ON A SPECTRAL PROBLEM FOR CERTAIN DIFFERENTIAL ALGEBRAIC EQUATIONS OF INDEX 1

A A ABRAMOV

alalabr@ccas.ru

V I UL'YANOVA

Dorodnicyn Computing Centre of the Russian Academy of Sciences
Vavilov St. 40, 119991 Moscow GSP-1, Russia

K BALLA

balla@sztaki.hu

Computer and Automation Research Institute
Hungarian Academy of Sciences
Kender St. 13-17, 1117 Budapest, Hungary

L F YUKHNO

yukhno@imamod.ru

Mathematical Modeling Institute of the Russian Academy of Sciences
Miusskaya Sq. 4a, 125047 Moscow, Russia

In this talk, several results described in [1] will be discussed. A self-adjoint linear homogeneous differential algebraic equation of index 1 is considered. One of the matrices occurring in the system depends on a spectral parameter (SP), in general, non-linearly. The self-adjoint homogeneous boundary conditions also may depend on the SP. We deal with the case when the dependence of problem data on the SP is of monotone type. A method is proposed and analyzed for computing the number of eigenvalues on a given interval of SP; this number takes the multiplicities into account. In the case when the boundary conditions are independent of the SP, an index, i.e. a serial number, is associated with each eigenvalue. A method for computing the eigenvalue with a prescribed index is given.

The results are closely related to ones considered in [2,3].

This work was supported by Russian Foundation for Basic Research (grants 02-01-00050, 02-01-00555) and Hungarian National Science Foundation (grant T029572).

REFERENCES

1. Abramov A.A., Balla K., Ul'yanova V.I., Yukhno L.F. On a nonlinear self-adjoint spectral problem for some differential algebraic equations of index 1. Comp. Maths Math Phys., Vol. 42, N. 7, 2002 (to appear).
2. Balla K., März R. A unified approach to linear differential algebraic equations and their adjoint equations. Preprint N 2000-18, Berlin: Humboldt-Univ., 2000.
3. Abramov A.A. Calculation of eigenvalues in a nonlinear spectral problem for the Hamiltonian systems of ordinary differential equations. Comp. Maths Math. Phys., Vol. 41, N 1, P. 27-36, 2001.