A fully diagonalized spectral method using generalized Laguerre functions on the half line

A fully diagonalized spectral method using generalized Laguerre functions is proposed and analyzed for solving elliptic equations on the half line. We first define the generalized Laguerre functions which are complete and mutually orthogonal with respect to an equivalent Sobolev inner product. Then the Fourier-like Sobolev orthogonal basis functions are constructed for the diagonalized Laguerre spectral method of elliptic equations. Besides, a unified orthogonal Laguerre projection is established for various elliptic equations. On the basis of this orthogonal Laguerre projection, we obtain optimal error estimates of the fully diagonalized Laguerre spectral method for both Dirichlet and Robin boundary value problems. Finally, numerical experiments, which are in agreement with the theoretical analysis, demonstrate the effectiveness and the spectral accuracy of our diagonalized method.     

Stair Laguerre pseudospectral method for differential equations on the half line

A stair Laguerre pseudospectral method is proposed for numerical solutions of differential equations on the half line. Some approximation results are established. A stair Laguerre pseudospcetral scheme is constructed for a model problem. The convergence is proved. The numerical results show that this new method provides much more accurate numerical results than the standard Laguerre spectral method. Dedicated to Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 65N35, 41A10. Li-lian Wang: The work of this author is partially supported by The Shanghai Natural Science Foundation N. 00JC14057, The Shanghai Natural Science Foundation for Youth N. 01QN85 and The Special Funds for Major Specialities of Shanghai Education Committee. Ben-yu Guo: The work of this author is partially supported by The Special Funds for Major State Basic Research Projects of China G1999032804, The Shanghai Natural Science Foundation N. 00JC14057 and The Special Funds for Major Specialities of Shanghai Education Committee.     

Modified Laguerre pseudospectral method refined by multidomain Legendre pseudospectral approximation

A modified Laguerre pseudospectral method is proposed for differential equations on the half-line. The numerical solutions are refined by multidomain Legendre pseudospectral approximation. Numerical results show the spectral accuracy of this approach. Some approximation results on the modified Laguerre and Legendre interpolations are established. The convergence of proposed method is proved.     

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.     

Laguerre polynomial approach for solving linear delay difference equations

This paper presents a numerical method for the approximate solution of mth-order linear delay difference equations with variable coefficients under the mixed conditions in terms of Laguerre polynomials. The aim of this article is to present an efficient numerical procedure for solving mth-order linear delay difference equations with variable coefficients. Our method depends mainly on a Laguerre series expansion approach. This method transforms linear delay difference equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equation. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments and performed on the computer algebraic system Maple.     

Solving the 2D Maxwell equations by a Laguerre spectral method

A spectral method for solving the 2D Maxwell equations with relaxation of electromagnetic parameters is presented. The method is based on an expansion of the solution in terms of Laguerre functions in time. The operation of convolution of functions, which is part of the formulas describing the relaxation processes, is reduced to a sum of products of the harmonics. The Maxwell equations transform to a system of linear algebraic equations for the solution harmonics. In the algorithm, an inner parameter of the Laguerre transformis used. With large values of this parameter, the solution is shifted to high harmonics. This is done to simplify the numerical algorithm and to increase the efficiency of the problem solution. Results of a comparison of the Laguerre method and a finite-difference method in accuracy both for a 2D medium structure and a layered medium are given. Results of a comparison of the spectral and finite-difference methods in efficiency for axial and plane geometries of the problem are presented.     

Laguerre approximation with negative integer and its application for the delay differential equation

In this paper, two kinds of novel algorithms based on generalized Laguerre approximation with negative integer are presented to solve the delay differential equations. The algorithms differ from the spectral collocation method by the high sparsity of the matrices. Moreover, the use of generalized Laguerre polynomials leads to much simplified analysis and more precise error estimates. The numerical results indicate the high accuracy and the stability of long-time calculation of suggested algorithm.     

A New Numerical Method for the Inversion of the Laplace Transform

This paper presents a new numerical method for the inversionof the Laplace transform, where only real values of the transformare known. This method, based on the approximation of the transformby truncated series of orthogonal functions, related to theJacobi or Laguerre polynomials, avoids the loss of significantfigures inherent in earlier methods.     

A method to evaluate the Hilbert transform on (0, +∞)

The author proposes a method to approximate the Hilbert transform on the real positive semiaxis by a suitable Lagrange interpolating polynomial. The method employs truncated Gaussian rules and uses the interlacing properties of the zeros of generalized Laguerre polynomials. The error estimate in a weighted uniform norm is proved and some numerical tests show the efficacy of the proposed procedure.     

Legendre-Laguerre coupled spectral element methods for second- and fourth-order equations on the half line

Some Legendre spectral element/Laguerre spectral coupled methods are proposed to numerically solve second- and fourth-order equations on the half line. The proposed methods are based on splitting the infinite domain into two parts, then using the Legendre spectral element method in the finite subdomain and Laguerre method in the infinite subdomain. C⁰ or C¹-continuity, according to the problem under consideration, is imposed to couple the two methods. Rigorous error analysis is carried out to establish the convergence of the method. More importantly, an efficient computational process is introduced to solve the discrete system. Several numerical examples are provided to confirm the theoretical results and the efficiency of the method.     

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.     

Stokes' first problem for a Newtonian fluid in a non‐Darcian porous half‐space using a Laguerre–Galerkin method

A Laguerre–Galerkin method is proposed and analysed for the Stokes' first problem of a Newtonian fluid in a non‐Darcian porous half‐space on a semi‐infinite interval. It is well known that Stokes' first problem has a jump discontinuity on boundary which is the main obstacle in numerical methods. By reformulating this equation with suitable functional transforms, it is shown that the Laguerre–Galerkin approximations are convergent on a semi‐infinite interval with spectral accuracy. An efficient and accurate algorithm based on the Laguerre–Galerkin approximations of the transformed equations is developed and implemented. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Laguerre Tau Methods for Solving Higher-Order Ordinary Differential Equations

In this paper, we present two numerical methods for solving higher-order differential equations using the Laguerre Tau method. These methods generate linear systems, which can be solved by Gauss elimination with maximal partial pivoting strategy. Results of some numerical experiments and theoretical analysis are presented.     

New unconditionally stable scheme for telegraph equation based on weighted Laguerre polynomials

This article proposes a new unconditionally stable scheme to solve one‐dimensional telegraph equation using weighted Laguerre polynomials. Unlike other numerical schemes, the time derivatives in the equation can be expanded analytically based on the Laguerre polynomials and basis functions. By applying a Galerkin temporal testing procedure and using the orthogonal property of weighted Laguerre polynomials, the time variable can be eliminated from computations, which results in an implicit equation. After solving the equation recursively one can obtain the numerical results of telegraph equation by using the expanded coefficients. Some numerical examples are considered to validate the accuracy and stability of this proposed scheme, and the results are compared with some existing numerical schemes.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1603–1615, 2017     

Numerical Methods for Approximating Continuous Probability Density Functions, over [0, {infty}), Using Moments

Three numerical methods are presented for the reconstructionof a continuous probability density function f(x) from givenvalues of the moments of the distribution. The first methodis obtained by assuming that f(x) may be expanded as an infiniteseries of generalized Laguerre polynomials . The use of ordinary Laguerre polynomials, corresponding to theparticular choice = 0, is related to a second method involvingthe numerical inversion of a Laplace transform. In the thirdmethod the principle of maximization of entropy, subject tothe known moment constraints, is used to reconstruct f(x). Thetype of fit to be expected from each method is illustrated bynumerical examples.     

LAGUERRE PSEUDOSPECTRAL METHOD FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

AbstractThe Laguerre Gauss-Radau interpolation is investigated. Some approximation results are obtained. As an example, the Laguerre pseudospectral scheme is constructed for the BBM equation. The stability and the convergence of proposed scheme are proved. The numerical results show the high accuracy of this approch.     

GI/G/1 Interdeparture time and queue-length distributions via the Laguerre transform

Queue length and interdeparture distributions for GI/G/1 are obtained using the Laguerre function expansion of the waiting time distribution. The expansion of the steady state waiting time distribution is obtained here by solving a small set of linear equations in the Laguerre function expansion coefficients. Examples show the accuracy of the results and illustrate purely numerical techniques for obtaining the necessary expansions of the arrival and service distributions.     

Numerical approximation of weakly singular integrals on the half line

This paper deals with the numerical approximation of a weakly singular integral transform by means of Laguerre nodes. Error estimates in a weighted uniform norm and some numerical tests are given.     

Application of Laguerre matrix polynomials to the numerical inversion of Laplace transforms of matrix functions

This paper presents an application of Laguerre matrix polynomial series to the numerical inversion of Laplace transforms of matrix functions.