Generalized Laguerre Interpolation and Pseudospectral Method for Unbounded Domains

Abstract

Abstract. In this paper, error estimates for generalized Laguerre-Gauss-type interpolations are derived in nonuniformly weighted Sobolev spaces weighted with ω α,β (x) = x α e −βx , α > −1, β > 0. Generalized Laguerre pseudospectral methods are analyzed and implemented. Two model problems are considered. The proposed schemes keep spectral accuracy and, with suitable choice of basis functions, lead to sparse and symmetric linear systems.

Jacobi and Laguerre quasi-orthogonal approximations and related interpolations

“…(3.33) of [22]) for α > −1, 1 ≤ k ≤ r ≤ N + 1 and r > α + 1 (or |α| < 1), However, it was only shown before that (cf.…”

“…According to (2.15)-(2.19) of [22], there exists a certain fixed number η > 0, and the constants c 1 ∼ π 2 2 and c 2 ∼ 8, such that • 2β According to (2.15)-(2.19) of [22], there exists a certain fixed number η > 0, and the constants c 1 ∼ π 2 2 and c 2 ∼ 8, such that • 2β …”

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

“…(i) truncate an unbounded domain to a bounded one and solve the problem on the bounded domain subject to artificial or transparent boundary conditions [22,26]; (ii) map the original problem on an unbounded domain to one on a bounded domain and use classic spectral methods to solve the new problem [9]; or equivalently, approximate the original problem by some non-classical functions mapped from the classic orthogonal polynomials/functions on a bounded domain [2,3,7,11,12,27,31,34]; (iii) directly approximate the original problem by genuine orthogonal functions such as Laguerre polynomials or functions on the unbounded domain [6,10,13,14,15,16,17,18,19,20,24,30,32,33,35].…”

Modified Laguerre spectral and pseudospectral methods for nonlinear partial differential equations in multiple dimensions

“…Maday, Bernaud-Thomas and Vandeven [1] , Funaro [2] , Guo and Shen [3] , Guo, Wang and Wang [4] , Guo and Zhang [5] , Guo and Xu [6] , Mastroianni and Monegato [7] , Shen [8] , Xu and Guo [9] established some results on the Laguerre orthogonal approximation and interpolation in weighted Sobolev spaces, with their applications to numerical solutions of differential equations defined on the half line. As we know, many practical problems are set up in multi-dimensional unbounded domains.…”