Showing papers on "Legendre polynomials published in 2009"

Making sense of the Legendre transform

Abstract: The Legendre transform is a powerful tool in theoretical physics and plays an important role in classical mechanics, statistical mechanics, and thermodynamics. In typical undergraduate and graduate courses the motivation and elegance of the method are often missing, unlike the treatments frequently enjoyed by Fourier transforms. We review and modify the presentation of Legendre transforms in a way that explicates the formal mathematics, resulting in manifestly symmetric equations, thereby clarifying the structure of the transform. We then discuss examples to motivate the transform as a way of choosing independent variables that are more easily controlled. We demonstrate how the Legendre transform arises naturally from statistical mechanics and show how the use of dimensionless thermodynamic potentials leads to more natural and symmetric relations.

Methods of harmonic synthesis for global geopotential models and their first-, second- and third-order gradients

Abstract: Four widely used algorithms for the computation of the Earth’s gravitational potential and its first-, second- and third-order gradients are examined: the traditional increasing degree recursion in associated Legendre functions and its variant based on the Clenshaw summation, plus the methods of Pines and Cunningham–Metris, which are free from the singularities that distinguish the first two methods at the geographic poles. All four methods are reorganized with the lumped coefficients approach, which in the cases of Pines and Cunningham–Metris requires a complete revision of the algorithms. The characteristics of the four methods are studied and described, and numerical tests are performed to assess and compare their precision, accuracy, and efficiency. In general the performance levels of all four codes exhibit large improvements over previously published versions. From the point of view of numerical precision, away from the geographic poles Clenshaw and Legendre offer an overall better quality. Furthermore, Pines and Cunningham–Metris are affected by an intrinsic loss of precision at the equator and suffer from additional deterioration when the gravity gradients components are rotated into the East-North-Up topocentric reference system.

On Legendre Curves in 3-Dimensional Normal Almost Paracontact Metric Manifolds

Abstract: The present paper is devoted to study the curvature and torsion of Frenet Legendre curves in 3-dimensional normal almost paracontact metric manifolds. Moreover, in this class of manifolds, properties of non-Frenet Legendre curves (with null tangents or null normals or null binormals) are obtained. Many examples of Legendre curves are constructed.

Legendre spectral Galerkin method for second-kind Volterra integral equations

Abstract: The Legendre spectral Galerkin method for the Volterra integral equations of the second kind is proposed in this paper. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors (in the L2 norm) will decay exponentially provided that the kernel function and the source function are sufficiently smooth. Numerical examples are given to illustrate the theoretical results.

Dolph–Chebyshev Beampattern Design for Spherical Arrays

Abstract: Spherical arrays with array processing based on spherical harmonics have been recently studied for a wide range of applications that require three-dimensional beamforming. In this correspondence, a Dolph-Chebychev beampattern design, widely used in array processing due to the direct control over main-lobe width and maximum sidelobe level, is developed for spherical arrays within the spherical harmonics framework. We show that due to the similarity between the Legendre polynomials that define the spherical array beampattern and the Chebyshev polynomials that define the desired Dolph-Chebyshev beampattern, spherical array weights can be computed directly and accurately given desired main-lobe width or sidelobe level.

Hermite-Legendre-Gauss-Lobatto Direct Transcription in Trajectory Optimization

Abstract: D IRECT methods have been widely applied for solving trajectory optimization problems [1–5]. Among the popular methods are the Hermite–Simpson method [6] and the Legendre pseudospectral (PS)method [7,8]. There has been some considerable interest in developing theory related to the Legendre PS method due to its high accuracy, although the Hermite–Simpson method continues to be applied to large-scale practical problems [9]. The main characteristic of the Hermite–Simpson method is the combination of reasonable accuracy with a highly sparse constraint Jacobian and Hessian matrix [10]. The PS method offers impressive accuracy (spectral accuracy) for smooth problems, but the constraint Jacobians are much denser than other methods. The sparsity of the constraint Jacobians can be increased in the PS method by using knots [11]. In addition to the Hermite–Simpson method, additional highorder methods have been proposed by Herman and Conway [2]. These methods are attractive from the point of view of accuracy, but in the framework proposed by Herman and Conway, they require detailed derivation when extended to arbitrary higher orders. For instance, in [2], the form of the constraints was derived via the symbolic manipulation software MAPLE. The purpose of this Note is to provide an alternative framework for arbitrary higher-order methods suitable for implementation on digital computers and in a reusable form. The optimal control problem is approximated by a discrete nonlinear programming problem (NLP) by expanding the state trajectories using local Hermite interpolating polynomials. For high accuracy, the collocation points are selected from the family of Gauss–Lobatto points. This also allows the integral performance index to be approximated viaGauss– Lobatto quadrature rules. For optimal control problems of the Bolza form, the natural choice of quadrature points are the Legendre– Gauss–Lobatto (LGL) points, because they are derived on the basis of a unity weight function, giving the highest accuracy for polynomial integrands. The generalization of the approach is referred to as the Hermite–Legendre–Gauss–Lobatto (HLGL) approach throughout the remainder of this Note.

Wave characteristics in functionally graded piezoelectric hollow cylinders

Abstract: Based on linear three-dimensional piezoelasticity, the Legendre orthogonal polynomial series expansion approach is used for determining the wave characteristics in hollow cylinders composed of the functionally graded piezoelectric materials (FGPM) with open circuit. The displacement and electric potential components, expanded in a series of Legendre polynomials, are introduced into the governing equations along with position-dependent material constants so that the solution of the wave equation is reduced to an eigenvalue problem. Dispersion curves for FGPM and the corresponding non-piezoelectric hollow cylinders are calculated to show the piezoelectric effect. The influence of the ratio of radius to thickness is discussed. Electric potential and displacement distributions are used to show the piezoelectric effect on the flexural torsional mode. The influence of the polarizing direction on the piezoelectric effect is illustrated. For the radial and axial polarization, the piezoelectric effect reacts mostly on the longitudinal mode. For circumferential polarization, the piezoelectric effect reacts mostly on the torsional mode. In the FGPM hollow cylinder, piezoelectricity can weaken the guided wave dispersion.

Optimal Control of Linear Time-Delayed Systems by Linear Legendre Multiwavelets

Abstract: This paper introduces the application of linear Legendre multiwavelets to the optimal control synthesis for linear time-delayed systems. Based on some useful properties of linear Legendre multiwavelets, integration, product and delay operational matrices are proposed to solve the linear time-delayed systems first. Then, a quadratic cost functional is approximated by those properties. By using Lagrange multipliers, the quadratic cost functional is minimized subject to the solution of the linear time-delayed system and an explicit formula for the optimal control is obtained. The effectiveness of the method and accuracy of the solution are shown in comparison with some other methods by illustrative examples.

Legendre multiwavelet Galerkin method for solving the hyperbolic telegraph equation

Abstract: Recently, it is found that telegraph equation is more suitable than ordinary diffusion equation in modeling reaction diffusion for such branches of sciences. In this article a numerical method for solving the one-dimensional hyperbolic telegraph equation is presented. The method is based upon Legendre multiwavelet approximations. The properties of Legendre multiwavelet are first presented. These properties together with Galerkin method are then utilized to reduce the telegraph equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

A combinatorial interpretation of the Legendre-Stirling numbers

Abstract: The Legendre-Stirling numbers were discovered in 2002 as a result of a problem involving the spectral theory of powers of the classical second-order Legendre differential expression. Specifically, these numbers are the coefficients of integral composite powers of the Legendre expression in Lagrangian symmetric form. Quite remarkably, they share many similar properties with the classical Stirling numbers of the second kind which, as shown by Littlejohn and Wellman, are the coefficients of integral powers of the Laguerre differential expression. An open question regarding the Legendre-Stirling numbers has been to obtain a combinatorial interpretation of these numbers. In this paper, we provide such an interpretation.

Composite generalized Laguerre-Legendre spectral method with domain decomposition and its application to Fokker-Planck equation in an infinite channel

Abstract: In this paper, we propose a composite generalized LaguerreLegendre spectral method for partial differential equations on two-dimensional unbounded domains, which are not of standard types. Some approximation results are established, which are the mixed generalized Laguerre-Legendre approximations coupled with domain decomposition. These results play an important role in the related spectral methods. As an important application, the composite spectral scheme with domain decomposition is provided for the Fokker-Planck equation in an infinite channel. The convergence of the proposed scheme is proved. An efficient algorithm is described. Numerical results show the spectral accuracy in the space of this approach and coincide well with theoretical analysis. The approximation results and techniques developed in this paper are applicable to many other problems on unbounded domains. In particular, some quasi-orthogonal approximations are very appropriate for solving PDEs, which behave like parabolic equations in some directions, and behave like hyperbolic equations in other directions. They are also useful for various spectral methods with domain decompositions, and numerical simulations of exterior problems.

Jacobi spectral Galerkin method for elliptic Neumann problems

Abstract: This paper is concerned with fast spectral-Galerkin Jacobi algorithms for solving one- and two-dimensional elliptic equations with homogeneous and nonhomogeneous Neumann boundary conditions. The paper extends the algorithms proposed by Shen (SIAM J Sci Comput 15:1489–1505, 1994) and Auteri et al. (J Comput Phys 185:427–444, 2003), based on Legendre polynomials, to Jacobi polynomials $P_{n}^{(\alpha,\beta)}(x)$ with arbitrary α and β. The key to the efficiency of our algorithms is to construct appropriate basis functions with zero slope at the endpoints, which lead to systems with sparse matrices for the discrete variational formulations. The direct solution algorithm developed for the homogeneous Neumann problem in two-dimensions relies upon a tensor product process. Nonhomogeneous Neumann data are accounted for by means of a lifting. Numerical results indicating the high accuracy and effectiveness of these algorithms are presented.

New construction of quaternary sequences with ideal autocorrelation from Legendre sequences

Abstract: In this paper, for an odd prime p, new quaternary sequences of even period 2p with ideal autocorrelation property are constructed using the Legendre sequences of period p. The distribution of autocorrelation function of the proposed quaternary sequences is also derived.

On-axis and far-field sound radiation from resilient flat and dome-shaped radiators.

Abstract: On-axis and far-field series expansions are developed for the sound pressure due to an arbitrary, circular symmetric velocity distribution on a flat radiator in an infinite baffle. These expansions are obtained by expanding the velocity distributions in terms of orthogonal polynomials R(2n) (0)(sigma/a)=P(n)(2(sigma/a)(2)-1) with P(n) the Legendre polynomials. The terms R(2n) (0) give rise to a closed-form expression for the pressure on-axis as well as for the far-field pressure. Furthermore, for a large number of velocity profiles, including those associated with the rigid piston, the simply supported radiator, and the clamped radiators as well as Gaussian radiators, there are closed-form expressions for the required expansion coefficients. In particular, for the rigid, simply supported, and clamped radiators, this results in explicit finite-series expressions for both the on-axis and far-field pressures. In the reverse direction, a method of estimating velocity distributions from (measured) on-axis pressures by matching in terms of expansion coefficients is proposed. Together with the forward far-field computation scheme, this yields a method for far-field loudspeaker assessment from on-axis data (generalized Keele scheme). The forward computation scheme is extended to dome-shaped radiators with arbitrary velocity distributions.

Sound radiation quantities arising from a resilient circular radiator.

Abstract: Power series expansions in ka are derived for the pressure at the edge of a radiator, the reaction force on the radiator, and the total radiated power arising from a harmonically excited, resilient, flat, circular radiator of radius a in an infinite baffle. The velocity profiles on the radiator are either Stenzel functions (1-(sigma/a)2)n, with sigma the radial coordinate on the radiator, or linear combinations of Zernike functions Pn(2(sigma/a)2-1), with Pn the Legendre polynomial of degree n. Both sets of functions give rise, via King's integral for the pressure, to integrals for the quantities of interest involving the product of two Bessel functions. These integrals have a power series expansion and allow an expression in terms of Bessel functions of the first kind and Struve functions. Consequently, many of the results in [M. Greenspan, J. Acoust. Soc. Am. 65, 608-621 (1979)] are generalized and treated in a unified manner. A foreseen application is for loudspeakers. The relation between the radiated power in the near-field on one hand and in the far field on the other is highlighted.

Minimal dynamical systems on a discrete valuation domain

Abstract: We consider isometric dynamical systems on a Legendre set of a discrete valuation domain with finite residual field We characterize the minimality of the system by using the structure sequence of the Legendre set and also find the corresponding adding machine to which such a minimal system is conjugate The minimality of affine maps acting on the domain or on the group of units is fully studied

Superconvergence of Discontinuous Galerkin Methods for Convection-Diffusion Problems

Abstract: Some discontinuous Galerkin methods for the linear convection-diffusion equation ?? u?+bu?=f are studied. Based on superconvergence properties of numerical fluxes at element nodes established in some earlier works, e.g., Celiker and Cockburn in Math. Comput. 76(257), 67---96, 2007, we identify superconvergence points for the approximations of u or q=u?. Our results are twofold: 1) For the minimal dissipation LDG method (we call it md-LDG in this paper) using polynomials of degree p, we prove that the leading terms of the discretization errors for u and q are proportional to the right Radau and left Radau polynomials of degree p+1, respectively. Consequently, the zeros of the right-Radau and left-Radau polynomials of degree p+1 are the superconvergence points of order p+2 for the discretization errors of the potential and of the gradient, respectively. 2) For the consistent DG methods whose numerical fluxes at the mesh nodes converge at the rate of O(h p+1), we prove that the leading term of the discretization error for q is proportional to the Legendre polynomial of degree p. Consequently, the approximation of the gradient superconverges at the zeros of the Legendre polynomial of degree p at the rate of O(h p+1). Numerical experiments are presented to illustrate the theoretical findings.

Density-functional theory with additional basic variables: Extended Legendre transform

Abstract: In a recent series of papers, Higuchi and Higuchi defined an extended constrained-search procedure by extending the Levy constrained search by adding additional constraints. As shown here, this procedure can be equivalently formulated in terms of Lieb's Legendre transformation functional. The Legendre transform approach has advantages in cases where the additional constraints are restrictive enough to cause problems with $N$-representability.

A Spectral Element Method to Price European Options. I. Single Asset with and without Jump Diffusion

Abstract: We develop a spectral element method to price single factor European options with and without jump diffusion. The method uses piecewise high order Legendre polynomial expansions to approximate the option price represented pointwise on a Gauss-Lobatto mesh within each element, which allows an exact representation of the non-smooth payoff function. The convolution integral is approximated by high order Gauss-Lobatto quadratures. A second order implicit/explicit (IMEX) approximation is used to integrate in time, with the convolution integral integrated explicitly. The method is spectrally accurate (exponentially convergent) in space for the solution and Greeks, and second-order accurate in time. The spectral element solution to the Black-Scholes equation is ten to one hundred times faster than commonly used second order finite difference methods.

Sound radiation quantities arising from a resilient circular radiator (without former Sec. VII, and with new order for references)

Abstract: Power series expansions in ka are derived for the pressure at the edge of a radiator, the reaction force on the radiator, and the total radiated power arising from a harmonically excited, resilient, flat, circular radiator of radius a in an infinite baffle. The velocity profiles on the radiator are either Stenzel functions (1− (σ/a)) with σ the radial coordinate on the radiator, or linear combinations of Zernike functions Pn(2(σ/a) 2 − 1) with Pn the Legendre polynomial of degree n. Both sets of functions give rise, via King’s integral for the pressure, to integrals for the quantities of interest involving the product of two Bessel functions. These integrals have a power series expansion and allow an expression in terms of Bessel functions of the first kind and Struve functions. Consequently, many of the results in [Piston radiator: Some extensions of the theory, J. Acoust. Soc. Am. 65(3), 1979] are generalized and treated in a unified manner. A foreseen application is for loudspeakers. The relation between the radiated power in the near-field on one hand and in the far-field on the other is highlighted.

Legendre multiscaling functions for solving the one‐dimensional parabolic inverse problem

Abstract: An inverse problem concerning diffusion equation with a source control parameter is investigated The approximation of the problem is based on the Legendre multiscaling basis The properties of Legendre multiscaling functions are first presented These properties together with Galerkin method are then utilized to reduce the inverse problem to the solution of algebraic equations Illustrative examples are included to demonstrate the validity and applicability of the new technique © 2009 Wiley Periodicals, Inc Numer Methods Partial Differential Eq, 2009

Geometric study for the Legendre duality of generalized entropies and its application to the porous medium equation

Abstract: We geometrically study the Legendre duality relation that plays an important role in statistical physics with the standard or generalized entropies For this purpose, we introduce dualistic structure defined by information geometry, and discuss concepts arising in generalized thermostatistics, such as relative entropies, escort distributions and modified expectations Further, a possible generalization of these concepts in a certain direction is also considered Finally, as an application of such a geometric viewpoint, we briefly demonstrate several new results on the behavior of the solution to a nonlinear diffusion equation called the porous medium equation

Estimates of radiation over clouds and dust aerosols: Optimized number of terms in phase function expansion

Abstract: The bulk-scattering properties of dust aerosols and clouds are computed for the community radiative transfer model (CRTM) that is a flagship effort of the Joint Center for Satellite Data Assimilation (JCSDA). The delta-fit method is employed to truncate the forward peaks of the scattering phase functions and to compute the Legendre expansion coefficients for re-constructing the truncated phase function. Use of more terms in the expansion gives more accurate re-construction of the phase function, but the issue remains as to how many terms are necessary for different applications. To explore this issue further, the bidirectional reflectances associated with dust aerosols, water clouds, and ice clouds are simulated with various numbers of Legendre expansion terms. To have relative numerical errors smaller than 5%, the present analyses indicate that, in the visible spectrum, 16 Legendre polynomials should be used for dust aerosols, while 32 Legendre expansion terms should be used for both water and ice clouds. In the infrared spectrum, the brightness temperatures at the top of the atmosphere are computed by using the scattering properties of dust aerosols, water clouds and ice clouds. Although small differences of brightness temperatures compared with the counterparts computed with 4, 8, 128 expansion terms are observed at large viewing angles for each layer, it is shown that 4 terms of Legendre polynomials are sufficient in the radiative transfer computation at infrared wavelengths for practical applications.