Inverse Sturm–Liouville problems with pseudospectral methods

Abstract

In this paper a technique to obtain a first approximation for singular inverse Sturm–Liouville problems with a symmetrical potential is introduced. The singularity, as a result of unbounded domain (−∞, ∞), is treated by considering numerically the asymptotic limit of the associated problem on a finite interval (−L, L). In spite of this treatment, the problem has still an ill-conditioned structure unlike the classical regular ones and needs regularization techniques. Direct computation of eigenvalues in iterative solution procedure is made by means of pseudospectral methods. A fairly detailed description of the numerical algorithm and its applications to specific examples are presented to illustrate the accuracy and convergence behaviour of the proposed approach.

Keywords:

• regular and singular inverse Sturm–Liouville problems
• pseudospectral method
• condition number
• regularization method

2010 AMS Subject Classifications:

• 31A25
• 65F18

Acknowledgements

This research was supported by a grant from TUBITAK, the Scientific and Technical Research Council of Turkey.