Showing papers on "Legendre polynomials published in 1994"
TL;DR: This paper presents some efficient algorithms based on the Legendre–Galerkin approximations for the direct solution of the second- and fourth-order elliptic equations using matrix-matrix multiplications for discrete variational formulations.
Abstract: This paper presents some efficient algorithms based on the Legendre–Galerkin approximations for the direct solution of the second- and fourth-order elliptic equations. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with sparse matrices for the discrete variational formulations. The complexities of the algorithms are a small multiple of $N^{d + 1} $ operations for a d-dimensional domain with $(N - 1)^d $ unknowns, while the convergence rates of the algorithms are exponential for problems with smooth solutions. In addition, the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix-matrix multiplications.
593 citations
TL;DR: In this paper, the random field is represented by a series of orthogonal functions, and is incorporated directly in the finite-element formulation and first-order reliability analysis, and its relationship with the Karhunen-Loeve expansion used in recent stochastic finite element studies is examined.
Abstract: A new approach for first‐order reliability analysis of structures with material parameters modeled as random fields is presented. The random field is represented by a series of orthogonal functions, and is incorporated directly in the finite‐element formulation and first‐order reliability analysis. This method avoids the difficulty of selecting a suitable mesh for discretizing the random field. A general continuous orthogonal series expansion of the random field is derived, and its relationship with the Karhunen‐Loeve expansion used in recent stochastic finite‐element studies is examined. The method is illustrated for a fixed‐end beam with bending rigidity modeled as a random field. A set of Legendre polynomials is used as the orthogonal base to represent the random field. Two types of correlation models are considered. The Karhunen‐Loeve expansion leads to a lower truncation error than does the Legendre expansion for a given number of terms, but one or two additional terms in the Legendre expansion yield...
186 citations
TL;DR: In this article, the activity coefficients at infinite dilution were derived from the data at low concentrations using a flexible Legendre polynomial, using a static apparatus with special attention paid to the dilute region.
Abstract: Precise isothermal P-x data at 50 C were measured for the different binary butanol-water systems using a static apparatus with special attention paid to the dilute region. Using these data, parameters for the different GE models were fitted and the activity coefficients at infinite dilution were derived from the data at low concentrations using a flexible Legendre polynomial.
122 citations
Journal Article•
TL;DR: In this article, the authors measured the isothermal P-x data at 50 o C for different binary butanol-water systems using a static apparatus with special attention paid to the dilute region.
Abstract: Precise isothermal P-x data at 50 o C were measured for the different binary butanol-water systems using a static apparatus with special attention paid to the dilute region. Using these data, parameters for the different G E models were fitted and the activity coefficients at infinite dilution were derived from the data at low concentrations using a flexible Legendre polynomial
101 citations
TL;DR: In this paper, a new collocation method for numerical solution of partial differential equations is presented, which uses the Chebyshev collocation points, but, because of the way the boundary conditions are implemented, it has all the advantages of the Legendre methods.
Abstract: A new collocation method for the numerical solution of partial differential equations is presented. This method uses the Chebyshev collocation points, but, because of the way the boundary conditions are implemented, it has all the advantages of the Legendre methods. In particular $L_2 $ estimates can be easily obtained for hyperbolic and parabolic problems.
75 citations
TL;DR: In this paper, it was shown that if a curve on a 3-dimensional Sasakian manifold has constant torsion + 1 and satisfies the initial conditions at one point for a Legendre curve, then it is a SIS curve.
Abstract: It is first observed that on a 3-dimensional Sasakian manifold the torsion of a Legendre curve is identically equal to +1. It is then shown that, conversely, if a curve on a Sasakian 3-manifold has constant torsion +1 and satisfies the initial conditions at one point for a Legendre curve, it is a Legendre curve. Furthermore, among contact metric structures, this property is characteristic of Sasakian metrics. For the standard contact structure onR3 with its standard Sasakian metric the curvature of a Legendre curve is shown to be twice the curvature of its projection to thexy-plane with respect to the Euclidean metric. Thus this metric onR3 is more natural for the study of Legendre curves than the Euclidean metric.
72 citations
TL;DR: The SSFPQL as mentioned in this paper code, which solves the steady state quasi-linear kinetic equation describing the ion distribution function during ion cyclotron heating in two velocity variables has been written.
Abstract: A new, very fast code, SSFPQL, which solves the steady state quasi-linear kinetic equation describing the ion distribution function during ion cyclotron heating in two velocity variables has been written. It uses an expansion of the distribution function in Legendre polynomials in pitch angle and cubic finite elements in the velocity variable. The success of the Legendre expansion for the ion cyclotron heating problem depends on a non-conventional representation of the quasi-linear diffusion coefficient based on the addition theorem of Bessel functions. By omitting toroidal trapping of energetic ions but otherwise including finite Larmor radius effects to all orders, SSFPQL offers a very efficient complement to wave codes and tokamak radial transport codes in view of the ultimate goals of a self-consistent modelling of ion cyclotron heating. In particular, the velocity space information supplied by SSFPQL can easily be used to take into account quasi-linear effects in the description of wave propagation and absorption with a reasonably modest numerical effort. A few applications of SSFPQL to first harmonic ion cyclotron heating of tritium in a typical ITER plasma are presented. At the power levels to be expected in ITER, the suprathermal ion population is strongly anisotropic: the parallel energy increase is typically only 20% of the total. The total energy in the ion tail is never very high, and its effects on the heating rate are rather modest; in particular, self-boosting of first harmonic heating by finite Larmor radius effects can hardly compensate for the poor efficiency of this heating method at ohmic temperatures. A reduction of the density at the beginning of the heating pulse, or the introduction of a low concentration 3He minority, might be needed to overcome this unfavourable situation. On the other hand, near ignition the quasi-linear increase of the fusion reactivity by first harmonic heating of tritium is not negligible and could lower the ignition temperature by a few keV
65 citations
TL;DR: In this article, the density functional theory in terms of field theoretical Legendre transformation formation is presented, where the ground state energy is first written as a functional of local probe coupled to the density operator.
Abstract: Theoretical foundation of density functional theory in terms of field theoretical Legendre trans formation is presented. The ground state energy is first written as a functional of local probe coupled to the density operator. The density functional is then defined by the functional Legendre transformation, which leads to a systematic formulation of density functional theory. Excitation spectrum is also determined within the same formalism in a unified way. The diagrammatic evaluation is most conveniently done by using auxiliary field method. Several generalizations of the formalism and extension to the case other than the density operator are also discussed. The density functional theory is now one.of the most commonly used methods in discussing various many particle systems. 1 l In this formalism the energy of the system is written as a functional of the density which is a function of single variable x instead of N coordinates X1 ~ XN of N particles. In spite of the usefulness, its theoretical formulation is rather involved and sometimes difficult to achieve a system atic approximation scheme. The purpose of this paper is to present a clear formulation of the density func tional theory in terms of full use of the Legendre transformation applied to the quantum system, especially to the system described by the second quantized field theory. 2 l It presents a theoretical basis of the density functional formalism which is both exact and systematic. These can be achieved by a straightforward application of our technique called on-shell expansion. The discussion is organized as follows: 1. Ground state 2. Excited states 3. Generalization to the case other than the density operator 4. Finite temperature case (equilibrium and non-equilibrium), etc. They are all formulated in terms of the field theoretical Legendre transformation in a unified way. We have in mind, in the following, atomic system with N electrons but the arguments can be applied to any many particle system with minor modifications. The essential feature of the formulation of the density functional theory in terms of Legendre transformation is the following. In the usual approach the ground state wave function lf! is used to connect the potential v and the density n;
64 citations
TL;DR: Stochastic temperature fluctuations of the cosmic background radiation (CBR) arising via the Sachs-Wolfe effect from gravitational wave perturbations produced in the early universe are examined, described by an angular correlation function $C(\gamma)$.
Abstract: We examine stochastic temperature fluctuations of the cosmic background radiation (CBR) arising via the Sachs-Wolfe effect from gravitational wave perturbations produced in the early universe. These temperature fluctuations are described by an angular correlation function $C(\gamma)$. A new (more concise and general) derivation of $C(\gamma)$ is given, and evaluated for inflationary-universe cosmologies. This yields standard results for angles $\gamma$ greater than a few degrees, but new results for smaller angles, because we do not make standard long-wavelength approximations to the gravitational wave mode functions. The function $C(\gamma)$ may be expanded in a series of Legendre polynomials; we use numerical methods to compare the coefficients of the resulting expansion in our exact calculation with standard (approximate) results. We also report some progress towards finding a closed form expression for $C(\gamma)$.
35 citations
TL;DR: In this article, an analytic solution to the incremental field equations and interface conditions governing the problem is derived using a formulation in terms of the isopotential incremental pressure measuring the increment of the hydrostatic initial pressure with respect to a particular level surface of the gravitational potential.
Abstract: SUMMARY We consider a spherical, isochemical, incompressible, non-rotating fluid planet and study infinitesimal, quasi-static, gravitational-viscoelastic perturbations, induced by surface loads, of a hydrostatic initial state. The analytic solution to the incremental field equations and interface conditions governing the problem is derived using a formulation in terms of the isopotential incremental pressure measuring the increment of the hydrostatic initial pressure with respect to a particular level surface of the gravitational potential. This admits the decoupling of the incremental equilibrium equation from the incremental potential equation. As result, two mutually independent (4 X 4) and (2 X 2) first-order ordinary differential systems in terms of the mechanical and gravitational quantities, respectively, are obtained, whose integration is algebraically easier than that of the conventional (6 X 6) differential system. In support of various types of application, we provide transfer functions, impulse-response functions and Green’s functions for the full range of incremental field quantities of interest in studies of planetary deformations. The functional forms in the different solution domains involve explicit expressions for the Legendre degrees n = 0, n = 1 and n 2 2, apply to any location in the interior or exterior of the planet and are valid for any type of generalized Maxwell viscoelasticity and for arbitrary surface loads.
31 citations
TL;DR: In this article, the optimal control of linear systems with quadratic cost functional is considered based on shifted Legendre approximation of each state variable that converts the linear quad ratic problem to a system of linear algebraic equations.
Abstract: The optimal control of linear systems with quadratic cost functional is considered. The method is based on shifted Legendre approximation of each state variable that converts the linear quadratic problem to a system of linear algebraic equations. An illustrative example is included to demonstrate the validity and applicability of the technique.
TL;DR: In this paper, a procedure for the reduction of derivatives of Nalewajski's Legendre transformations is presented. But it only includes derivatives of the four representations and does not include derivatives of all the representations.
Abstract: The ground-state energy of an electronic system is a functional of the number of electrons (N) and the external potential (v): E = E[N, v], this is the energy representation for ground states. In 1982, Nalewajski defined the Legendre transforms of this representation, taking advantage of the strict concavity of E with respect to their variables (concave respect v and convex respect N), and he also constructed a scheme for the reduction of derivatives of his representations. Unfortunately, N and the electronic density (ρ) were the independent variables of one of these representations, but ρ depends explicitly on N. In this work, this problem is avoided using the energy per particle (ϵ) as the basic variable. In this case ϵ is a strict concave functional respect to both of his variables, and the Legendre transformations can be defined. A procedure for the reduction of derivatives is generated for the new four representations and, in contrast to the Nalewajski's procedure, it only includes derivatives of the four representations. Finally, the reduction of derivatives is used to test some relationships between the hardness and softness kernels. © 1994 John Wiley & Sons, Inc.
TL;DR: In this paper, the authors give local normal forms of generic implicit first order ordinary differential equations with independent first integrals, which are then reduced to classifying a certain class of divergent diagrams of map-germs; integral diagrams.
Abstract: In this paper we give local normal forms of generic implicit first order ordinary differential equations with independent first integrals. The classification problem in this case is reduced to classifying a certain class of divergent diagrams of map-germs; integral diagrams. The main tools are Legendre singularity theory and differential analysis. We also discuss the relation between previous works and the results obtained in this paper.
CERN1
TL;DR: In this paper, the functional flow equations for the Legendre effective action, with respect to changes in a smooth cutoff, are approximated by a derivative expansion; no other approximation is made.
Abstract: The functional flow equations for the Legendre effective action, with respect to changes in a smooth cutoff, are approximated by a derivative expansion; no other approximation is made. This results in a set of coupled non-linear differential equations. The corresponding differential equations for a fixed point action have at most a countable number of solutions that are well defined for all values of the field. We apply the technique to the fixed points of one-component real scalar field theory in three dimensions. Only two non-singular solutions are found: the gaussian fixed point and an approximation to the Wilson fixed point. The latter is used to compute critical exponents, by carrying the approximation to second order. The results appear to converge rapidly.
TL;DR: In this paper, two different Nystrom interpolants for the numerical solution of the following singular integral equation (formule...), 0 < x ≤ 1, arising from a problem of determining the distribution of stress in a thin elastic plate in the vicinity of a cruciform crack were considered.
Abstract: We consider two different Nystrom interpolants for the numerical solution of the following singular integral equation (formule...), 0 < x ≤ 1, arising from a problem of determining the distribution of stress in a thin elastic plate in the vicinity of a cruciform crack. These interpolants originate from the discretization of the integral by two different quadrature formulas of interpolatory type based on the zeros of Legendre orthogonal polynomials. The first quadrature is of product type and integrates exactly the kernel; the second one is the well-known Gauss-Legendre formula. First we derive uniform convergence estimates for the two basic quadrature rules
TL;DR: In this article, the Born-Infeld equation in two dimensions is generalized to higher dimensions and Lorentz invariance is retained, and the resulting system is completely integrable via linearization by Legendre transformation.
Abstract: The Born-Infeld equation in two dimensions is generalized to higher dimensions. Lorentz invariance is retained, and the resulting system is completely integrable via linearization by Legendre transformation. This generalization retains homogeneity in second derivatives of the field.
TL;DR: In this paper, an advection-diffusion model is discretized by a spectral Legendre method augmented by trial/test functions with local support (bubbles), and numerical analysis shows that the correction is equivalent to adding a spectrally accurate streamline-upwind stabilization term.
TL;DR: In this paper, the cumulants and factorial moments of photon distribution for squeezed and correlated light are calculated in terms of Chebyshev, Legendre and Laguerre polynomials.
TL;DR: In this paper, the authors introduce the notion of front hypersurfaces by the property that the Nash modification projects to the hypersurface itself finitely to one, which is the closure of the lifting of the regular points set to the projective cotangent bundle.
Abstract: BY Goo ISHIKAWA0. IntroductionLegendre varieties and Lagrange varieties appear in many areas, for instance, geo-metric optics [A][J2], generalized Cauchy problem for Hamilton-Jacobi equations [G2][13], projective geometry [SI], microlocal analysis [P][DP], moduli problem of vectorbundles on complex surfaces [Y], symplectic topology [Gl] and so on.In this survey we treat Legendre and Lagrange varieties admitting some parametriza-tions in complex analytic or C°° category. Then our study fits with the framework ofthe theory of singularities of differentiate mappings [AGV][B][D][GWPL][W].First we introduce the notion of a "front hypersurface" by the property that the Nashmodification projects to the hypersurface itself finitely to one. The Nash modification, inthis case, is the closure of the lifting of the regular points set to the projective cotangentbundle of the manifold where the hypersurface lies in: The projective cotangent bundleis identified with the totality of contact elements (tangent hyperplanes) of the basespace and it has the natural contact structure [A][SI]. The tangent hyperplanes to theregular points of a front hypersurface form a Legendre submanifold, that is, the maximaldimensional integral submanifold of the contact distribution defined over the projectivecotangent bundle and the closure of this natural lifting might be regarded as a Legendrevariety. In fact, a definition of Legendre variety is that it contains an open dense Legendresubmanifold. The Legendre variety thus obtained by Nash modification has singularitiesin general. If the Nash modification is non-singular, then the hypersurface turns out theprojection of a Legendre submanifold. Then the front hypersurface is called a wave frontset [A][Z1]. Remark that, for a generic Legendre submanifold, the projection is finiteto one. In the above definition of front hypersurfaces we allow singularities for Nashmodification, and to make the definition non-trivial, we add the finiteness condition.(See [LT] for the general theory of limits of tangent spaces.)We utilize parametrizations of varieties to formulate the notion above mentionedas follows: A mapping / from an n-dimensional manifold N to an n + 1-dimensionalmanifold B (say, of class (7°° or complex analytic) is called a front mapping if the set ofregular points of / is dense in TV, and, for each point x £ TV, the images of the tangentspaces of regular points converge to a tangent hyperplane T
TL;DR: In this paper, a new method of gathering statistics for Monte Carlo methods, Legendre polynomial weighted sampling (LPWS), is presented, which requires only a minimum of particles to extract higher-order derivative information about a particle's distribution function.
Abstract: A new method of gathering statistics for Monte Carlo methods, Legendre polynomial weighted sampling (LPWS), is presented. LPWS requires only a minimum of particles to extract higher‐order derivative information about a particle’s distribution function. In this technique, when calculating a particle’s distribution function, higher‐order derivative information about the Monte Carlo particles is recorded along with just counting the number of particles in a bin. The distribution function is then constructed from this information. Specifically, in this paper, second‐order Legendre polynomial weighted sampling is employed. Legendre polynomial weighted sampling is demonstrated by calculating the electron energy distribution functions in an inductively coupled plasma reactor.
TL;DR: In this paper, the cumulants and factorial moments of photon distribution for squeezed and correlated light are calculated in terms of Chebyshev, Legendre and Laguerre polynomials.
Abstract: The cumulants and factorial moments of photon distribution for squeezed and correlated light are calculated in terms of Chebyshev, Legendre and Laguerre polynomials. The phenomenon of strong oscillations of the ratio of the cumulant to factorial moment is found.
TL;DR: The adjusted spherical harmonic analysis (ASHA) as discussed by the authors was developed for the regional modeling of the geomagnetic field /3/, which is used in this paper for the same purpose.
TL;DR: In this article, the behavior of eigenvalues for the second-order Legendre spectral (Galerkin) differentiation operator are extended to the eigen values of the corresponding Legendre pseudospectral collocation operator.
TL;DR: In this paper, a new high-order refined shear deformation theory based on Reissner's mixed variational principle in conjunction with the state-space concept is used to determine the deflections and stresses for rectangular cross-ply composite plates.
TL;DR: In this article, the impedance boundary method of moments (IBMOM) is proposed to accurately and efficiently compute the propagation characteristics including the number of guided modes of general graded-index dielectric slab waveguide structures.
Abstract: An impedance boundary method of moments (IBMOM) is proposed to accurately and efficiently compute the propagation characteristics including the number of guided modes of general graded-index dielectric slab waveguide structures. The method is based on Galerkin's procedure in the method of moments and employs the exact impedance boundary condition at the interfaces between the graded-index region and constant-index cladding. Legendre polynomials are utilized in the field expansion. Computational results are shown for waveguides with various inhomogeneous refractive index profiles. The results indicate that typically five Legendre polynomials are sufficient for accurate solutions of the dominant TE and TM modes in optical waveguides having a finite region of inhomogeneous refractive index. Diffused optical waveguides with untruncated index profiles as well as coupled dielectric waveguides can be accurately analyzed using ten Legendre polynomials. >
TL;DR: In this paper, the average orientation of a linear dipole in a strong electric field is estimated based on a zeroth-order perturbation treatment of vibration in two dimensions.
TL;DR: 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.
TL;DR: This paper shows a method of obtaining general and orthogonal moments, specifically Legendre and Zernicke moments, from the Radon Transform data of a two-dimensional function.
TL;DR: In this article, a linear Schrodinger type equation in a rectangular domain with mixed Dirichlet-Neumann boundary conditions was solved by combining a Legendre type spectral method in the first direction and a leap-frog scheme in the other one.
Abstract: This paper deals with a linear Schrodinger type equation in a rectangular domain with mixed Dirichlet-Neumann boundary conditions. The well-posedness of the continuous problem is proved, then a discrete problem is defined by combining a Legendre type spectral method in the first direction and a leap-frog scheme in the other one. The numerical analysis of the discretization is performed and error estimates are given. Numerical tests are presented.