Showing papers on "Legendre polynomials published in 1993"

Calculus of variations and optimal control theory

Abstract: The Euler equation. A necessary condition for the solution of (16.1). An alternative form of the Euler equation. The Legendre condition. A necessary condition for the solution of (16.1). Sufficient conditions for the solution of (16.1). Transversality condition. Adding condition (16.5) gives sufficient conditions.

Legendre pseudospectral viscosity method for nonlinear conservation laws

Abstract: In this paper, the Legendre spectral viscosity (SV) method for the approximate solution of initial boundary value problems associated with nonlinear conservation laws is studied. The authors prove that by adding a small amount of SV, bounded solutions of the Legendre SV method converge to the exact scalar entropy solution. The convergence proof is based on compensated compactness arguments, and therefore applies to certain $2 \times 2$ systems. Finally, numerical experiments for scalar as well as the one-dimensional system of gas dynamics equations are presented, which confirm the convergence of the Legendre SV method. Moreover, these numerical experiments indicate that by post-processing the SV approximation, one can recover the entropy solution within spectral accuracy.

Projection Pursuit Indexes Based on Orthonormal Function Expansions

Abstract: Projection pursuit describes a procedure for searching high-dimensional data for “interesting” low-dimensional projections via the optimization of a criterion function called the projection pursuit index. By empirically examining the optimization process for several projection pursuit indexes, we observed differences in the types of structure that maximized each index. We were especially curious about differences between two indexes based on expansions in terms of orthogonal polynomials, the Legendre index, and the Hermite index. Being fast to compute, these indexes are ideally suited for dynamic graphics implementations. Both Legendre and Hermite indexes are weighted L 2 distances between the density of the projected data and a standard normal density. A general form for this type of index is introduced that encompasses both indexes. The form clarifies the effects of the weight function on the index's sensitivity to differences from normality, highlighting some conceptual problems with the Legen...

Discrete variable representations of differential operators

Abstract: By making use of known properties of orthogonal polynomials the discrete variable representation (DVR) method [J. C. Light, I. P. Hamilton, and J. V. Lill, J. Chem. Phys. 82, 1400 (1985)] has been rederived. Simple analytical formulas have been obtained for the matrix elements of DVRs of differential operators which may appear in the rovibrational Hamiltonian of a molecule. DVRs corresponding to Hermite, Laguerre, generalized Laguerre, Legendre, and Jacobi polynomial bases and to the Lanczos basis for Morse oscillator, that is, to basis sets often used in calculating rovibrational energy levels, have been discussed.

Variational theory for spatial rods

Abstract: The simplest theory of spatial rods is presented in a variational setting and certain necessary conditions for minimizers of the potential energy are derived. These include the Weierstrass and Legendre inequalities, which require that the vector describing curvature and twist belong to a domain of convexity of the strain energy function.

The achievable accuracy in estimating the instantaneous phase and frequency of a constant amplitude signal

Abstract: The approach is based on modeling the signal phase by a polynomial function of time on a finite interval. The phase polynomial is expressed as a linear combination of the Legendre basis polynomials. First, the Cramer-Rao bound (CRB) of the instantaneous phase and frequency of constant-amplitude polynomial-phase signals is derived. Then some properties of the CRBs are used to estimate the order of magnitude of the bounds. The analysis is extended to signals whose phase and frequency are continuous but not polynomial. The CRB can be achieved asymptotically if the estimation of the phase coefficients is done by maximum likelihood. The maximum-likelihood estimates are used to show that the achievable accuracy in phase and frequency estimation is determined by the CRB of the polynomial coefficients and the deviation of true phase and frequency from the polynomial approximations. >

Asymptotic approximations for symmetric elliptic integrals

Abstract: Symmetric elliptic integrals, which have been used as replacements for Legendre's integrals in recent integral tables and computer codes, are homogeneous functions of three or four variables. When some of the variables are much larger than the others, asymptotic approximations with error bounds are presented. In most cases they are derived from a uniform approximation to the integrand. As an application the symmetric elliptic integrals of the first, second, and third kinds are proved to be linearly independent with respect to coefficients that are rational functions.

Rotational behavior in intermediate energy heavy ion collisions

Abstract: The rotational behavior together with in-plane collective flow for the intermediate energy $^{40}\mathrm{Ar}$${+}^{27}$Al reaction is investigated by analyzing the rapidity-dependent azimuthal distributions with the Boltzmann-Uehling-Uhlenbeck (BUU) model. The azimuthal distributions are fitted by a Legendre polynomial expansion up to the second order. By incorporating the uncertainties in the experimental reaction plane determination into our calculations, quantitative agreement between the calculations and data is obtained. It is found that the rotational behavior at midrapidity depends strongly on the impact parameter. Information about the in-plane collective flow is also extracted from the azimuthal distributions. Both the rotational behavior and the in-plane collective flow depend strongly on the in-medium nucleon-nucleon cross section and nuclear equation of state.

Eigenvalue solution for the ion-collisional effects on ion-acoustic and entropy waves

Abstract: The linearized ion Fokker–Planck equation is solved as an eigenvalue problem under the condition of collisionless electrons in the quasineutral limit (φ=0) for ionization‐temperature ratios, ZTe/Ti=2, 4, and 8 for entropy waves and ionization‐temperature ratios, ZTe/Ti=4, 8, 16, 32, 48, 64, and 80 for ion‐acoustic waves. The perturbed ion distribution function for the ion‐acoustic and entropy waves is formed from a Legendre polynomial expansion of eigenvectors and can be used to calculate collisionally dependent macroscopic quantities in the plasma such as gamma (Γ=Cp/Cv), the ratio of specific heats, and the ion thermal conductivity (κi).

Inelastic F–H2 scattering

Abstract: We present a first experimental study of inelastic, rotational excitation cross-sections in a reactive collision. By preliminary model calculations we show that these measurements provide an ideal method for probing the anisotropy in the rate-determining entrance-channel valley of the potential-energy surface for the F–H2 reaction. A comparison with the non-reactive Ne–D2, H2 scattring system reveals the importance of a surprisingly large fourth-order Legendre term in the anisotropic expansion of the F–H2 potential. This particular shape feature is related to the incipient chemical reaction and is absent in the more ellipsoidal shape of non-reactive potentials for rare gas–H2 systems.

Atomic integrals containing r 23λr 13μr 12ν with λ, μ, ν ≥ −2

Abstract: In this paper, the efficient evaluation of the atomic integrals I=∫r 1 a r 2 b r 3 c r λ 23 r μ 13 r υ 12 e -αr1-βr2-γr3 dτ one or two factors r ij -2 is described. These integrals are necessary for a lower-bound calculation for Li-like systems using the method of variance minimization or Temple's formula

Magnetic field evaluation for disk conductors

Abstract: A method for calculating the magnetic field from a sector or an entire disk conductor in which a radial current flows is described. New analytic expressions for the components of the magnetic vector potential and field are given in closed form. These consist of suitably polynomial, inverse trigonometric and logarithmic functions of the coordinates of the point considered and of the dimensions of the conductor, and of Legendre's elliptic integrals of the first, second and third kind, having the same independent variables. The relations presented can be usefully adopted as efficient and time-saving algorithms in magnetic field and eddy current analysis. >

A generalized Legendre polynomial/sparse matrix approach for determining the distribution function in non-polar semiconductors

Abstract: Until recently, the Legendre polynomial (LP) expansion method for solving the Boltzmann transport equation was limited to the use of two or three Legendre polynomials. In this work we generalize the method to include an arbitrarily high order LP expansion. The expansion method consists of representing the angular dependence of the distribution function about the field direction in terms of an infinite series of Legendre polynomials with unknown coefficients. The expansion is then substituted into the Boltzmann transport equation. With the use of orthogonality and the LP recurrence relations, an infinite system of equations is then generated from the original Boltzmann equation. This system is then solved numerically, using sparse matrix algebra, for the unknown coefficients of the LP expansion. Once the coefficient are determined, the complete distribution function is readily constructed. In an example calculation the Boltzmann equation is solved to 40th order of the LP expansion. Finally, resulting values for the energy distribution, as well as average energy and average velocity, are shown to agree with Monte Carlo simulation results.

Stress-strain state of non-thin plates and shells. Generalized theory (survey)

Abstract: Thus, this part of our survey has presented the main approaches that have been taken to the construction of two-dimensional (in terms of the space coordinates) equations of a generalized theory of plates and shells. The solutions of these equations represent a certain approximation of the solution of the initial three-dimensional problem. They are based on expansion of the sought functions into Fourier series in Legendre polynomials of the thickness coordinate. Studies completed on the basis of the given variants of plate and shell theory were systematized and analyzed. In terms of the method of its construction, the theory involves a regular process of replacing the solution of the three-dimensional problem by the solution (or sequence of solutions) of two-dimensional boundary-value problems or initial-boundary-value problems. Numerical results illustrating the convergence of the successive approximation were presented. It should be noted that to make comparison with the results of classical or applied theories, several of the studies cited here presented solutions of problems for thin plates and shells with allowance only for the initial terms of expansions of the stress and displacement components into base functions (Legendre polynomials).

Spatially transient hot electron distributions in silicon determined from the chambers path integral solution of the Boltzmann transport equation

Abstract: A new technique is presented for determining the spatially transient hot electron distribution function in silicon. The Boltzmann transport equation is solved deterministically using a modificationm of the Chambers path integral solution. This formulation is shown to be relatively simple to implement, retains all essential physics and avoids the use of Legendre polynomial expansions. The accuracy of the technique is confirmed by comparison with Monte Carlo simulations. This method is computationally very efficient and should find applications in the study of hot carrier effects in silicon.

A legendre technique for solving time-varying linear quadratic optimal control problems

Abstract: A method for the optimal control of linear time-varying systems with a quadratic cost functional is proposed. The state and control variables are expanded in the shifted Legendre series, and an algorithm is provided for approximating the system dynamics, boundary conditions and performance index. The necessary condition of optimality is then derived as a system of linear algebraic equations. Numerical examples are included to demonstrate the validitiy and applicability of the technique.

Thickness Expansions for Higher-Order Effects in Vibrating Cylindrical Shells

Abstract: By expanding displacements in spatial coordinates, integral expressions for strain and kinetic energy are converted to quadratic sums involving time dependent generalized coordinates. Hamilton's principle provides ordinary differential equations for these coordinates. This viewpoint yields physical insight into the mechanisms of energy storage and avoids the geometrically thin assumption. A set of Legendre polynomials multiplied by a radial factor represent the radial dependences of displacement components, while circumferential variations are represented by sinusoidal functions

Error analysis of the Tau method: dependence of the approximation error on the choice of perturbation term

Abstract: We consider a system of ordinary differential equations with constant coefficients and deduce asymptotic estimates for the Tau Method approximation error vector per step for different choices of the perturbation term Hn(x). The cases considered are Legendre polynomials, Chebyshev polynomials, powers of x and polynomials of the form ( x 2 − r 2 ) n , −r ⩽ x ⩽ r . The first two are standard choices for the Tau Method, for Chebyshev and Legendre series expansion techniques and also for collocation; the third one realizes the classical power series expansion techniques in the framework of the Tau Method and the last is related to the trial functions used in weighted residuals methods; we shall refer to it as the weighted residuals choice. We show that the resulting Tau Method implementations can be arranged into the following scale of increasing error estimates at the end point x = r : Legendre For the interesting case of Legendre Tau approximations, we offer upper and lower error bounds for the end point of the interval of approximation. In particular, this last estimates solve a conjecture on increased accuracy at the end point of the interval of approximation formulated by Lanczos in 1956. Such conjecture has equivalent forms for other polynomial methods for the numerical solution of differential equations. Although formulated in the convenient framework of the recursive Tau Method (see Ortiz [1]), the results given here apply, without essential modifications, to Chebyshev or Legendre series expansion techniques for differential equations, collocation and spectral methods. We give numerical examples which confirm the sharpness of the lemmas and theorems given in this paper. Finally, we discuss in an example the application of our results to the analysis of singularly perturbed differential equations.

The legendre polynomials under a left definite energy norm

Abstract: The Legendre operator ly = -((1—x 2)y′)′ + ¼y, under boundary constraints lim x→±1 (1—x 2)y′(x) = 0 is shown to be self-adjoint in a space with inner product The spectrum is discrete {(n + 1/22)}∞ n=0 with eigenfunctions {P n(x)}∞ n=0. The spectral resolutions (eigenfunction expansions), well known in L 2(-1, 1), hold in the new space as well.

Error bounds for Gauss-Kronrod quadrature formulae of analytic functions

Abstract: We consider the Gauss-Kronrod quadrature formula for the Legendre weight function. On certain spaces of analytic functions its error term is a continuous linear functional. We derive easy to compute estimates for the norm of the error functional, which lead to bounds for the error functional itself. The efficiency of these bounds is illustrated with some numerical examples.

On the computation of discrete Legendre polynomial coefficients

Abstract: A new and fast method to find the discrete Legendre polynomial (DLP) coefficients is presented. The method is based on forming a simple matrix using addition only and then multiplying two elements of the matrix to compute the DLP coefficients.

Higher Order Modeling of Plates by p ‐Version of Finite Element Method

Abstract: A new p‐version finite element based on higher order theory is developed for the two‐dimensional modeling of thin‐to‐thick plates undergoing three‐dimensional (3‐D) deformations. The special case of cylindrical bending is also considered. In each case, the displacement is expressed as the product of two functions—one in terms of the out‐of‐plane coordinate and the other in terms of in‐plane coordinates. The displacement fields are based on integrals of Legendre polynomials. The computer implementation allows arbitrary variations of p‐level in the plane of the plate and in the thickness direction up to a maximum value of 8. A number of test problems on thin‐to‐thick isotropic and orthotropic plates are considered to evaluate the performance of the proposed scheme. Convergence characteristics of pointwise values of displacement and stress, as well as that of total potential energy, are studied. The superior performance of the scheme is established by comparing the results with those in the published literature.

Evaluation of angular integrals using matrix algebra: Applications to the alpha -decay integrals Kll'm and the scattering of deformed nuclei.

Abstract: In many physical problems one is faced with the calculation of matrix elements of the form 〈lm|f(\ensuremath{\theta})|l\ensuremath{'}m〉. We use matrix algebra to derive a powerful technique for the numerical evaluation of such integrals. We obtain two different integration formulas: one which may be used if f has no particular symmetry and a second which exploits reflection symmetry of this operator. Our final results are essentially generalizations of the Gauss quadrature formula especially suited to the evaluation of matrix elements of the above form. We demonstrate this with an application to the angular part of the barrier transmission coefficient for the \ensuremath{\alpha} decay of a heavy nucleus. The derivation involves a unitary transformation to the eigenstates of the Legendre polynomials ${\mathrm{P}}_{1}$ or ${\mathrm{P}}_{2}$ in a truncated angular momentum space. The same transformation also relates the eigenchannels of the adiabatic treatment of the scattering of deformed nuclei to the corresponding physical channels.