ON A SPECTRAL PROBLEM FOR CERTAIN DIFFERENTIAL ALGEBRAIC EQUATIONS OF INDEX 1
A A ABRAMOV
V I UL'YANOVA
Dorodnicyn Computing Centre of the Russian Academy of Sciences
Vavilov St. 40, 119991 Moscow GSP-1, Russia
Computer and Automation Research Institute
Hungarian Academy of Sciences
Kender St. 13-17, 1117 Budapest, Hungary
L F YUKHNO
Mathematical Modeling Institute of the Russian Academy of Sciences
Miusskaya Sq. 4a, 125047 Moscow, Russia
In this talk, several results described in  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).
- 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).
- Balla K., März R. A unified approach to linear
differential algebraic equations and their adjoint equations.
Preprint N 2000-18, Berlin: Humboldt-Univ., 2000.
- 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.
EPSRC Gregynog Workshop, 21-26 July 2002