Numerical Solutions of Singular Eigenvalue Problems for Systems of Ordinary Differential Equations

Christa Simon

July 27, 2009, noon KG 712

This thesis is concerned with the computation of eigenvalues and eigenfunctions of singular eigenvalue problem arising in ordinary dierential 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 the thesis the existing code bvpsuite designed for the solution of boundary value problems was extended by a module for the computation of eigenvalues and eigenfunctions.

For the solution of infinite interval problems a transformation of the independent variable is carried out in such a way that the 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.

The second solution approach represents a finite difference 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.