Sobolev orthogonal Legendre rational spectral methods for problems on the half line

Abstract

Modified Legendre rational spectral methods for solving second-order differential equations on the half line are proposed. Some Sobolev orthogonal Legendre rational basis functions are constructed, which lead to the diagonalization of discrete systems. Accordingly, both the exact solutions and the approximate solutions can be represented as infinite and truncated Fourier series. Numerical results demonstrate the effectiveness and the spectral accuracy of this approach. KEYWORDS elliptic boundary value problems, modified Legendre rational spectral methods, numerical results, Sobolev orthogonal functions MSC CLASSIFICATION 65N35; 41A20; 32C45; 35J25; 35K05 INTRODUCTIONSpectral method is an extremely effective tool for solving various problems in science and engineering numerically. For unbounded domains, we usually employ the Hermite or Laguerre polynomials/functions as the basis functions to approximate the potential problems; see, eg, previous studies. 1-11 However, since the Laguerre and Hermite Gauss points are too concentrated near zero, the approximation results are usually not ideal for algebraic decay functions, especially where the points are far away from zero. Alternately, we may use rational functions to approximate the problems on unbounded domains. Christov, 12 Boyd, 13,14 and Guo et al [15][16][17][18] developed various rational spectral methods on infinite intervals. Clearly, the distribution of the Gauss points for the rational functions is more reasonable, and the approximation results are also ideal especially for algebraic decay solutions. Nevertheless, the utilization of rational spectral methods usually lead to a full or sparse algebraic system as mentioned in Section 3 of this paper, the condition numbers increase as (N 2 ) for the second order problems.As is well known, the Fourier functions are the eigenfunctions of the Laplace operator, whose kth derivatives are orthogonal with respect to each other. This property results in the diagonalization of the resulting algebraic systems of the corresponding Fourier spectral methods for periodic problems; cf previous studies. 2,3,10 For nonperiodic problems, the usual spectral methods only lead to a highly sparse or full algebraic system. It is really meaningful and worth looking forward to find out a class of basis functions such that the resulting algebraic systems of the spectral methods for nonperiodic problems are still diagonal; cf previous studies. 19 Recently, Liu et al 20,21 constructed the Fourier-like Sobolev orthogonal basis functions based on generalized Laguerre functions and applied them to the second-and fourth-order elliptic equations on the half line. Motivated by previous studies, 19-21 the main purpose of this paper is to construct the Fourier-like Legendre rational basis functions on the half line, such that they are mutually orthogonal with respect to certain Sobolev inner product. On this basis, we further propose the diagonalized Legendre rational spectral methods for the second-order elliptic boundary value problems with D...