Search results for "Laguerre polynomials"
showing 9 items of 9 documents
Asymptotics for thenth-degree Laguerre polynomial evaluated atn
1992
We investigate the asymptotic behaviour of ? n (n),n?? where ? n (x) denotes the Laguerre polynomial of degreen. Our results give a partial answer to the conjecture ?? n (n)>1 forn>6, made in 1984 by van Iseghem. We also show the connection between this conjecture and the continued fraction approximants of $$6\sqrt {{3 \mathord{\left/ {\vphantom {3 \pi }} \right. \kern-\nulldelimiterspace} \pi }} $$ .
A Computational Technique for Solving Singularly Perturbed Delay Partial Differential Equations
2021
Abstract In this work, a matrix method based on Laguerre series to solve singularly perturbed second order delay parabolic convection-diffusion and reaction-diffusion type problems involving boundary and initial conditions is introduced. The approximate solution of the problem is obtained by truncated Laguerre series. Moreover convergence analysis is introduced and stability is explained. Besides, a test case is given and the error analysis is considered by the different norms in order to show the applicability of the method.
Analytical solution for multisingular vortex Gaussian beams: The mathematical theory of scattering modes
2016
We present a novel procedure to solve the Schr\"odinger equation, which in optics is the paraxial wave equation, with an initial multisingular vortex Gaussian beam. This initial condition has a number of singularities in a plane transversal to propagation embedded in a Gaussian beam. We use the scattering modes, which are solutions of the paraxial wave equation that can be combined straightforwardly to express the initial condition and therefore permit to solve the problem. To construct the scattering modes one needs to obtain a particular set of polynomials, which play an analogous role than Laguerre polynomials for Laguerre-Gaussian modes. We demonstrate here the recurrence relations need…
Indefinite integrals for some orthogonal polynomials obtained using integrating factors
2020
A method has been presented recently for deriving integrals of special functions using two kinds of integrating factor for the homogeneous second-order linear differential equations which many spec...
Hybrid approximation for solutions of high-order integro-differential equations including variable delay
2020
Abstract In this study, a numerical technique with hybrid approximation is developed for solving high-order linear integro-differential equations including variable delay under the initial conditions. These type of problems are of applications in mathematical physics, mechanics, natural sciences, electronics and computer science. The aim of this work is to investigate an approximation with the matrix forms of Taylor and Laguerre polynomials along with standard collocation points. By the reduction of the solution of this problem with regard to the matrix relations, the solution of a system of algebraic equations has been obtained. The usefulness of this algorithm has been demonstrated by num…
Adaptive estimation of Laguerre models with time-varying delay
2000
Abstract An Orthonormal Basis Functions (OBF) approach is effectively used in adaptive parameter estimation of linear(ized) open-loop stable, possibly nonminimum phase plants with time-varying delay. In particular, discrete Laguerre models are considered in detail. A special attention is paid to the numerical conditioning issue in case of ’poor’ excitation of a plant under control, where OBF models are of particular value. Closed-loop predictive control simulations confirm the usefulness of adaptive OBF modelling, especially for systems with time-varying delays.
Creating highly squeezed vacua in hybrid Laguerre-Gauss modes
2009
In this communication we study the above threshold quantum properties of a degenerate optical parametric oscillator (DOPO) tuned to a given transverse mode family at the signal frequency. We will show that under this configuration DOPOs are versatile sources of nonclassical light, in which one could be able to generate highly squeezed vacua with the non trivial shapes of Hybrid Laguerre-Gauss modes.
Laguerre Matrix-Collocation Method to Solve Systems of Pantograph Type Delay Differential Equations
2020
In this study, an improved matrix method based on collocation points is developed to obtain the approximate solutions of systems of high-order pantograph type delay differential equations with variable coefficients. These kinds of systems described by the existence of linear functional argument play a critical role in defining many different phenomena and particularly, arise in industrial applications and in studies based on biology, economy, electrodynamics, physics and chemistry. The technique we have used reduces the mentioned delay system solution with the initial conditions to the solution of a matrix equation with the unknown Laguerre coefficients. Thereby, the approximate solution is…
Topological protection of highly entangled non-Gaussian two-photon states
2021
Abstract We study theoretically the evolution of entangled non-Gaussian two-photon states in disordered topological lattices. Specifically, we consider spatially entangled two-photon states, modulated by Laguerre polynomials up to the 3rd order, which feature ring-shaped spatial and spectral correlation patterns. Such states are discrete analogs of photon-subtracted squeezed states, which are ubiquitous in optical quantum information processing or sensing applications. We find that, in general, a higher degree of entanglement coincides with a loss of topological protection against disorder, this is in line with previous results for Gaussian two-photon states. However, we identify a particul…