A spectral solution of the Sturm–Liouville equation: comparison of classical and nonclassical basis sets

This paper considers a spectral method of solution of the Sturm–Liouville equation and the associated Schroedinger equation. The main objective is to develop a collocation method based on quadrature (collocation) points generated from nonclassical polynomials. The polynomials and associated quadrature points are calculated with Gautschiʹs Stieltjes procedure from some specified weight function. The particular spectral method used here with nonclassical basis sets is referred to as the Quadrature Discretization Method (QDM). The QDM and a related weighted QDM are applied to several Sturm–Liouville and Schroedinger equations and the results are compared with the traditional spectral methods based on Chebyshev and Legendre quadrature points. The results are also compared with the results of other workers wherever available. The QDM was found to give the most rapid convergence relative to other methods for the problems studied.