Terms Of Legendre Polynomials Research Articles

Legendre polynomial solutions of high-order linear Fredholm integro-differential equations

In this study, a Legendre collocation matrix method is presented to solve high-order Linear Fredholm integro-differential equations under the mixed conditions in terms of Legendre polynomials. The proposed method converts the equation and conditions to matrix equations, by means of collocation points on the interval [−1, 1], which corresponding to systems of linear algebraic equations with Legendre coefficients. Thus, by solving the matrix equation, Legendre coefficients and polynomial approach are obtained. Also examples that illustrate the pertinent features of the method are presented and by using the error analysis, the results are discussed.

Frequency spectra of bilaminar prolate spheroidal shells.

The kinetic and strain energy densities were derived for the vibration of a bilaminar prolate spheroidal shell of constant thickness. The bilaminar shell is composed of two bonded concentric prolate spheroidal layers of different material properties and thicknesses. The elastic strain energy density has seven independent kinematic variables: three displacements, two thickness shears, and two thickness stretches. Continuity of displacements is enforced at the interfacial reference surface. The shell has constant thickness h=h1+h2, where h1 and h2 denote the thicknesses of the respective layers. The reference surface eccentricity is defined by 1/a, where a is its shape parameter. Using appropriate comparison functions in terms of Legendre polynomials that satisfy the boundary conditions for a closed spheroidal shell, the system is solved using the Galerkin method. Numerical results of the frequency spectra are presented for various ratios of h1/h2 and various material properties for the two layers. Initial results are presented for a=100 (a nearly spherical shape). [Work supported by the ONR/ASEE Summer Faculty Research Program.]

Algebraic approach to electronic spectroscopy and dynamics

Lie algebra, Zassenhaus, and parameter differentiation techniques are utilized to break up the exponential of a bilinear Hamiltonian operator into a product of noncommuting exponential operators by the virtue of the theory of Wei and Norman [J. Math. Phys. 4, 575 (1963); Proc. Am. Math. Soc., 15, 327 (1964)]. There are about three different ways to find the Zassenhaus exponents, namely, binomial expansion, Suzuki formula, and q-exponential transformation. A fourth, and most reliable method, is provided. Since linearly displaced and distorted (curvature change upon excitation/emission) Hamiltonian and spin-boson Hamiltonian may be classified as bilinear Hamiltonians, the presented algebraic algorithm (exponential operator disentanglement exploiting six-dimensional Lie algebra case) should be useful in spin-boson problems. The linearly displaced and distorted Hamiltonian exponential is only treated here. While the spin-boson model is used here only as a demonstration of the idea, the herein approach is more general and powerful than the specific example treated. The optical linear dipole moment correlation function is algebraically derived using the above mentioned methods and coherent states. Coherent states are eigenvectors of the bosonic lowering operator a and not of the raising operator a(+). While exp(a(+)) translates coherent states, exp(a(+)a(+)) operation on coherent states has always been a challenge, as a(+) has no eigenvectors. Three approaches, and the results, of that operation are provided. Linear absorption spectra are derived, calculated, and discussed. The linear dipole moment correlation function for the pure quadratic coupling case is expressed in terms of Legendre polynomials to better show the even vibronic transitions in the absorption spectrum. Comparison of the present line shapes to those calculated by other methods is provided. Franck-Condon factors for both linear and quadratic couplings are exactly accounted for by the herein calculated linear absorption spectra. This new methodology should easily pave the way to calculating the four-point correlation function, F(tau(1),tau(2),tau(3),tau(4)), of which the optical nonlinear response function may be procured, as evaluating F(tau(1),tau(2),tau(3),tau(4)) is only evaluating the optical linear dipole moment correlation function iteratively over different time intervals, which should allow calculating various optical nonlinear temporal/spectral signals.

Determination of the orientation distribution function in aligned single wall nanotube polymer nanocomposites by polarized Raman spectroscopy

This work focuses on the derivation of the orientation distribution function (ODF) for a uniaxial-axially symmetric system using polarized Raman spectroscopy. A numerical methodology is proposed to determine the ODF that is formulated in terms of Legendre polynomials and the principle of maximum information entropy. The ultimate goal is to quantify the alignment of single wall nanotubes (SWNTs) in a polymer matrix using the experimental information from the Raman intensity. Some of the mathematical and numerical steps in the determination of ODF, not clarified in the current literature, are shown in this work. The proposed numerical methodology to obtain the ODF is illustrated using an experimental case. Electric field–aligned SWNT-urethane dimethacrylate/1,6-hexanediol dimethacrylate nanocomposites are investigated at different processing conditions to bring forward factors that may enhance the alignment of SWNT inclusions in the polymer. ODF results indicate that the higher electric field frequencies produce a good alignment of the SWNT inclusions; a result also corroborated by optical microscopy imaging and electrical conductivity measurements.

Surface electromagnetic waves supported by nano conducting layers with inhomogeneities in the conductivity profile

Conducting interfaces and nano conducting layers can support surface electromagnetic waves. Uniform charge layers of non-zero thickness and their asymptotic behavior toward conducting interfaces of infinitely small thicknesses, where the thin charge layer is modeled via a surface conductivity σs, are already studied. Here, the possible effects of inhomogeneity in the conductivity profile of the thin conducting layers are investigated for the first time and a new approximate yet accurate enough analytical formulation for mode extraction in such structures is given. In order to rigorously analyze the structure and justify the proposed approximate formulation, the Galerkin’s method with Legendre polynomial basis functions is applied, i.e. the transverse electric field for the TE polarized surface waves and the transverse magnetic field for the TM polarized surface waves are each expanded in terms of Legendre polynomials and then each eigenmode; subjected to appropriate boundary conditions, is sought in the complete space spanned by Legendre basis functions. The proposed approximate solution is then proved to be accurate. In particular, sinusoidal fluctuations are introduced into formerly uniform conductivity profiles and it is numerically demonstrated that surface electromagnetic waves supported by nano conducting layers are not much sensitive to the very shape of conductivity profiles.

Three-dimensional diffraction analysis of gratings based on Legendre expansion of electromagnetic fields

Three-dimensional vectorial diffraction analysis of gratings is presented based on Legendre polynomial expansion of electromagnetic fields. In contrast to conventional rigorous coupled wave analysis (RCWA) in which the solution is obtained using state variables representation of the coupled wave amplitudes, here the solution of first-order coupled Maxwell's equations is expanded in terms of Legendre polynomials, where Maxwell's equations are analytically projected in the Hilbert space spanned by Legendre polynomials. This approach yields well-behaved algebraic equations for deriving diffraction efficiencies and electromagnetic field profiles without facing the problem of numerical instability. The proposed approach can be applied in the analysis of two cases: first, arbitrarily oriented planar gratings with slanted yet homogeneous fringes; second, nonslanted but longitudinally inhomogeneous gratings. The method is then applied to various test cases within the above-mentioned two categories, comparison to other methods already reported in the literature is made, and the presented approach is justified. Different aspects of the proposed method such as numerical stability and convergence rate are also investigated. Special attention is given to how the resonant frequency of frequency selective structures varies with introducing tilt angles and/or longitudinal inhomogenity in the permittivity profile.

The effects of different expansions of the exit distribution on the extrapolation length for linearly anisotropic scattering

Abstract HN and singular eigenfunction methods are used to determine the neutron distribution everywhere in a source-free half space with zero incident flux for a linearly anisotropic scattering kernel. The singular eigenfunction expansion of the method of elementary solutions is used. The orthogonality relations of the discrete and continuous eigenfunctions for linearly anisotropic scattering provides the determination of the expansion coefficients. Different expansions of the exit distribution are used: the expansion in powers of μ, the expansion in terms of Legendre polynomials and the expansion in powers of 1/(1 + μ). The results are compared to each other. In the second part of our work, the transport equation and the infinite medium Green function are used. The numerical results of the extrapolation length obtained for the different expansions is discussed.

Collisional–radiative model of neon discharge: determination of E/N in the positive column of low pressure discharge

A method for the determination of the reduced electric field strength (E/N) in a neon discharge from the optical emission spectra was developed. This method is based on a collisional–radiative model, which was used to calculate the emission spectra of the neon plasma. In the model, populations of 30 excited levels of the neon atom were studied. Various elementary collision processes were taken into account: electron impact excitation, de-excitation and ionization of neon atoms, emission and absorption of radiation, metastable–metastable collisions, metastable and radiative dimer production, Penning ionization, etc. To determine the rates of electron collisions, the Boltzmann kinetic equation for the electron distribution function (EDF) was solved for given E/N, and thus the dependence of the emission spectra on E/N could be determined. The EDF was expanded in terms of Legendre polynomials and the first two terms of this approximation were taken into account. The theoretical emission spectra were fitted by the non-linear least-squares method to the measured spectra with respect to the unknown parameter E/N.The method was applied to the study of low pressure dc glow discharge in neon. In such a discharge the calculated reduced electric field strength could be compared with the independent results of simultaneously performed electric probe measurements of E/N. Generally, close agreement of the calculated values with the experimental data was achieved.

Planar Diffraction Analysis of Homogeneous and Longitudinally Inhomogeneous Gratings Based on Legendre Expansion of Electromagnetic Fields

Planar grating diffraction analysis based on Legendre expansion of electromagnetic fields is reported. In contrast to conventional RCWA in which the solution is obtained using state variables representation of the coupled wave amplitudes; here, the solution is expanded in terms of Legendre polynomials. This approach, without facing the problem of numerical instability and inevitable round off errors, yields well-behaved algebraic equations for deriving diffraction efficiencies, and can be employed for analysis of different types of gratings. Thanks to the recursive properties of Legendre polynomials, for longitudinally inhomogeneous gratings, wherein differential equations with non-constant coefficients are encountered, it can also be used to analyze the whole structure at one stroke. Although this is the only case for which the presented approach is efficient from both aspects of stability and computation load, the presented approach is applied to different test cases, and justified by comparison of the results to those obtained using previously reported methods. The method is general, and can handle many different cases like thick gratings, non-Bragg incidence, and cases in which higher diffracted orders or evanescent orders corresponding to real eigenvalues, have to be included in the solution of the Maxwell's equations. In deriving the formulation, a rigorous approach is followed

Validation of first-order diffraction theory for the traveltimes and amplitudes of propagating waves

Ultrasonic measurements of acoustic wavefields scattered by single spheres placed in a homogenous background medium (water) are presented. The dimensions of the spheres are comparable to the wavelength and the wavelength and represent both positive (rubber) and negative (teflon) velocity anomalies with respect to the background medium. The sensitivity of the recorded wavefield to scattering in terms of traveltime delay and amplitude variation is investigated. The results validate a linear (first-order) diffraction theory for wavefields propagating in heterogeneous media with anomaly contrasts on the order of [Formula: see text]. The diffraction theory is compared further with the exact results known from literature for scattering from an elastic sphere, formulated in terms of Legendre polynomials. To investigate the 2D case, the first-order scattering theory is tested against 2D elastic finite-difference calculations. As the presented theory involves a volume integral, it is applicable to any geometric shape, and the scattering object does not need to be spherical or any other specific symmetrical shape. Furthermore, it can be implemented easily in seismic data inversion schemes, which is illustrated with examples from seismic crosswell tomography and a reflection experiment.

Conduction through an assembly of spherical particles at low liquid contents

The steady-state conduction of heat through the solid phase of a porous material consisting of an assembly of simple cubic packed uniform spheres at low liquid contents with heat conducted from one sphere to the next through annular liquid lenses around the points of contact, is analysed in terms of Legendre polynomials assuming the liquid bridges to behave as isothermal hemispheres around point sources and sinks. The conductance of a sphere is approximately the same when the surface area of the hemisphere is the same as that of the liquid lens assumed to be an isothermal disc for which numerical results of isotherms are compared with the analytical results.

A matrix method for solving high-order linear difference equations with mixed argument using hybrid legendre and taylor polynomials

A numerical method for solving the higher order linear difference equations with variable coefficients and mixed argument under the mixed conditions is presented. The method is based on the hybrid Legendre and Taylor polynomials. The solution is obtained in terms of Legendre polynomials. IIIustrative examples are included to demonstrate the validity and applicability of the technique.

Approximation properties of mixed series in terms of Legendre polynomials on the classes

The study of the approximation properties on the class of partial sums of a mixed series in terms of Legendre polynomials is continued.

Parametrization of theH2(γ,p)nreaction between 185 and 420 MeV

A simple parametrization of the $^{2}\mathrm{H}$$(\ensuremath{\gamma},p)n$ cross-section data from LEGS, Mainz, and Bonn at laboratory photon energies between 187.5 and 420 MeV and the LEGS and Mainz beam asymmetry data between 185 and 412 MeV has been done. The fit represents both the energy and angular dependence of the cross section and asymmetry in terms of Legendre polynomials. A reduced ${\ensuremath{\chi}}^{2}$ of 1.641 is obtained for the cross-section data with a total of 20 parameters. Three of the parameters are renormalization factors introduced to permit the three cross-section data sets to float relative to one another. The resulting renormalization factors are found to be well within the quoted systematic uncertainties for each experiment. The energy dependence of the parameters is in good agreement with fits done to the Mainz data at individual energies. The asymmetries are fit with an additional eight parameters with a reduced ${\ensuremath{\chi}}^{2}$ of 1.229.

Diffuse reflectance of TiO2 pigmented paints: Spectral dependence of the average pathlength parameter and the forward scattering ratio

Diffuse reflectance spectra of paint coatings with different pigment concentrations, normally illuminated with unpolarized radiation, have been measured. A four-flux radiative transfer approach is used to model the diffuse reflectance of TiO2 (rutile) pigmented coatings through the solar spectral range. The spectral dependence of the average pathlength parameter and of the forward scattering ratio for diffuse radiation, are explicitly incorporated into this four-flux model from two novel approximations. The size distribution of the pigments has been taken into account to obtain the averages of the four-flux parameters: scattering and absorption cross sections, forward scattering ratios for collimated and isotropic diffuse radiation, and coefficients involved in the expansion of the single particle phase function in terms of Legendre polynomials.

Particle−Interface Interaction across a Nonpolar Medium in Relation to the Production of Particle-Stabilized Emulsions

Quantitative theory of the particle-interface interaction across a nonpolar medium is developed. We consider a spherical dielectric particle (phase 1), which is immersed in a nonpolar medium (phase 2), near its boundary with a third dielectric medium (phase 3). The interaction originates from electric charges at the particle surface (e.g., the surface of a silica particle immersed in oil). The theoretical problem is solved exactly, in terms of Legendre polynomials, for arbitrary values of the dielectric constants of the three phases. As a result, expressions for calculating the interaction force and energy are derived. These expressions generalize the known theory of the electrostatic image force (acing on point charges) to the case of particles that have finite size and uniform surface charge density. For typical parameter values (silica or glass particles immersed in tetradecane), the image-force interaction becomes significant for particles of radius R > 30 nm. At fixed relative particle-to-interface distance, the force increases with the cube of the particle radius. In general, this is a strong and long-range interaction. For micrometer-sized particles, the interaction energy could be on the order of 10(5) k(B)T at close contact, and, in addition, the interaction range could be about 10(5) particle radii. The sign of the interaction depends on the difference between the dielectric constants of phases 2 and 3. When phase 3 has a smaller dielectric constant (e.g., air), the interface repels the particle. In contrast, when phase 3 has a greater dielectric constant (e.g., water), the interaction is attractive. Especially, water drops attract charged hydrophobic particles dispersed in the oily phase, and thus favor the formation of reverse particle-stabilized (Pickering) emulsions. The particle-interface interaction across the oily phase is insensitive to the concentration of electrolyte in the third, aqueous phase.

Superconformal Ward identities and their solution

Superconformal Ward identities are derived for the four point functions of chiral primary BPS operators for N = 2 , 4 superconformal symmetry in four dimensions. Manipulations of arbitrary tensorial fields are simplified by introducing a null vector so that the four point functions depend on two internal R-symmetry invariants as well as two conformal invariants. The solutions of these identities are interpreted in terms of the operator product expansion and are shown to accommodate long supermultiplets with free scale dimensions and also short and semi-short multiplets with protected dimensions. The decomposition into R-symmetry representations is achieved by an expansion in terms of two variable harmonic polynomials which can be expressed also in terms of Legendre polynomials. Crossing symmetry conditions on the four point functions are also discussed.

Rotationally inelastic collisions of LiH (X 1Σ +) with H: State-to-state inelastic rotational cross-section

The potential energy surface describing the interaction between the hydrogen atom and the rotating LiH molecule has been calculated very accurately and presented in our previous work (J. Mol. Struct. (THEOCHEM) 678 (2004) 11). The rigid rotor potential already fitted analytically and expanded in terms of Legendre polynomials, is employed here to evaluate the pure state-to-state rotational cross-sections over a range of energies of astrophysical interest varying from 10 to 3000 cm −1. An exact close coupling and CS calculation of the state to state rotational cross-section was done first in the range of small energy, from 10 to 100 cm −1. The comparison of both results shows the good agreement between them and leads to a generalisation of such calculation for higher energy using the CS approximation. We determine the state to state rotational cross-section for transitions from j=0, 1, 2, 3, 4 to all accessible final states j′, at total energy varying between 10 and 3000 cm −1. The presented cross-sections were found to be very large even for low energy and large Δ jj′ . Thus indicates the important probability to produce the heated LiH molecule in excited states by collisions with H and explain the strong anisotropy discussed in our previous paper.

Nonlinear response of a dipolar system with rotational diffusion to an oscillatingfield

The rotational diffusion equation for a dipole in the presence of an oscillating fieldis solved by expansion of the orientational distribution function in terms ofLegendre polynomials and harmonics. The nonlinear response of the averagedipole moment is studied as a function of field strength and frequency. Outsidethe linear regime the in-phase and out-of-phase response as functions of frequencydo not satisfy Kramers–Kronig relations. A comparison is made with thenonlinear response calculated from approximate macroscopic relaxation equationsproposed by Shliomis and by Martsenyuk et al. The response of a macroscopicsystem of interacting dipoles is calculated in the mean-field approximation for aspherical sample.

A Spherical Inclusion with Inhomogeneous Interface in Conduction

ABSTRACTA series solution is presented for a spherical inclusion embedded in an infinite matrix under a remotely applied uniform intensity. Particularly, the interface between the inclusion and the matrix is considered to be inhomegeneously bonded. We examine the axisymmetric case in which the interface parameter varies with the cone angle θ. Two kinds of imperfect interfaces are considered: an imperfect interface which models a thin interphase of low conductivity and an imperfect interface which models a thin interphase of high conductivity. We show that, by expanding the solutions of terms of Legendre polynomials, the field solution is governed by a linear set of algebraic equations with an infinite number of unknowns. The key step of the formulation relies on algebraic identities between coefficients of products of Legendre series. Some numerical illustrations are presented to show the correctness of the presented procedures. Further, solutions of the boundary-value problem are employed to estimate the effective conductivity tensor of a composite consisting of dispersions of spherical inclusions with equal size. The effective conductivity solely depends on one particular constant among an infinite number of unknowns.