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.