Calculation of the eigenvalues of the problems generated by the arbitrary even order Sturm -- Liouville operators

S. I. Kadchenko ; L. S. Ryazanova ; I. E. Kadchenko

Journal of computational and engineering mathematics, Tome 7 (2020) no. 3, pp. 34-44.

Résumé

The problem of calculating the eigenvalues of discrete semi-bounded differential operators is one of the important problems of numerical analysis. Despite of the simple formulation, for solving many problems that are encountered in practice, it is impossible to propose a single algorithm for calculating eigenvalues. Known methods of calculating the eigenvalues of linear differential operators are based on reducing problems to discrete models, using mainly grid methods or projection methods, which reduce the problem of finding the spectral characteristics of systems of linear algebraic equations. In view of the fact that poor separation of eigenvalues of matrices obtained from the corresponding systems of equations, the use of traditional methods of solving requires a very significant amount of calculations.This often leads to the need of solving the ill-posed problems. Generally, the choice of algorithms for approximate finding of eigenvalues of matrices is mainly determined by their type. This greatly narrows the possibilities of using computational methods to find their eigenvalues of matrices. It's important to note that the problem of finding the all points of the spectrum for high-order matrices does not have a satisfactory numerical solution yet. Using the numerical method of regularized traces and the Galerkin method, we previously obtained the linear formulas for calculating approximate eigenvalues of discrete semi-bounded operators. They allow you to find approximate eigenvalues with an any ordinal number. However, there are no computational difficulties that occur in other methods. The comparison of the results of computational experiments has shown that the eigenvalues found by linear formulas using the Galerkin method, as well as the known eigenvalues of spectral problems, are in line with each other. The article examines the possibility of using linear formulas obtained in the authors' articles to find the eigenvalues of the Sturm – Liouville operators of arbitrary even order. The considered examples show that the eigenvalues found from linear formulas and known asymptotic formulas are computationally the same.

Détail

Keywords: eigenvalues and eigenfunctions, discrete, self-adjoint, semibounded operators, the Galerkin method, asymptotic formula.