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

TL;DR

This paper introduces a fully diagonalized spectral method based on generalized Laguerre functions for solving elliptic equations on the half line, providing optimal error estimates and demonstrating spectral accuracy through numerical experiments.

Contribution

The paper develops a new fully diagonalized Laguerre spectral method with orthogonal basis functions and establishes optimal error estimates for elliptic equations on the half line.

Findings

Method achieves spectral accuracy in numerical experiments.

Optimal error estimates are proven for Dirichlet and Robin problems.

Numerical results confirm theoretical analysis.

Abstract

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.