Approximation of eigenvalues of Sturm-Liouville problems defined on a semi-infinite domain

In this paper, we describe how to approximate numerically the eigenvalues of a Sturm-Liouville problem defined on a semi-infinite interval. The key idea is to transform the problem in such a way as to compress the semi-infinite interval in a finite interval by applying a suitable change of the independent variable. Then, we approximate each derivative in the Sturm-Liouville equation thus obtained with finite difference schemes. Consequently, we convert the Sturm-Liouville problem into an algebraic eigenvalue problem. The numerical results of the experiments show that the proposed approach is promising.