Error Analysis of the Strang Time-Splitting Laguerre–Hermite/Hermite Collocation Methods for the Gross–Pitaevskii Equation

The aim of this paper is to carry out a rigorous error analysis for the Strang splitting Laguerre–Hermite/Hermite collocation methods for the time-dependent Gross–Pitaevskii equation (GPE). We derive error estimates for full discretizations of the three-dimensional GPE with cylindrical symmetry by the Strang splitting Laguerre–Hermite collocation method, and for the d-dimensional GPE by the Strang splitting Hermite collocation method.     

The eigenvalues of Hermite and rational spectral differentiation matrices

Summary We derive expressions for the eigenvalues of spectral differentiation matrices for unbounded domains. In particular, we consider Galerkin and collocation methods based on Hermite functions as well as rational functions (a Fourier series combined with a cotangent mapping). We show that (i) first derivative matrices have purely imaginary eigenvalues and second derivative matrices have real and negative eigenvalues, (ii) for the Hermite method the eigenvalues are determined by the roots of the Hermite polynomials and for the rational method they are determined by the Laguerre polynomials, and (iii) the Hermite method has attractive stability properties in the sense of small condition numbers and spectral radii.     

Efficient Laguerre and Hermite Spectral Methods for Odd-Order Differential Equations in Unbounded Domains

Laguerre dual-Petrov-Galerkin spectral methods and Hermite Galerkin spectral methods for solving odd-order differential equations in unbounded domains are proposed. Some Sobolev bi-orthogonal basis functions are constructed which lead to the diagonalization of discrete systems. Numerical results demonstrate the effectiveness of the suggested approaches.     

Hardy's inequalities for Hermite and Laguerre expansions

Hardy's inequalities are proved for higher-dimensional Hermite and special Hermite expansions of functions in Hardy spaces. Inequalities for multiple Laguerre expansions are also deduced.

     

The bilinear Hilbert transform acting on Hermite and Laguerre functions

We obtain several formulas for the action of the bilinear Hilbert transform on pairs of Hermite and Laguerre functions. The results can be given as linear combinations of products of Hermite or Laguerre functions. We show also that for the generalized bilinear Hilbert transforms the results cannot be expressed in such a simple way.     

Asymptotics on Laguerre or Hermite polynomial expansions and their applications in Gauss quadrature

In this paper, we present asymptotic analysis on the coefficients of functions expanded in forms of Laguerre or Hermite polynomial series, which shows the decay of the coefficients and derives new error bounds on the truncated series. Moreover, by applying the asymptotics, new estimates on the errors for Gauss–Laguerre, Radau–Laguerre and Gauss–Hermite quadrature are deduced. These results show that Gauss–Laguerre-type and Gauss-Hermite-type quadratures are nearly of same convergence rates.     

A Collocation Finite Element Method for Integration of the Boundary Layer Equations

The author proposes a collocation finite element procedure using the Hermite basis functions. The particular choice of the trial functions gives a physical interpretation to all the nodal values. In order to demonstrate the versatility of collocation, three classical problems from boundary layer theory are solved numerically. The calculations verify that the method can produce accurate information at low cost.     

Bilateral generating functions and operational methods

Bilateral generating functions are those involving products of different types of polynomials. We show that operational methods offer a powerful tool to derive these families of generating functions. We study cases relevant to products of Hermite polynomials with Laguerre, Legendre and other polynomials. We also propose further extensions of the method which we develop here.     

Hagedorn Wavepackets in Time-Frequency and Phase Space

The Hermite functions are an orthonormalbasis of the space of square integrable functions with favourable approximation properties. Allowing for a flexible localization in position and momentum, the Hagedorn wavepackets generalize the Hermite functions also to several dimensions. Using Hagedorn’s raising and lowering operators, we derive explicit formulas and recurrence relations for the Wigner and FBI transform of the wavepackets and show their relation to the Laguerre polyomials.     

Application of the frontal algorithm in the collocation method

The author reports on a numerical experimentation with the collocation finite element procedure using Hermite basis functions and the frontal elimination technique to solve some large-scale problems where up to 1000 linear equations are involved. Several test cases, including some applications to engineering problems, are presented. The implementation of the frontal technique applied to collocation is discussed to some extent.     

Asymptotic Approximations to the Nodes and Weights of Gauss–Hermite and Gauss–Laguerre Quadratures

Asymptotic approximations to the zeros of Hermite and Laguerre polynomials are given, together with methods for obtaining the coefficients in the expansions. These approximations can be used as a stand‐alone method of computation of Gaussian quadratures for high enough degrees, with Gaussian weights computed from asymptotic approximations for the orthogonal polynomials. We provide numerical evidence showing that for degrees greater than 100, the asymptotic methods are enough for a double precision accuracy computation (15–16 digits) of the nodes and weights of the Gauss–Hermite and Gauss–Laguerre quadratures.     

Integral inequalities for some special functions

Several integral inequalities for the classical hypergeometric, confluent hypergeometric, and confluent hypergeometric limit functions are given. The related results for Bessel and Whittaker functions as well as for Laguerre, Hermite, and Jacobi polynomials are discussed.     

High-precision Gauss–Turán quadrature rules for Laguerre and Hermite weight functions

Procedures and corresponding Matlab software are presented for generating Gauss–Turán quadrature rules for the Laguerre and Hermite weight functions to arbitrarily high accuracy. The focus is on the solution of certain systems of nonlinear equations for implicitly defined recurrence coefficients. This is accomplished by the Newton–Kantorovich method, using initial approximations that are sufficiently accurate to be capable of producing n-point quadrature formulae for n as large as 42 in the case of the Laguerre weight function, and 90 in the case of the Hermite weight function.     

On regularity of twisted spherical means and special Hermite expansions

Regularity properties of twisted spherical means are studied in terms of certain Sobolev spaces defined using Laguerre functions. As an application we prove a localisation theorem for special Hermite expansions.     

New Rational Interpolation Basis Functions on the Unbounded Intervals and Their Applications

Based on the rational system of Legendre rational functions, we construct two set of new interpolation basis functions on the unbounded intervals. Their explicit expressions are derived, and fast and stable algorithms are provided for computing the new basis functions. As applications, new rational collocation methods based on these new basis functions are proposed for solving various second-order differential equations on the unbounded domains. Numerical experiments illustrate that our new methods are more effective and stable than the existing collocation methods.     

On multiindices and multivariables presentation of the Voigt functions

This paper aims at presenting multiindices and multivariables study of the unified (or generalized) Voigt functions which play an important rôle in the several diverse field of physics such as astrophysical spectroscopy and the theory of neutron reactions. Some expressions (representations) of these functions are given in terms of familiar special functions of multivariables. Further representations and series expansions involving multidimensional classical polynomials (Laguerre and Hermite) of mathematical physics are established.     

The Combinatorics of Laguerre,Charlier, and Hermite Polynomials

This paper describes several combinatorial models for Laguerre, Charlier, and Hermite polynomials, and uses them to prove combinatorially some classical formulas. The so-called “Italian limit formula” (from Laguerre to Hermite), the Appel identity for Hermite polynomials, and the two Sheffer identities for Laguerre and Charlier polynomials are proved. We also give bijective proofs of the three-term recurrences. These three families form the bottom triangle in R. Askey's chart classifying hypergeometric orthogonal polynomials.     

Some progress in spectral methods

In this paper,we review some results on the spectral methods.We frst consider the Jacobi spectral method and the generalized Jacobi spectral method for various problems,including degenerated and singular diferential equations.Then we present the generalized Jacobi quasi-orthogonal approximation and its applications to the spectral element methods for high order problems with mixed inhomogeneous boundary conditions.We also discuss the related spectral methods for non-rectangular domains and the irrational spectral methods for unbounded domains.Next,we consider the Hermite spectral method and the generalized Hermite spectral method with their applications.Finally,we consider the Laguerre spectral method and the generalized Laguerre spectral method for many problems defned on unbounded domains.We also present the generalized Laguerre quasi-orthogonal approximation and its applications to certain problems of non-standard type and exterior problems.     

Incomplete 2D Hermite polynomials: properties and applications

Incomplete forms of two-variable two-index Hermite polynomials are introduced. Their link with Laguerre polynomials is discussed and it is shown that they are a useful tool to study quantum mechanical harmonic oscillator entangled states. The possibility of developing the theory of complete 2D Hermite polynomials from the point of view of the incomplete forms is analyzed too. The orthogonality properties of the associated harmonic-oscillator functions are finally discussed.