On a modified Numerov's method for inverse Sturm–Liouville problems

ABSTRACT

In this paper, we discuss the convergence of modified Numerov's method in Q. Gao, X.L. Cheng and Z.D. Huang [Modified Numerov's method for inverse Sturm–Liouville problems, J. Comput. Appl. Math. 253 (2013), pp. 181–199] for computing symmetric potentials from finite Dirichlet eigenvalues. A sufficient condition for convergence of the estimate to the true potential is given and the rate of convergence is investigated. The proof relies on the asymptotics of eigenvalues of the Sturm–Liouville operator and the errors in the finite difference eigenvalues obtained by Numerov's approach. Some numerical experiments are presented to confirm the theoretically predicted convergence properties.

Keywords:

• Modified Numerov's method
• Sturm–Liouville operator
• inverse eigenvalue problem
• correction
• convergence

2010 AMS Subject Classifications:

• 34L16
• 65F05
• 65F15
• 65F18

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

We greatly acknowledge the valuable comments of the anonymous reviewers. This work was supported by the National Natural Science Foundation of China [grant number 11526086], the Fundamental Research Funds for the Central Universities [grant number CCNU16A05013] and the China Postdoctoral Science Foundation [grant number 2016M592356].