A corrected spectral method for Sturm-Liouville problems with unbounded potential at one endpoint
For researchers in numerical analysis and applied mathematics, this provides an improved spectral method for a class of singular Sturm-Liouville problems, though the contribution is incremental.
This paper develops a spectral Galerkin method using Legendre polynomials to approximate eigenvalues of Sturm-Liouville problems with unbounded potential at one endpoint, and introduces low-cost correction procedures that improve accuracy for problems with unsmooth eigenfunctions. Numerical experiments confirm the method's effectiveness.
In this paper, we shall derive a spectral matrix method for the approximation of the eigenvalues of (weakly) regular and singular Sturm-Liouville problems in normal form with an unbounded potential at the left endpoint. The method is obtained by using a Galerkin approach with an approximation of the eigenfunctions given by suitable combinations of Legendre polynomials. We will study the errors in the eigenvalue estimates for problems with unsmooth eigenfunctions in proximity of the left endpoint. The results of this analysis will be then used conveniently to determine low-cost and effective procedures for the computation of corrected numerical eigenvalues. Finally, we shall present and discuss the results of several numerical experiments which confirm the effectiveness of the approach.