldner andgeodesic Riemann coordinates based upon theAiry total deformation (energy) measure is presented in a corollary and numerically tested inTable 4.1. In a local strip [-l E,l E] = [-2°, +2°], [b S,b N] = [-2°, +2°]Riemann normal coordinates generate the smallest distortion, next are theparallel Soldner coordinates; the largest distortion by far is met by theconformal Gauβ-Kruger coordinates. Thus it can be concluded that for mapping of local areas of the biaxial ellipsoid surface the oblique azimuthal projection of geodesic type/Riemann polar/normal coordinates has to be favored with respect to others.

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>

A novel use of differential equations to fit exponential functions to experimental data

A procedure for fitting multi-exponential functions to experimental data is described. It is fast, requires no initial parameter estimates and is particularly suited to sums of several closely spaced exponentials. The method comprises the application of three well tried numerical techniques: (i) the signal is smoothed by representing it as an abbreviated Legendre series; (ii) the coefficients of a certain kind of differential equation are determined such that it's solution is the closest fit to the smoothed signal; and (iii) the amplitudes of the exponential components are determined, given the calculated values of the exponential rate constants. The method is computationally efficient, since determination of amplitudes and exponents involves the use of linear techniques, and therefore does not require multiple iterations, and the smoothed signal is contained in a handful of coefficients rather than as a lengthy time series. The severe ill-conditioning that is unavoidable in this problem is contained within the well-understood procedures of inverting a matrix and determining the roots of a polynomial. This method is particularly appropriate for analysis of data that may be modelled by a scheme of linked first-order reactions, describing for example the stochastic behaviour of ion channels, a chemical reaction, or the uptake and distribution of a drug within body compartments.

Improved methods for prediction of electromagnetic noise radiated by electrical machines

Improved methods for predicting the electromagnetic noise radiated by electrical machines are presented. The preferred method is based on the calculation of the sound-pressure and stator-vibration velocity distributions on a finite-cylindrical acoustic model. It results in a simple formula for calculating the acoustic power from the relative sound intensity and the stator-surface vibration, and is therefore eminently suitable for use during routine design. The method eliminates the need to calculate the sound-pressure and air-velocity distributions around a machine, as is required for existing methods. It can also account for the effect of an axial variation of the radial vibration. In an alternative method the relative sound intensity is calculated from a spherical acoustic model in which the radial vibration of a machine is expressed as a function of the circumferential mode order in the form of a Fourier series rather than a Legendre series, as in previous analyses. This makes it possible to use Fourier series throughout the various stages leading up to the prediction of electromagnetic noise. Predictions of the relative sound intensity from the different models are compared, and the sound-pressure distribution and the acoustic power are investigated both theoretically and experimentally on a TEFC 3-phase induction motor. It is shown that methods based on a spherical acoustic model are only geometrical and physical approximations, whilst a finite-cylindrical acoustic model is more generally applicable.

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.

Stellar core collapse - A Boltzmann treatment of neutrino-electron scattering

Neutrino-electron scattering plays a major role in the deleptonization of the iron core during the gravitational collapse of a presupernova star and, hence, plays a major role in the success or failure of the shock ejection mechanism for Type II supernovae. In this paper we present the first simulation of realistic gravitational collapse in which neutrino-electron scattering is not aproximated in the neutrino transport equation with either a truncated Legendre series or a Fokker-Planck approximation. We begin with a 1.17 M ○. iron core extracted from a Nomoto-Hashimoto 13 Ma presupernova star

A legendre technique for solving time-varying linear quadratic optimal control problems

A method for the optimal control of linear time-varying systems with a quadratic cost functional is proposed. The state and control variables are expanded in the shifted Legendre series, and an algorithm is provided for approximating the system dynamics, boundary conditions and performance index. The necessary condition of optimality is then derived as a system of linear algebraic equations. Numerical examples are included to demonstrate the validitiy and applicability of the technique.

Excitation of a fluid-filled, submerged spherical shell by a transient acoustic wave

A mathematical formulation and method of solution for the title problem are developed, and numerical results are presented for excitation by a plane step wave. The formulation is based on the familiar equations of motion for a thin spherical shell and the wave equation for the internal and external fluid domains. The Laplace transform is invoked, and the usual separation of variables method produces modal equations for each component of the Fourier–Legendre series solution. The modal equations are then restructured to facilitate transform inversion, yielding delayed differential equations in time for each response mode of the shell-fluid system, which are integrated numerically in time. Complete response solutions then follow by modal superposition, with special techniques being employed to improve modal convergence. Transient response histories are shown for a step-wave-excited steel shell containing water and submerged in water. Also, these results are compared with their counterparts for an empty submerged steel shell.

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.

Orientation-dependent CF 3 emission observed in the Ar( 3P) + CF 3H → CF* 3 (1E′, 2A″ 2) + H + Ar reaction

The orientation-dependent CF 3 emission in the visible region observed in the Ar( 3P) + CF 3H reaction was investigated. The Legendre polynomial and Engel-Levine function expansions were applied to obtain an orientational opacity function, in which the lowest six terms of the Legendre series were found to be important to reproduce the observed orientational dependence. In both cases, these opacity functions reveal three reactive sites. The most favorable one is the attack of the Ar( 3P) atom at the CF 3 group in the collinear approach and the second one correlates with the direct approach to the F atom. The least favorable site is centered at around 140° in the region of H atom attack. There is a cone of small reactivity in the collinear direction at the H-end attack. Although the exchange interaction cannot completely explain the relative reactivity at each reaction site, the shape of the opacity function was attributed to the exchange interaction between the 6a 1 molecular orbital of CF 3H and the 3p empty core orbital of Ar( 3P).

Ionization of molecular hydrogen by proton impact. I. Single ionization.

Single-ionization cross sections of the proton-hydrogen molecule collision are calculated in semiclassical approximation by using atomic and Heitler-London-like two-center molecular wave functions for the ground state of H{sub 2}. In transition matrix elements, calculated with two-center wave functions, the factors depending on {vert bar}{bold r}{minus}{bold R}{sub 0}/2{vert bar} ({bold R}{sub 0} is the vector of the molecular axis) are expanded into Legendre series. The calculated total and differential cross sections are compared to the measured data, and their dependence on target wave functions is discussed. The success of the simple approach, which uses twice the cross sections calculated for the hydrogen atom, is explained with the dominance of monopole contributions in the expansion in {bold R}{sub 0}. While for homonuclear diatomic molecules the differential cross sections could be well described by the additivity rule, for other molecules the validity of such approaches is questionable.

A general solution of axisymmetric problem of arbitrary thick spherical shell and solid sphere

In this paper, the axisymmetric problems of arbitrary thick spherical shell and solid sphere are studied directly from equilibrium equations of three-dimensional problem, and the general solutions in forms of Legendre series for thick spherical shell and solid sphere are given by using the method of weighted residuals.