A Diagonalized Legendre Rational Spectral Method for Problems on the Whole Line

Abstract

A diagonalized Legendre rational spectral method for solving second and fourth order differential equations are proposed. Some Fourier-like Sobolev orthogonal 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 of this approach.