Showing papers on "Legendre polynomials published in 2004"
TL;DR: The pBasex algorithm as mentioned in this paper reconstructs the original Newton sphere of expanding charged particles from its two-dimensional projection by fitting a set of basis functions with a known inverse Abel integral, adapted to the polar symmetry of the photoionization process to optimize the energy and angular resolution.
Abstract: We present an inversion method called pBasex aimed at reconstructing the original Newton sphere of expanding charged particles from its two-dimensional projection by fitting a set of basis functions with a known inverse Abel integral. The basis functions have been adapted to the polar symmetry of the photoionization process to optimize the energy and angular resolution while minimizing the CPU time and the response to the cartesian noise that could be given by the detection system. The method presented here only applies to systems with a unique axis of symmetry although it can be adapted to overcome this restriction. It has been tested on both simulated and experimental noisy images and compared to the Fourier-Hankel algorithm and the original Cartesian basis set used by [Dribinski et al.Rev. Sci. Instrum. 73, 2634 (2002)], and appears to give a better performance where odd Legendre polynomials are involved, while in the images where only even terms are present the method has been shown to be faster and simpler without compromising its accuracy.
602 citations
TL;DR: In this article, an uncertainty quantification scheme based on generalized polynomial chaos (PC) representations is constructed, which is applied to a model problem involving a simplified dynamical system and to the classical problem of Rayleigh-Benard instability.
463 citations
TL;DR: A new hierarchical basis of arbitrary order for integral equations solved with the method of moments derived from orthogonal Legendre polynomials which are modified to impose continuity of vector quantities between neighboring elements while maintaining most of their desirable features is presented.
Abstract: This paper presents a new hierarchical basis of arbitrary order for integral equations solved with the method of moments (MoM). The basis is derived from orthogonal Legendre polynomials which are modified to impose continuity of vector quantities between neighboring elements while maintaining most of their desirable features. Expressions are presented for wire, surface, and volume elements but emphasis is given to the surface elements. In this case, the new hierarchical basis leads to a near-orthogonal expansion of the unknown surface current and implicitly an orthogonal expansion of the surface charge. In addition, all higher order terms in the expansion have two vanishing moments. In contrast to existing formulations, these properties allow the use of very high-order basis functions without introducing ill-conditioning of the resulting MoM matrix. Numerical results confirm that the condition number of the MoM matrix obtained with this new basis is much lower than existing higher order interpolatory and hierarchical basis functions. As a consequence of the excellent condition numbers, we demonstrate that even very high-order MoM systems, e.g., tenth order, can be solved efficiently with an iterative solver in relatively few iterations.
247 citations
TL;DR: This paper introduces a new set of translation and scale invariants of Legendre moments based on Legendre polynomials, which remain unchanged for translated, elongated, contracted and reflected non-symmetrical as well as symmetrical images.
172 citations
Journal Article•
TL;DR: In this article, a version of the calculus of variations on time scales is introduced, which includes as special cases the classical calculus of variation and the discrete calculus of the variations, and necessary and sufficient conditions for weak local minima are established.
Abstract: We introduce a version of the calculus of variations on time scales, which includes as special cases the classical calculus of variations and the discrete calculus of variations. Necessary conditions for weak local minima are established, among them the Euler condition, the Legendre condition, the strengthened Legendre condition, and the Jacobi condition. AMS (MOS) Subject Classication. 39A10.
161 citations
TL;DR: A method for finding the optimal control of a linear time varying delay system with quadratic performance index with block-pulse functions plus Legendre polynomials is discussed.
Abstract: A method for finding the optimal control of a linear time varying delay system with quadratic performance index is discussed. The properties of the hybrid functions which consists of block-pulse functions plus Legendre polynomials are presented. The operational matrices of integration, delay and product are utilized to reduce the solution of optimal control to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.
125 citations
TL;DR: The Slepian series as discussed by the authors is an orthogonal expansion of the spheroidal wave function (PSWF) that is potentially optimal for discontinuous functions such as the square wave among others.
118 citations
TL;DR: In this paper, a numerical technique for solving some physical problems on a semi-infinite interval is presented, which is based on a rational Legendre tau method and the operational matrices of derivative and product of rational linear Legendre functions are used to reduce the solution of these physical problems to the solutions of systems of algebraic equations.
Abstract: A numerical technique for solving some physical problems on a semi-infinite interval is presented. Two nonlinear examples are proposed. In the first example the Volterra's population model growth is formulated as a nonlinear differential equation, and in the second example the Lane–Emden nonlinear differential equation is considered. The approach is based on a rational Legendre tau method. The operational matrices of derivative and product of rational Legendre functions are presented. These matrices together with the tau method are utilized to reduce the solution of these physical problems to the solution of systems of algebraic equations. The method is easy to implement and yields very accurate results.
117 citations
TL;DR: It is shown that when used as the spatial discretization for time-dependent partial differential equations in combination with explicit time-marching, prolate functions allow a longer stable timestep than Legendre polynomials and that it is almost trivial to modify existing pseudospectral and spectral element codes to use the prolate basis.
103 citations
TL;DR: In this article, it was shown that the redshift-space two-point correlation function can be expanded into tripolar spherical harmonics of zero total angular momentum S(1,2, ) for wide-angle surveys from the point of view of symmetries.
Abstract: We explore linear redshift distortions in wide-angle surveys from the point of view of symmetries. We show that the redshift-space two-point correlation function can be expanded into tripolar spherical harmonics of zero total angular momentum S(1,2,). The coefficients of the expansion B are analogous to the Cl of the angular power spectrum and express the anisotropy of the redshift-space correlation function. Moreover, only a handful of B are nonzero: the resulting formulae reveal a hidden simplicity comparable to the distant observer limit. The B depend on spherical Bessel moments of the power spectrum and f = Ω0.6/b. In the plane-parallel limit, the results of Kaiser and Hamilton are recovered. The general formalism is used to derive useful new expressions. We present a particularly simple trigonometric polynomial expansion, which is arguably the most compact expression of wide-angle redshift distortions. These formulae are suitable for inversion because of the orthogonality of the basis functions. An alternative Legendre polynomial expansion was obtained as well. This can be shown to be equivalent to the results of Szalay and coworkers. The simplicity of the underlying theory will admit similar calculations for higher order statistics as well.
102 citations
TL;DR: The hybrid Legendre and Block-Pulse functions on interval [0,1) are used to solve the linear integro-differential equation system, and the quadrature formulae for the calculation of inner products of any functions are constructed.
TL;DR: In this article, the authors compare three statistical ODFs that define the alignment by spreading from a mean value; in particular, the Gaussian, Fisher and Bingham distributions, with an ODF resulting from pure vertical compaction of a sediment.
Abstract: The elastic properties and anisotropy of shales are strongly influenced by the degree of alignment of the grain scale texture. In general, an orientation distribution function (ODF) can be used to describe this alignment, which, in practice, can be characterized by two Legendre coefficients. We discuss various statistical ODFs that define the alignment by spreading from a mean value; in particular, the Gaussian, Fisher and Bingham distributions. We compare the statistical models with an ODF resulting from pure vertical compaction (no shear strain) of a sediment. The compaction ODF may be used to estimate how the elastic properties and anisotropy evolve due to burial of clayey sediments.
Our study shows that the three statistical ODFs produce almost identical correspondence between the two Legendre coefficients as a function of the spreading parameter, so that the spreading parameter of one ODF can be converted to the spreading parameter of another ODF. In most cases it is then sufficient to apply the spreading parameter for the ODF instead of the two Legendre coefficients. The effect of compaction on the ODF gives a slightly different correspondence between the two Legendre coefficients from that for the other models. In principle, this opens up the possibility of distinguishing anisotropy effects due to compaction from those due to other processes.
We also study reflection amplitudes versus angle of incidence (AVA) for all wave modes, where shales having various ODFs overlie an isotropic medium. The AVA responses are modelled using both exact and approximation formulae, and their intercepts and gradients are compared. The modelling shows that the S-wave velocity is sensitive to any perturbation in the spreading parameter, while the P-wave velocity becomes increasingly sensitive to a perturbation of a less ordered system. Similar observations are found for the AVA of the P-P and P-SV waves. Modelling indicates that a combined use of the amplitude versus offset of P-P and P-SV reflected waves may reveal certain grain scale alignment properties of shale-like rocks.
TL;DR: In this article, the main purpose of the present paper is to build a Hamiltonian theory for fields which is consistent with the principles of relativity, and the main point is to stress out the interplay between the Lepage-Dedecker (LP) description and the De DonderWeyl (DDW) one.
Abstract: The main purpose in the present paper is to build a Hamiltonian theory for fields which is consistent with the principles of relativity. For this we consider detailed geometric pictures of Lepage theories in the spirit of Dedecker and try to stress out the interplay between the Lepage-Dedecker (LP) description and the (more usual) De DonderWeyl (DDW) one. One of the main points is the fact that the Legendre transform in the DDW approach is replaced by a Legendre correspondence in the LP theory (this correspondence behaves differently: ignoring the singularities whenever the Lagrangian is degenerate).
TL;DR: In this article, an alternative method for calculating the magnetic field from a set of permanent magnets in a permanent-magnet motor was presented, which uses a cylindrical coordinate system to model the geometry of the structure enclosing the magnets.
Abstract: We present an alternative method for calculating the magnetic field from a set of permanent magnets in a permanent-magnet motor. The method uses a cylindrical coordinate system to model the geometry of the structure enclosing the magnets. A Fourier series expansion yields an alternative to the more familiar multipole expansion given in spherical coordinates. The expansion is developed by using Green's function in cylindrical coordinates. A technique called charge simulation allows computation of an equivalent point charge distribution. Finally, Coulomb's law is applied to express the magnetic scalar potential in a mathematically tractable form.
TL;DR: In this article, a theory of forward glory scattering is developed for a state-to-state chemical reaction whose scattering amplitude can be expanded in a Legendre partial wave series and two transitional approximations are derived that are valid for angles on, and close to, the axial caustic associated with the glory.
Abstract: The theory of forward glory scattering is developed for a state-to-state chemical reaction whose scattering amplitude can be expanded in a Legendre partial wave series. Two transitional approximations are derived that are valid for angles on, and close to, the axial caustic associated with the glory. These are the integral transitional approximation (ITA) and the semiclassical transitional approximation (STA), which is obtained when the stationary phase method is applied to the ITA. Both the ITA and STA predict that the scattering amplitude for glory scattering is proportional to a Legendre function of real degree or, to a very good approximation, a Bessel function of order zero. A primitive semiclassical approximation (PSA) is also derived that is valid at larger angles, away from the caustic direction, but which is singular on the caustic. The PSA demonstrates that glory structure arises from nearside–farside (NF) interference, in an analogous way to the two-slit experiment. The main result of the paper is a uniform semiclassical approximation (USA) that correctly interpolates between small angles, where the ITA and STA are valid, and larger angles where the PSA is valid. The USA expresses the scattering amplitude in terms of Bessel functions of order zero and unity, together with N and F cross sections and phases. In addition, various subsidiary approximations are derived. The input to the theory consists of accurate quantum scattering matrix elements. The theory also has the important attribute that it provides physical insight by bringing out semiclassical and NF aspects of the scattering. The theory is used to show that the enhanced small angle scattering in the F + H2(vi = 0, ji = 0, mi = 0) → FH(vf = 3, jf = 3, mf = 0) + H reaction is a forward glory, where vi, ji, mi and vf, jf, mf are initial and final vibrational, rotational and helicity quantum numbers respectively. The forward angle scattering for the H + D2(vi = 0, ji = 0, mi = 0) → HD(vf = 3, jf = 0, mf = 0) + D reaction is also analysed and shown to be a forward glory, in agreement with a simpler treatment by D. Sokolovski (Chem. Phys. Lett., 2003, 370, 805), which is a special case of the STA.
TL;DR: A method for finding the solution of time-delay systems using a hybrid function which consists of block-pulse functions plus Legendre polynomials and the operational matrices of product and delay are introduced.
Posted Content•
TL;DR: The main purpose of as mentioned in this paper is to build a Hamiltonian theory for fields which is consistent with the principles of relativity, in which the Legendre transform in the dDW approach is replaced by a Legendre correspondence in the Lepage-Dedecker (LP) theory.
Abstract: The main purpose in the present paper is to build a Hamiltonian theory for fields which is consistent with the principles of relativity. For this we consider detailed geometric pictures of Lepage theories in the spirit of Dedecker and try to stress out the interplay between the Lepage-Dedecker (LP) description and the (more usual) de Donder-Weyl (dDW) one. One of the main points is the fact that the Legendre transform in the dDW approach is replaced by a Legendre correspondence in the LP theory (This correspondence behaves differently: ignoring the singularities whenever the Lagrangian is degenerate).
TL;DR: In this paper, a new mathematical model and code for radiative heat transfer of particulate media with anisotropic scattering for 2-D rectangular enclosure is developed, based on the coupling of finite volume method for the solution of radiative transfer equation with Mie equations for the evaluation of scattering phase function.
TL;DR: In this paper, the authors present results for 2D models of rapidly rotating main sequence stars for the case where the an-gular velocity Ω is constant throughout the star.
Abstract: We present results for 2-dimensional models of rapidly rotating main sequence stars for the case where the an- gular velocity Ω is constant throughout the star. The algorithm used solves for the structure on equipotential surfaces and iteratively updates the total potential, solving Poisson's equation by Legendre polynomial decomposition; the algorithm can readily be extended to include rotation constant on cylinders. We show that this only requires a small number of Legendre polynomials to accurately represent the solution. We present results for models of homogeneous zero age main sequence stars of mass 1, 2, 5, 10 Mwith a range of angular velocities up to break up. The models have a composition X = 0.70, Z = 0.02 and were computed using the OPAL equation of state and OPAL/Alexander opacities, and a mixing length model of convection modified to include the effect of rotation. The models all show a decrease in luminosity L and polar radius Rp with increasing angular velocity, the magnitude of the decrease varying with mass but of the order of a few percent for rapid rotation, and an increase in equatorial radius Re. Due to the contribution of the gravitational multipole moments the parameter Ω 2 R 3/GM can exceed unity in very rapidly rotating stars and Re/Rp can exceed 1.5.
TL;DR: In this article, an annular sector solid hierarchical finite element is presented and applied to three-dimensional free vibration analysis of annular sectors, where the element's displacements are expressed in terms of a fixed number of linear polynomial shape functions plus a variable number of shape functions which are forms of shifted Legendre orthogonal polynomials.
TL;DR: In this article, the complex process of three-dimensional flow interaction around a square jet in cross-flow has been studied, and the simulation was performed for a moderate value of the Reynolds number ( Re = 225), and for a jet-to-free-stream velocity ratio of 2.5.
Abstract: The present computational study is devoted to unfolding the complex process of three–dimensional flow interaction around a square jet in cross–flow. The aim is to provide a clear understanding about the structural development of the entire vortical flow field, which may immensely enhance our knowledge regarding mutual interaction among various vortical structures that takes place around the jet. Careful attempts have been made to capture the detailed mechanism of formation of the near–field horseshoe–vortex system and the roll–up process of the hovering vortices. The rolled–up shear–layer hovering vortices, which wrap around the front and the lateral jet–cross–flow interface, are observed to initiate the Kelvin–Helmholtz–like instability. The present study also clearly displays the inception process of the counter–rotating vortex pair (CVP) from the shear layers that develop on the two lateral side walls of the jet pipe. In order to better understand the complete flow–interaction process and the governing flow physics, the simulation was performed for a moderate value of the Reynolds number ( Re = 225), and for a jet–to–free–stream velocity ratio of 2.5. The interaction process between the streamwise wall vortices and the developed upright (or spin–off, or zipper) vortices in the downstream boundary layer is observed to contribute substantially in the structural development of the jet wake. The upright vortices were seen to originate from the tornado–like critical points on the channel floor shear layer, and subsequently the vortices lift themselves away from the channel floor to merge ultimately with the evolving CVP. Importantly, such merging processes are observed to locally enhance the CVP strength. Following the topological theory of Legendre, the depicted map of computed critical points and the separation lines helps to provide additional insight into the flow mechanism. The computed results clearly demonstrate the entire vortical flow–interaction process to its totality, including all the recent experimental predictions that are made for such flows. Notably, as it was experimentally verified for round jets in cross–flow, in the present configuration too, the flow separation on the channel floor is found to be the basic source of inception of the wall and the upright vortices. The separated flow in the vicinity of different wall vortical corelines joins to form the upright vortices.
TL;DR: The New-Tau method is presented, using a modified form of the Chebyshev or Legendre and Gegenbauer polynomials, to give a factorization of the operators arising from the application of the Tau method.
TL;DR: In this paper, a comparison between results for the electron velocity distribution function (evdf), and transport and rate coefficients of an electron swarm obtained under different assumptions for the space and angular dependence of the evdf was made in neon at a constant and homogeneous reduced electric field in the range 10 Td ≤ E/N ≤ 500 Td taking into account the production of electrons in ionizing collisions.
Abstract: We present a comparison between results for the electron velocity distribution function (evdf), and transport and rate coefficients of an electron swarm obtained under different assumptions for the space and angular dependence of the evdf. Several solution techniques for the Boltzmann equation as well as Monte Carlo simulations have been tested. The comparison is made in neon at a constant and homogeneous reduced electric field in the range 10 Td ≤ E/N ≤ 500 Td taking into account the production of electrons in ionizing collisions. The results show that to obtain an accurate description of the electron swarm we need to take into account the variation in space of the electron density in the representation of the evdf. In what regards the angular dependence on velocity we discuss criteria to estimate the importance of the anisotropy of the evdf for any gas. Depending on the solution technique and on the E/N value, we find good to excellent agreement between the Boltzmann results obtained with a half-range method, a multi-term Legendre expansion, an elliptic approximation and the Monte Carlo results. The accuracy of the transport and rate coefficients obtained with each approach is evaluated and it is found that although the two-term velocity expansion is not sufficiently accurate to be used for cross section fitting, the corresponding rate and transport coefficients can generally be used in discharge modelling.
TL;DR: This study suggests a new approach to the high-order DLTs achieved by substitution of a simple alternative factorization of the associated Legendre functions in place of that used previously, which improves stability significantly for a wide range of useful problem sizes.
Abstract: This paper proposes and investigates a method for reducing potential numerical instability in the fast spherical harmonic transform algorithms proposed previously in [5,11]. The key objective of this study is a numerically reliable fast algorithm for computing the discrete Legendre transform (DLT); that is, the projection of sampled data onto the associated Legendre functions within a specified range of degrees. A simple divide-and-conquer approach derives from a factorization of high-degree Legendre functions into Legendre functions of lower-degree, exploiting the fact that the complexity of projection onto Legendre functions decreases with decreasing degree. Combining the resulting fast DLT algorithms for each relevant order of associated Legendre function results in an O(Nlog 2
N) algorithm for computing the spherical harmonic expansion of a function sampled at N points on the sphere. While fast DLT algorithms of this form are exact in exact arithmetic, actual (finite precision) implementations of the earlier variants display instabilities which generally grow with the the order of the associated Legendre function in the transform. Here we return to the basic algorithm, present a slight modification of the general schema and examine the error mechanism for the higher-order cases. This study suggests a new approach to the high-order DLTs achieved by substitution of a simple alternative factorization of the associated Legendre functions in place of that used previously. This technique improves stability significantly for a wide range of useful problem sizes and may be used with any of the variants of the basic algorithm previously proposed. We present a description of the use of the new Legendre decomposition in the basic fast algorithm along with numerical experiments demonstrating the large advances in stability and efficiency of the new approach over the previous results.
TL;DR: The continuous Legendre wavelets on the interval [0, 1) constructed by M. Razzaghi and S. Yousefi are used to solve the linear second kind integro-differential equations and construct the quadrature formulae for the calculation of inner products of any functions, which are required in the approximation for the Integro- differential equations.
Abstract: In this article, we use the continuous Legendre wavelets on the interval [0, 1) constructed by [M. Razzaghi and S. Yousefi, The Legendre wavelets operational matrix of integration, International Journal of Systems Science, 32(4) (2001) 495–502.] to solve the linear second kind integro-differential equations and construct the quadrature formulae for the calculation of inner products of any functions, which are required in the approximation for the integro-differential equations. Then we reduce the integro-differential equation to the solution of linear algebraic equations. E-mail: ahadi@khuisf.ac.ir
TL;DR: In this paper, an efficient higher-order method of moments (MoM) solution of volume integral equations is presented, which is based on higher order hierarchical Legendre basis functions and higher order geometry modeling.
Abstract: [1] An efficient higher-order method of moments (MoM) solution of volume integral equations is presented. The higher-order MoM solution is based on higher-order hierarchical Legendre basis functions and higher-order geometry modeling. An unstructured mesh composed of 8-node trilinear and/or curved 27-node hexahedral elements is used for accurate representation of the scattering dielectric object. The permittivity of the object is allowed to vary continuously as a function of position inside each element. It is shown that the condition number of the resulting MoM matrix is reduced by several orders of magnitude in comparison to existing higher-order hierarchical basis functions. Consequently, an iterative solver can be applied even for high expansion orders. Numerical results demonstrate excellent agreement with the analytical Mie series solution for a dielectric sphere as well as with results obtained by other numerical methods.
TL;DR: In this paper, a local angular momentum (LAM) and a local impact parameter (LIP) are defined for a state-to-state reaction, whose scattering amplitude can be expanded in a Legendre partial wave series.
Abstract: A local angular momentum (LAM) and a local impact parameter (LIP) is defined for a state-to-state reaction, whose scattering amplitude can be expanded in a Legendre partial wave series. A nearside–farside decomposition is incorporated into the LAM-LIP theory, which is also extended to resummed partial wave series. The input to a LAM-LIP analysis consists of a set of accurate, or approximate, quantum scattering matrix elements. Application is made to the benchmark F + H2(vi = 0, ji = 0, mi = 0) → FH(vf = 3, jf = 3, mf = 0) + H reaction, where vi, ji, mi and vf, jf, mf are initial and final vibrational, rotational and helicity quantum numbers respectively. It is demonstrated that a LAM-LIP analysis of structure in the angular scattering provides valuable information on the dynamics of chemical reactions.
TL;DR: Bounds on ratios of consecutive zeros of Gauss and confluent hypergeometric functions are derived as well as an inequality involving the geometric mean ofZeros of Bessel functions.
TL;DR: An approximate method for solving the diffusion equation with nonlocal boundary conditions is proposed, based upon constructing the double shifted Legendre series to approximate the required solution using Legendre tau method.
Abstract: An approximate method for solving the diffusion equation with nonlocal boundary conditions is proposed. The method is based upon constructing the double shifted Legendre series to approximate the required solution using Legendre tau method. The differential and integral expressions which arise in the diffusion equation with nonlocal boundary conditions are converted into a system of linear algebraic equations which can be solved for the unknown coefficients. Numerical examples are included to demonstrate the validity and applicability of the method and a comparison is made with existing results.
TL;DR: Optimal error estimates featuring explicit expressions on the mapping parameters for several popular mappings are derived and play an important role in numerical analysis of mapped Legendre spectral and pseudospectral methods for differential equations.
Abstract: A general framework is introduced to analyze the approximation properties of mapped Legendre polynomials and of interpolations based on mapped Legendre--Gauss--Lobatto points. Optimal error estimates featuring explicit expressions on the mapping parameters for several popular mappings are derived. These results not only play an important role in numerical analysis of mapped Legendre spectral and pseudospectral methods for differential equations but also provide quantitative criteria for the choice of parameters in these mappings.