Laguerre Approximations Research Articles

Convergence analysis of Laguerre approximations for analytic functions

Laguerre spectral approximations play an important role in the development of efficient algorithms for problems in unbounded domains. In this paper, we present a comprehensive convergence rate analysis of Laguerre spectral approximations for analytic functions. By exploiting contour integral techniques from complex analysis, we prove that Laguerre projection and interpolation methods of degree n n converge at the root-exponential rate O ( exp ⁡ ( − 2 ρ n ) ) O(\exp (-2\rho \sqrt {n})) with ρ > 0 \rho >0 when the underlying function is analytic inside and on a parabola with focus at the origin and vertex at z = − ρ 2 z=-\rho ^2 . As far as we know, this is the first rigorous proof of root-exponential convergence of Laguerre approximations for analytic functions. Several important applications of our analysis are also discussed, including Laguerre spectral differentiations, Gauss-Laguerre quadrature rules, the scaling factor and the Weeks method for the inversion of Laplace transform, and some sharp convergence rate estimates are derived. Numerical experiments are presented to verify the theoretical results.

Elastic properties of Laguerre approximations of random foams

Stochastic models are valuable tools to study the mechanical behaviour of foams whose characteristic features include random cellular morphology and cells of different size (polydispersity). Random Laguerre tessellations, additively weighted generalisations of Voronoi tessellations, possess these features which makes them natural models for foams. This work studies how well a random foam can be represented by a Laguerre tessellation. For this means the Surface Evolver is used to simulate random, polydisperse soap froths which are subsequently reconstructed by Laguerre tessellations. By design, the resulting Laguerre approximations resemble the cellular morphology of the underlying froth. The Young’s modulus of low-density solid foams with open cells is then calculated for random, polydisperse foams based on Laguerre approximations and simulated soap froths. For the special case of monodisperse foams, we compare Laguerre approximated foams with the well known Kelvin and Weaire–Phelan foams.

Problems in the Calculus of Variations on Unbounded Intervals—Fourier–Laguerre Analysis and Approximations

In this paper, we consider a class of variational problems on an unbounded interval of the real axis. This type of problems arises, e.g., in quantum mechanics and asymptotic controllability. The problem is treated in a Hilbert space setting with uniformly and non-uniformly weighted Sobolev spaces as state spaces. We provide sufficient conditions that a function from a weighted Sobolev space can be expanded into a Fourier–Laguerre series converging together with its distributional derivative pointwisely and uniformly. With this result, the considered variational problem is transformed into a problem in the sequence space of Fourier–Laguerre coefficients. We develop a Fourier–Laguerre method in these spaces in order to construct a polynomial approximation scheme for the solution of this problem.

Van der Waals Interactions Between Two Hydrogen Atoms: The Slater--Kirkwood Method Revisited

We examine a technique of Slater and Kirkwood [Phys. Rev., 37 (1931), 682] which provides an exact resolution of the asymptotic behavior of the van der Waals attraction between two hydrogen atoms. We modify their technique to make the problem more tractable analytically and more easily solvable by numerical methods. Moreover, we prove rigorously that this approach provides an exact solution for the asymptotic electron correlation. The proof makes use of recent results [Anapolitanos, On van der Waals Forces, Ph.D. thesis, University of Toronto, Toronto, 2011] (see also [Anapolitanos and Sigal, Long Range Behaviour of van der Waals Force, arXiv.org, 2013]. We provide visual representations of the asymptotic electron correlation (entanglement) based on the use of Laguerre approximations.

Laguerre pseudospectral approximation to the Thomas–Fermi equation

We propose an iterative method to solve the nonlinear Thomas–Fermi equation based on Laguerre pseudospectral approximation. Different from the direct use of pseudospectral methods for solving the problem in the literature, we represent the solution of the Thomas–Fermi equation as sum of two parts due to its singularity at the origin. Both parts regard the initial slope as a parameter unknown to be determined. One “singular” part is a power-series expansion. The other “smooth” part satisfies a nonlinear two-point boundary value problem. We use a Laguerre pseudospectral method to discretize and solve it numerically by the Newton iteration. Moreover, an introduced new boundary condition is employed innovatively to construct an outer iteration to determine the initial slope. We show numerically this separation of solution as two parts is necessary for improvement of accuracy by direct Laguerre approximations of the Thomas–Fermi equation. Comparisons of our numerical results with those obtained by other methods show effectiveness and accuracy of our approach.

Numerical Solution for the Fractional Wave Equation Using Pseudo-Spectral Method Based on the Generalized Laguerre Polynomials

In this paper, an efficient numerical method is considered for solving the fractional wave equation (FWE). The fractional derivative is described in the Caputo sense. The method is based on Laguerre approximations. The properties of Laguerre polynomials are utilized to reduce FWE to a system of ordinary differential equations, which is solved by the finite difference method. An approximate formula of the fractional derivative is given. Special attention is given to study the convergence analysis and estimate an error upper bound of the presented formula. Numerical solutions of FWE are given and the results are compared with the exact solution.

Generalized Laguerre approximations and spectral method for the Camassa–Holm equation

Generalized Laguerre approximations and spectral method for the Camassa–Holm equation

Pade and Laguerre approximations applied to the active queue management model of internet protocol

This paper considers the problem of pure time delay in the Active Queue Management model, the delay element is approximated by using Pade and Laguerre presentations. Hankel norm is used to determine the order of truncation. Quantitative error bounds on delay approximations are given and calculated. The approximated models are compared with the exact model with delay to determine the efficiency of the presented methods. The example presented in illustration is the model of Congestion control in Internet Protocol.

The Laguerre spectral method for solving Neumann boundary value problems

In this paper, we propose a Laguerre spectral method for solving Neumann boundary value problems. This approach differs from the classical spectral method in that the homogeneous boundary condition is satisfied exactly. Moreover, a tridiagonal matrix is employed, instead of the full stiffness matrix encountered in the classical variational formulation of such problems. For analyzing the numerical errors, some basic results on Laguerre approximations are established. The convergence is proved. The numerical results demonstrate the efficiency of this approach.

H-infinity error bounds in approximating time-delay systems

In this paper, upper and lower bounds on approximating time-delay systems are proposed. Bounds on the infinity norm of the weighted error are obtained when the approximating function is a general rational function, an all-pass function or Pad and Laguerre approximations. In addition, approximations of the weighted errors for both Pade and Laguerre approximating functions are developed. Examples are presented in illustration.

H-infinity error bounds in approximating time-delay systems

In this paper, upper and lower bounds on approximating time-delay systems are proposed. Bounds on the infinity norm of the weighted error are obtained when the approximating function is a general rational function, an all-pass function or Pade and Laguerre approximations. In addition, approximations of the weighted errors for both Pade and Laguerre approximating functions are developed. Examples are presented in illustration.

Spectral Laguerre method for the approximate solution of time dependent problems

This paper presents some efficient algorithms based on the spectral Laguerre approximations of temporal derivatives for time dependent problems.

Laguerre and Kautz shift approximations of delay systems

Certain Laguerre and Kautz shift-induced, finite-dimensional approximations of an important class of delay systems are studied. A substantially complete set of lower and upper H infinity and H2/L2 error bounds is given. Furthermore, exact asymptotic H infinity error formulas are provided. The Laguerre formula provides surprisingly good finite-dimensional models of delay systems in the most transparent manner possible.

Reduced-order models using optimal Laguerre approximations

The letter describes a model order-reduction prodcedure that is based on an optimally truncated Laguerre series expansion. Some of the advantages of the prodcedure are that it simultaneously minimises the error in the impulse response and the energy between the original system and the low-order model.

Gaussian-filter approximation

A new approximation to the Gaussian-filter response is presented that uses a power-series-economisation technique with Cheby̅shev polynomials of the first kind. For high-order filters, the new method appears to yield a closer approximation to the Gaussian response than is given by the Taylor or Laguerre approximations.

Choice of time scale in Laguerre approximations using signal measurements

This correspondence investigates the choice of time scale or pole location that minimizes the L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> error in approximating a given signal with a Laguerre function basis. For a pair of measurements related to how rapidly the signal decays, a pole location is derived that minimizes the maximum error encountered for the equivalence class of signals with these measurements.