INSTITUTE FOR ANALYSIS AND SCIENTIFIC COMPUTING VIENNA UNIVERSITY OF TECHNOLOGY Numerical Solution of Singular Eigenvalue Problems for ODEs with a Focus on Problems Posed on Semi-Infinite Intervals

This work is concerned with the computation of eigenvalues and eigenfunctions of singular eigenvalue problems (EVPs) arising in ordinary differential equations. Two different numerical methods to determine values for the eigenparameter such that the boundary value problem has nontrivial solutions are considered. The first approach incorporates a collocation method. In the course of this work the existing code bvpsuite designed for the solution of boundary value problems was extended by a module for the computation of eigenvalues and eigenfunctions. The second solution approach represents a matrix method. A code for first order problems is realized in such a way that problems of higher order can also be solved after a transformation to the first order formulation. Since many eigenvalue problems are of second order, for example Sturm-Liouville problems , we also implemented a code for second order problems and present an empirical error analysis. For the solution of semi-infinite interval problems a transformation of the independent variable is carried out in such a way that the boundary value problem (BVP) originally posed on a semi-infinite interval is reduced to a singular problem posed on a finite interval. The implementation of this transformation is also incorporated into the bvpsuite package. The time-independent Schrödinger equation serves as an illustrating example.