A Numerical Method for Solving Inverse Problems Generated by the Perturbed Self-Adjoint Operators

S. I. Kadchenko

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 6 (2013) no. 4, pp. 15-25

Résumé

Based on the methods of regularized traces and Bubnov–Galerkin's method a new method for the solution of inverse problems is developed in spectral characteristics perturbed self-adjoint operators. Simple formulas for calculating the eigenvalues of discrete operators without the roots of the corresponding secular equation are found. Computation of eigenvalues of a perturbed self-adjoint operator can be started with any of their numbers, regardless of whether the previous numbers of eigenvalues are known or not. Numerical calculations for eigenvalues of the Sturm–Liouville's operator show that the proposed formulas for large numbers of eigenvalues give more accurate results than the Bubnov–Galerkin's method. In addition, the obtained formulas allow us to calculate the eigenvalues of perturbed self-adjoint operator with very large numbers, where the use of the Bubnov–Galerkin's method becomes difficult. It can be used in problems of hydrodynamic stability theory, if you want to find signs of the real or imaginary parts of the eigenvalues with large numbers. An integral Fredholm equation of the first kind, restoring the value of the perturbing operator in the nodal points of the sample, is obtained. The method is tested on inverse problems for the Sturm–Liouville's problem. The results of numerous calculations have shown its computational efficiency.

Détail

Keywords: the inverse spectral problem; perturbation theory; discrete and self-adjoint operators; eigenvalues; eigenfunctions; incorrectly formulated problems.