Algorithm for numerical solution of inverse spectral problems generated by Sturm--Liouville operators of an arbitrary even order

S. I. Kadchenko ; L. S. Ryazanova ; Yu. R. Dzhiganchina

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 14 (2021) no. 2, pp. 52-63

Résumé

The article is devoted to the construction of algorithm for solving inverse spectral problems generated by Sturm–Liouville differential operators of an arbitrary even order. The goal of solving inverse spectral problems is to recover operators from their spectral characteristics and spectral characteristics of auxiliary problems. In the scientific literature, we can not find examples of the numerical solution of inverse spectral problems for the Sturm–Liouville operator of higher than the second order. However, their solution is caused by the need to construct mathematical models of many processes arising in science and technology. Therefore, the development of computationally efficient algorithm for the numerical solution of inverse spectral problems generated by the Sturm–Liouville operators of an arbitrary even order is of great scientific interest. In this article, we use linear formulas obtained earlier in order to find the eigenvalues of discrete semi-bounded operators and develop algorithm for solving inverse spectral problems for Sturm–Liouville operators of an arbitrary even order. The results of the performed computational experiments show that the use of the algorithm developed in the article makes it possible to recover the values of the potentials in the Sturm–Liouville operators of any necessary even order.

Détail

Keywords: eigenvalues and eigenfunctions, discrete, self-adjoint and semi-bounded operators, Galerkin method, ill-posed problems, Fredholm integral equations of the first kind, asymptotic formulas.