# Mathematical Modelling of Finding the Values of Eigenfunctions for the Electrical Oscillations in the Extended Line Problem Using the Method of Regularized Traces

S.N. KakushkinThis paper describes a new numerical method for computing the values of the eigenfunctions of perturbed self-adjoint operators. The new method is based on the method of regularized traces. A mathematical model for calculating the eigenfunction values of the spectral problem concerning electrical oscillations in the extended line is developed. The elaborated algorithms make it possible to calculate the values of the eigenfunction of the perturbed operator whether the previous values are known or not. We've obtained the estimates of functional series residual sum'suspended', the corrections of the perturbation theory of perturbed self-adjoint operators, and proved their convergence. Effective algorithms for finding'suspended', perturbation theory corrections are discovered for the numerical implementation of the method. The numerical experiments on the calculation of the values of a problem on its own electrical oscillations in the extended lines show that the method is consistent with the other well-known methods of A.N.Krylov and A.M.Danilevsky. The method of regularized traces proved its reliability and high efficiency.Full text

- Keywords
- Sturm - Liouville problem, eigenvalues, eigenfunctions, perturbation theory, the method of regularized traces.
- References
- 1. Sviridyuk G.A. Solubility of the Thermal Convection of Viscoelastic Incompressible Fluid. Soviet Mathematics (Izvestiya VUZ. Matematika), 1990, vol. 34, no. 12, pp. 80-86.

2. Dubrovckiy V.V., Kadchenko S.I., Kravchenko V.F., Sadovnichiy V.A. A New Method of Calculation of the First Eigenvalues of the Spectral Problem of Hydrodynamic Stability Theory Cramped Viscous Fluid Between Two rotating Cylinders. Doklady Akademii nauk [Doklady Mathematics], 2001, vol. 381, no. 3, pp. 320-324. (in Russian)

3. Valeev N.F., Rabtsevich S.A., Nugumanov E.R. The Problem of Determining the Parameters of the Boundary Conditions of the Sturm - Liouville Operator on the Spectrum. Vestnik Samarskogo Gosudarstvennogo Universiteta. Estestvenno-Nauchnaya Seriya, 2009, no. 6 (72), pp. 12-20. (in Russian)

4. Kadchenko S.I., Kakushkin S.N. The Calculation of the Values of Natural Functions of Discrete Semi-Bounded from Below by the Operators of Regularized Traces. Vestnik Samarskogo Gosudarstvennogo Universiteta. Estestvenno-Nauchnaya Seriya, 2012, no. 6 (97), pp. 13-21. (in Russian)

5. Kadchenko S.I., Kakushkin S.N. The Algorithm for Finding the Values of the Eigenfunctions of Self-Adjoint Operators Perturbed by Regularized Traces. Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming & Computer Software", 2012, no. 40 (299), issue 14, pp. 71-76. (in Russian)