Asymptotic solutions of singularly perturbed linear differential-algebraic equations with periodic coefficients

S. Radchenko; V. Samoilenko; P. Samusenko

doi:10.30970/ms.59.2.187-200

S. Radchenko Department of Information Technology and Mathematical Sciences, Borys Grinchenko Kyiv University Kyiv, Ukraine
V. Samoilenko Department of Mathematical Physics, Taras Shevchenko National University of Kyiv Kyiv, Ukraine
P. Samusenko Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute” Кyiv, Ukraine

DOI: https://doi.org/10.30970/ms.59.2.187-200

Keywords: Asymptotic solution, differential-algebraic equations, singular perturbed system

Abstract

The paper deals with the problem of constructing asymptotic solutions for singular perturbed linear differential-algebraic equations with periodic coefficients. The case of multiple roots of a characteristic equation is studied. It is assumed that the limit pencil of matrices of the system has one eigenvalue of multiplicity n, which corresponds to two finite elementary divisors and two infinite elementary divisors whose multiplicity is greater than 1.
A technique for finding the asymptotic solutions is developed and n formal linearly independent solutions are constructed for the corresponding differential-algebraic system. The developed algorithm for constructing formal solutions of the system is a nontrivial generalization of the corresponding algorithm for constructing asymptotic solutions of a singularly perturbed system of differential equations in normal form, which was used in the case of simple roots of the characteristic equation.
The modification of the algorithm is based on the equalization method in a special way the coefficients at powers of a small parameter in algebraic systems of equations, from which the coefficients of the formal expansions of the searched solution are found. Asymptotic estimates for the terms of these expansions with respect to a small parameter are also given.
For an inhomogeneous differential-algebraic system of equations with periodic coefficients, existence and uniqueness theorems for a periodic solution satisfying some asymptotic estimate are proved, and an algorithm for constructing the corresponding formal solutions of the system is developed. Both critical and non-critical cases are considered.

Author Biographies

S. Radchenko, Department of Information Technology and Mathematical Sciences, Borys Grinchenko Kyiv University Kyiv, Ukraine

Department of Information Technology and Mathematical Sciences, Borys Grinchenko Kyiv University
Kyiv, Ukraine

V. Samoilenko, Department of Mathematical Physics, Taras Shevchenko National University of Kyiv Kyiv, Ukraine

Department of Mathematical Physics, Taras Shevchenko National University of Kyiv
Kyiv, Ukraine

P. Samusenko, Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute” Кyiv, Ukraine

Department of Mathematical Analysis and Probability Theory,
National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”
Кyiv, Ukraine

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