Showing papers on "Legendre polynomials published in 2010"
TL;DR: This paper presents a proof that for the case of one dimensional linear advection the spectral difference method is stable for all orders of accuracy in a norm of Sobolev type, provided that the interior fluxes collocation points are placed at the zeros of the corresponding Legendre polynomial.
Abstract: While second order methods for computational simulations of fluid flow provide the basis of widely used commercial software, there is a need for higher order methods for more accurate simulations of turbulent and vortex dominated flows. The discontinuous Galerkin (DG) method is the subject of much current research toward this goal. The spectral difference (SD) method has recently emerged as a promising alternative which can reduce the computational costs of higher order simulations. There remains some questions, however, about the stability of the SD method. This paper presents a proof that for the case of one dimensional linear advection the SD method is stable for all orders of accuracy in a norm of Sobolev type, provided that the interior fluxes collocation points are placed at the zeros of the corresponding Legendre polynomial.
172 citations
TL;DR: In this article, a discrete-time fractional calculus of variations is introduced, and the first and second order necessary optimality conditions are established, showing that the solutions of the fractional problems coincide with the corresponding non-fractional variational problems when the order of the discrete derivatives is an integer value.
Abstract: We introduce a discrete-time fractional calculus of variations. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler-Lagrange and Legendre type conditions are given. They show that the solutions of the fractional problems coincide with the solutions of the corresponding non-fractional variational problems when the order of the discrete derivatives is an integer value.
116 citations
TL;DR: In this article, a discrete-time fractional calculus of variations is introduced, and the Euler-Lagrange and Legendre type conditions are established for the first and second order necessary optimality conditions.
Abstract: We introduce a discrete-time fractional calculus of variations.
First and second order necessary optimality conditions are
established. Examples illustrating the use of the new Euler--Lagrange
and Legendre type conditions are given. They show that the solutions
of the fractional problems coincide with the solutions of the
corresponding non-fractional variational problems when the order of
the discrete derivatives is an integer value.
113 citations
Journal Article•
TL;DR: In this article, it was shown that there are at most finitely many complex points on the Legendre elliptic curve that have finite order and that two points on such a curve can be found in finite order.
Abstract: We prove that there are at most finitely many complex $\lambda
eq 0,1$ such that two points on the Legendre elliptic curve $Y^2 = X(X-1)(X-\lambda)$ with coordinates $X = 2,3$ both have finite order This is a very special case of some conjectures on unlikely intersections in semiabelian schemes
109 citations
TL;DR: The experimental results show that the proposed descriptors are more robust to noise and have better discriminative power than the methods based on geometric or complex moments.
Abstract: Processing blurred images is a key problem in many image applications. Existing methods to obtain blur invariants which are invariant with respect to centrally symmetric blur are based on geometric moments or complex moments. In this paper, we propose a new method to construct a set of blur invariants using the orthogonal Legendre moments. Some important properties of Legendre moments for the blurred image are presented and proved. The performance of the proposed descriptors is evaluated with various point-spread functions and different image noises. The comparison of the present approach with previous methods in terms of pattern recognition accuracy is also provided. The experimental results show that the proposed descriptors are more robust to noise and have better discriminative power than the methods based on geometric or complex moments.
96 citations
TL;DR: An orthogonal basis on the square [−1, 1] × [‐1, 2] generated by Legendre polynomials is introduced, and an associated expression for the expansion of a Riemann integrable function is defined.
Abstract: We introduce an orthogonal basis on the square [−1, 1] × [-1, 1] generated by Legendre polynomials on [−1, 1], and define an associated expression for the expansion of a Riemann integrable function. We describe some properties and derive a uniform convergence theorem. We then present several numerical experiments that indicate that our methods are more efficient and have better convergence results than some other methods. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010
94 citations
TL;DR: The method is shown to generate realizations of complex spatial patterns, reproduce bimodal data distributions, data variograms, and high-order spatial cumulants of the data, and it is shown that the available hard data dominate the simulation process and have a definitive effect on the simulated realizations.
Abstract: Spatially distributed and varying natural phenomena encountered in geoscience and engineering problem solving are typically incompatible with Gaussian models, exhibiting nonlinear spatial patterns and complex, multiple-point connectivity of extreme values. Stochastic simulation of such phenomena is historically founded on second-order spatial statistical approaches, which are limited in their capacity to model complex spatial uncertainty. The newer multiple-point (MP) simulation framework addresses past limits by establishing the concept of a training image, and, arguably, has its own drawbacks. An alternative to current MP approaches is founded upon new high-order measures of spatial complexity, termed “high-order spatial cumulants.” These are combinations of moments of statistical parameters that characterize non-Gaussian random fields and can describe complex spatial information. Stochastic simulation of complex spatial processes is developed based on high-order spatial cumulants in the high-dimensional space of Legendre polynomials. Starting with discrete Legendre polynomials, a set of discrete orthogonal cumulants is introduced as a tool to characterize spatial shapes. Weighted orthonormal Legendre polynomials define the so-called Legendre cumulants that are high-order conditional spatial cumulants inferred from training images and are combined with available sparse data sets. Advantages of the high-order sequential simulation approach developed herein include the absence of any distribution-related assumptions and pre- or post-processing steps. The method is shown to generate realizations of complex spatial patterns, reproduce bimodal data distributions, data variograms, and high-order spatial cumulants of the data. In addition, it is shown that the available hard data dominate the simulation process and have a definitive effect on the simulated realizations, whereas the training images are only used to fill in high-order relations that cannot be inferred from data. Compared to the MP framework, the proposed approach is data-driven and consistently reconstructs the lower-order spatial complexity in the data used, in addition to high order.
91 citations
TL;DR: In this article, the authors revisited the derivation of the disturbing function in Legendre polynomial, with a special focus on the secular system, and provided explicit expansions of the system for orders in the ratio of semi-major axis up to ten in the planar case and five in the spatial case.
Abstract: Since the original work of Hansen and Tisserand in the XIXth century, there have been many variations in the analytical expansion of the three-body disturbing function in series of the semi-major axis ratio. With the increasing number of planetary systems of large eccentricity, these expansions are even more interesting as they allow us to obtain for the secular systems finite expressions that are valid for all eccentricities and inclinations. We revisited the derivation of the disturbing function in Legendre polynomial, with a special focus on the secular system. We provide here expressions of the disturbing function for the planar and spatial case at any order with respect to the ratio of the semi-major axes. Moreover, for orders in the ratio of semi-major axis up to ten in the planar case and five in the spatial case, we provide explicit expansions of the secular system, and simple algorithms with minimal computation to extend this to higher order, as well as the algorithms for the computation of non secular terms.
89 citations
TL;DR: In this paper, by using the properties of Legendre polynomials, the authors proved some congruences for ρ = 0, ρ ∈ {p-1/2}.
Abstract: Let $p$ be an odd prime. In the paper, by using the properties of Legendre polynomials we prove some congruences for $\sum_{k=0}^{\frac{p-1}2}\binom{2k}k^2m^{-k}\mod {p^2}$. In particular, we confirm several conjectures of Z.W. Sun. We also pose 13 conjectures on supercongruences.
76 citations
TL;DR: The main aim of this paper is to apply the Legendre polynomials for the solution of the linear Fredholm integro-differential-difference equation of high order, which is usually difficult to solve analytically.
Abstract: The main aim of this paper is to apply the Legendre polynomials for the solution of the linear Fredholm integro-differential-difference equation of high order. This equation is usually difficult to solve analytically. Our approach consists of reducing the problem to a set of linear equations by expanding the approximate solution in terms of shifted Legendre polynomials with unknown coefficients. The operational matrices of delay and derivative together with the tau method are then utilized to evaluate the unknown coefficients of shifted Legendre polynomials. Illustrative examples are included to demonstrate the validity and applicability of the presented technique and a comparison is made with existing results.
74 citations
TL;DR: It is shown that singular points, which result in a discontinuous orientation field, can be modelled by the zero-poles of Legendre polynomials, which addresses one of the main problems it addresses is smoothing orientation data while preserving details in high curvature areas, especially singular points.
TL;DR: The Inverse Polynomial Reconstruction Method is generalized by reconstructing a function from its m lowest Fourier coefficients as an algebraic polynomial of degree at most n-1(m>=n) and approximate Legendre coefficients of the function are computed by solving a linear least squares problem.
TL;DR: In this paper, a set of shape mode equations is derived to describe unsteady motions of a sessile drop actuated by electrowetting, and a modified boundary condition is obtained by combining the contact angle model and the normal stress condition at the surface.
Abstract: A set of shape mode equations is derived to describe unsteady motions of a sessile drop actuated by electrowetting. The unsteady, axially symmetric, and linearized flow field is analyzed by expressing the shape of a drop using the Legendre polynomials. A modified boundary condition is obtained by combining the contact angle model and the normal stress condition at the surface. The electrical force is assumed to be concentrated on one point (i.e., three-phase contact line) rather than distributed on the narrow surface of the order of dielectric layer thickness near the contact line. Then, the delta function is used to represent the wetting tension, which includes the capillary force, electrical force, and contact line friction. In previous work [J. M. Oh et al., Langmuir 24, 8379 (2008)], the capillary forces of the air-substrate and liquid-substrate interfaces were neglected, together with the contact-line friction. The delta function is decomposed into a weighted sum of the Legendre polynomials so that e...
TL;DR: In this article, the authors revisited the derivation of the disturbing function in Legendre polynomial, with a special focus on the secular system, and provided explicit expansions of the system for orders in the ratio of semi-major axis up to ten in the planar case and five in the spatial case.
Abstract: Since the original work of Hansen and Tisserand in the XIXth century, there have been many variations in the analytical expansion of the three-body disturbing function in series of the semi-major axis ratio. With the increasing number of planetary systems of large eccentricity, these expansions are even more interesting as they allow us to obtain for the secular systems finite expressions that are valid for all eccentricities and inclinations. We revisited the derivation of the disturbing function in Legendre polynomial, with a special focus on the secular system. We provide here expressions of the disturbing function for the planar and spatial case at any order with respect to the ratio of the semi-major axes. Moreover, for orders in the ratio of semi-major axis up to ten in the planar case and five in the spatial case, we provide explicit expansions of the secular system, and simple algorithms with minimal computation to extend this to higher order, as well as the algorithms for the computation of non secular terms.
TL;DR: A combinatorial interpretation of the coefficients of the polynomial (1-x)^3^k^+^1@?"n"="0^~{{n+kn}}x^n analogous to that of the Eulerian numbers, where {{nk}}are Everitt, Littlejohn, and Wellman's Legendre-Stirling numbers of the second kind.
Abstract: We first give a combinatorial interpretation of Everitt, Littlejohn, and Wellman's Legendre-Stirling numbers of the first kind. We then give a combinatorial interpretation of the coefficients of the polynomial (1-x)^3^k^+^1@?"n"="0^~{{n+kn}}x^n analogous to that of the Eulerian numbers, where {{nk}}are Everitt, Littlejohn, and Wellman's Legendre-Stirling numbers of the second kind. Finally we use a result of Bender to show that the limiting distribution of these coefficients as n approaches infinity is the normal distribution.
TL;DR: In this paper, an analytical theory with numerical simulations to study the orbital motion of lunar artificial satellites is presented, considering the problem of an artificial satellite perturbed by the non-uniform distribution of mass of the Moon and by a third body in elliptical orbit (Earth is considered).
Abstract: In this paper we present an analytical theory with numerical simulations to study the orbital motion of lunar artificial satellites. We consider the problem of an artificial satellite perturbed by the non-uniform distribution of mass of the Moon and by a third-body in elliptical orbit (Earth is considered). Legendre polynomials are expanded in powers of the eccentricity up to the degree four and are used for the disturbing potential due to the third-body. We show a new approximated equation to compute the critical semi-major axis for the orbit of the satellite. Lie-Hori perturbation method up to the second-order is applied to eliminate the terms of short-period of the disturbing potential. Coupling terms are analyzed. Emphasis is given to the case of frozen orbits and critical inclination. Numerical simulations for hypothetical lunar artificial satellites are performed, considering that the perturbations are acting together or one at a time.
TL;DR: A Legendre quadrilateral spectral element approximation is developed for the Black-Scholes equation to price European options with one underlying asset and stochastic volatility and shows that it gives exponential convergence in the solution and the Greeks to the level of time and boundary errors in a domain of financial interest.
Abstract: We develop a Legendre quadrilateral spectral element approximation for the Black-Scholes equation to price European options with one underlying asset and stochastic volatility. A weak formulation of the equations imposes the boundary conditions naturally along the boundaries where the equation becomes singular, and in particular, we use an energy method to derive boundary conditions at outer boundaries for which the problem is well-posed on a finite domain. Using Heston's analytical solution as a benchmark, we show that the spectral element approximation along with the proposed boundary conditions gives exponential convergence in the solution and the Greeks to the level of time and boundary errors in a domain of financial interest.
TL;DR: A novel algorithm that permits the fast and accurate computation of the Legendre image moments is introduced in this paper, based on the block representation of an image and on a new image representation scheme, the Image Slice Representation (ISR) method.
Posted Content•
TL;DR: In this article, the authors show that these orthogonal bases form the Appell system and coincide with those constructed by S. Bock and K. Guerlebeck, and obtain simple expressions of elements of these bases in terms of the Legendre polynomials.
Abstract: Spaces of homogeneous spherical monogenics in dimension 3 can be considered naturally as sl(2,C)-modules. As finite-dimensional irreducible sl(2,C)-modules, they have canonical bases which are, by construction, orthogonal. In this note, we show that these orthogonal bases form the Appell system and coincide with those constructed recently by S. Bock and K. Guerlebeck. Moreover, we obtain simple expressions of elements of these bases in terms of the Legendre polynomials.
TL;DR: In this article, the authors apply variational principles in the context of geometrothermodynamics, and explore the physical meaning of geodesic curves in E as describing quasi-static processes that connect different equilibrium states.
TL;DR: In this paper, an improved Legendre-spectral method was used to approximate solutions of singular initial value problems (IVPs) of the Lane-Emden type in second-order ODEs.
Abstract: In this paper, approximate solutions of singular initial value problems (IVPs) of the Lane-Emden type in second-order ordinary differential equations (ODEs) are obtained by an improved Legendre-spectral method. The Legendre-Gauss points are used as collocation nodes and Lagrange interpolation is employed in the Volterra term. The results reveal that the method is effective, simple and accurate.
TL;DR: The orthonormal aberration polynomials for an anamorphic system with a circular pupil, obtained by the Gram-Schmidt orthogonalization of the 2D Legendre polynmials, are not separable in the two coordinates.
Abstract: The classical aberrations of an anamorphic optical imaging system, representing the terms of a power-series expansion of its aberration function, are separable in the Cartesian coordinates of a point on its pupil. We discuss the balancing of a classical aberration of a certain order with one or more such aberrations of lower order to minimize its variance across a rectangular pupil of such a system. We show that the balanced aberrations are the products of two Legendre polynomials, one for each of the two Cartesian coordinates of the pupil point. The compound Legendre polynomials are orthogonal across a rectangular pupil and, like the classical aberrations, are inherently separable in the Cartesian coordinates of the pupil point. They are different from the balanced aberrations and the corresponding orthogonal polynomials for a system with rotational symmetry but a rectangular pupil.
TL;DR: Several techniques of varying efficiency are investigated, which treat all singularities present in the triatomic vibrational kinetic energy operator given in orthogonal internal coordinates of the two distances-one angle type, and confirm that PO-DVRs should be constructed employing relaxed potentials and PO- DVRs can be useful for optimizing quadrature points for calculations applying large coordinate intervals and describing large-amplitude motions.
Abstract: Several techniques of varying efficiency are investigated, which treat all singularities present in the triatomic vibrational kinetic energy operator given in orthogonal internal coordinates of the two distances–one angle type. The strategies are based on the use of a direct-product basis built from one-dimensional discrete variable representation (DVR) bases corresponding to the two distances and orthogonal Legendre polynomials, or the corresponding Legendre-DVR basis, corresponding to the angle. The use of Legendre functions ensures the efficient treatment of the angular singularity. Matrix elements of the singular radial operators are calculated employing DVRs using the quadrature approximation as well as special DVRs satisfying the boundary conditions and thus allowing for the use of exact DVR expressions. Potential optimized (PO) radial DVRs, based on one-dimensional Hamiltonians with potentials obtained by fixing or relaxing the two non-active coordinates, are also studied. The numerical calculations employed Hermite-DVR, spherical-oscillator-DVR, and Bessel-DVR bases as the primitive radial functions. A new analytical formula is given for the determination of the matrix elements of the singular radial operator using the Bessel-DVR basis. The usually claimed failure of the quadrature approximation in certain singular integrals is revisited in one and three dimensions. It is shown that as long as no potential optimization is carried out the quadrature approximation works almost as well as the exact DVR expressions. If wave functions with finite amplitude at the boundary are to be computed, the basis sets need to meet the required boundary conditions. The present numerical results also confirm that PO-DVRs should be constructed employing relaxed potentials and PO-DVRs can be useful for optimizing quadrature points for calculations applying large coordinate intervals and describing large-amplitude motions. The utility and efficiency of the different algorithms is demonstrated by the computation of converged near-dissociation vibrational energy levels for the H+3 molecular ion.
TL;DR: In this paper, the authors employ a new integral form of similarity conditions to the error analysis of truncation techniques and present a comparative theoretical and numerical analysis of the Delta function method [15], Delta-fit method [7], and Delta-M method [21].
Abstract: Accurate radiance computations with highly peaked phase functions is a challenging problem. The developed truncation methods replace the peak of phase function using different approximations in the cone of forward scattering. The main goal of this paper is to employ a new integral form of similarity conditions to the error analysis of truncation techniques. This analysis emphasizes two main error sources of these methods from (1) truncation of Legendre series, and (2) truncation of the forward cone for peaked phase functions. The first error has an oscillating pattern and is effectively suppressed by the single scattering correction. The second, often overlooked, error manifests itself as a bias which weakly depends on the number of Legendre terms used in the solution unless it becomes comparable to the total order of Legendre expansion series. This paper presents a comparative theoretical and numerical error analysis of the Delta function method [15], Delta-fit method [7], and Delta-M method [21]. The Delta-M method, combined with the single scattering correction, is shown to provide the best overall accuracy for the intensity computations.
02 Nov 2010
TL;DR: In this article, by using the properties of Legendre polynomials, the authors proved congruences for Σ p −1 2 k = 0 (2k k m −k (mod p 2 ).
Abstract: Let p be an odd prime. In this paper, by using the properties of Legendre polynomials we prove some congruences for Σ p―1 2 k=0 ( 2k k m ―k (mod p 2 ). In particular, we confirm several conjectures of Z.W. Sun. We also pose 13 conjectures on supercongruences.
TL;DR: In this paper, a refined translation-scale Legendre moment invariants are obtained through the exact computation of original Legendre moments which completely remove approximation, and the performance of descriptors is evaluated using a set of standard images.
TL;DR: In this article, a hybrid of block-pulse and Legendre polynomials are used as a basis to expand a part of the integrand, r f ( r ), appearing in the Hankel transform integral.
TL;DR: The convergence of the single-step and multi-domain versions of the proposed Legendre-Gauss collocation method are analyzed, and it is shown that the scheme enjoys high order accuracy and can be implemented in a stable and efficient manner.
Abstract: In this paper, we introduce an efficient Legendre-Gauss collocation method for solving nonlinear delay
differential equations with variable delay.
We analyze the convergence of the single-step and multi-domain versions of the proposed method, and
show that the scheme enjoys
high order accuracy and can be implemented in a stable and efficient manner.
We also make numerical comparison with other
methods.
TL;DR: In this article, a dynamic solution for the propagation of harmonic waves in piezoelectric-piezomagnetic FGM spherical curved plates is presented based on the Legendre orthogonal polynomial series expansion approach, where material properties are assumed to vary in the direction of the thickness according to a known variation law.
Abstract: Piezoelectric-piezomagnetic functionally graded materials (FGM), with a gradual change of the mechanical, electric and magnetic properties, have great promise for different applications. Based on the Legendre orthogonal polynomial series expansion approach, a dynamic solution is presented for the propagation of harmonic waves in piezoelectric-piezomagnetic FGM spherical curved plates. The material properties are assumed to vary in the direction of the thickness according to a known variation law. The dispersion curves of the piezoelectric-piezomagnetic FGM spherical curved plate and the corresponding non-piezoelectric, non-piezomagnetic plates are calculated to show the influences of the piezoelectricity and piezomagnetism. Electric potential and magnetic potential distributions are also obtained to illustrate the different influences of the piezoelectricity and piezomagnetism. Finally, a spherical curved plate at different ratio of radius to thickness is calculated to show the influence f the ratio on th...
TL;DR: This Note fills the key gap of costate computation for Chebyshev PS methods by furthering themethod of Fahroo andRoss [21], and combines some recent results from Clenshaw– Curtis integration [27], the unification principles proposed by Fahro and Ross [1,29,30], and the new results of Gong et al.
Abstract: AMONG the various pseudospectral (PS) methods for optimal control [1], only the Legendre PS method has been mathematically proven to guarantee the feasibility, consistency, and convergence of the approximations [2–5]. As exemplified by its experimental andflight applications in national programs [6–10], it is not surprising that the Legendre PS method has become the method of choice [11–19] in both industry and academia for solving optimal control problems. Efforts to improve the Legendre PS methods by using either other polynomials [20–22] or point distributions [23,24] have not yet resulted in any rigorous framework for convergence of these approximations [24,25]. Compared to Legendre PS methods, Chebyshev PS methods [21,22] for optimal control are somewhat more attractive for a number of reasons. When a function is approximated, it is well known that a Chebyshev expansion is very close to the best polynomial approximation in the infinity norm [26,27]. In addition, Chebyshev polynomials have an attractive computational advantage in terms of the computation of Chebyshev–Gauss–Lobatto (CGL) nodes. Unlike the Legendre–Gauss–Lobatto (LGL) nodes, CGL nodes can be evaluated in closed form [26]. Thus, a Chebyshev PS method offers the possibility of rapid computation because it does not require the use of advanced numerical linear algebra techniques that are necessary for the calculation of LGL nodes [21]. A similar numerical advantage applies to the computation of the derivative via a fast Chebyshev differentiation scheme that is similar to a fastFourier-transform (FFT) computation. In the same spirit, integration is also fast because of the connection between the Clenshaw–Curtis integration and the FFT [27]. Despite these attractive properties, Chebyshev PS methods have not advanced beyond the works of [21,22]. This is, in part, due to the absence of a covector mapping theorem that is crucial for the computation of the costates and other covectors. The computation of costates and other covectors is important in solving practical optimal control problems as it provides ameans for verification and validation of the computed solution [25]. Beyond verification and validation, information about covectors can also be used to facilitate the design of guidance and control algorithms [28]. In this Note, we fill the key gap of costate computation for Chebyshev PSmethods by furthering themethod of Fahroo andRoss [21]. We do this by combining some recent results from Clenshaw– Curtis integration [27], the unification principles proposed by Fahroo and Ross [1,29,30], and the new results of Gong et al. [23].