Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

Holomorphic Regularization of Weakly Nonlinear Singularly Perturbed Problems

Author(s):

Vasily Ivanovich Kachalov

FGBOU VO "NIU "MPEI"
(Federal state budget educational institution of
higher professional education "national research University "MPEI"),
Associate Professor of the Department of mathematics,
PhD in physics and mathematics, Associate Professor

vikachalov@rambler.ru

Yury Sergeevich Fedorov

FGBOU VO "NIU "MPEI"
(Federal state budget educational institution of
higher professional education "national research University "MPEI"),
associate Professor of the Department of mathematics

Abstract:

The holomorphic regularization method, which is a logical continuation of the method of regularization S. A.Lomov, is used to build pseudoholomorphic solutions of weakly nonlinear singularly perturbed systems of differential equations, i.e. those solutions which can be represented as convergent in the usual sense (not asymptotic) series in powers of the small parameter. The existence of first integrals of singularly perturbed systems holomorphic by the small parameter is proved and, thus, the Poincare theorem about the decomposition is generalized. From this and the implicit function theorem the Lagrange stability of the solutions of the system of equations of characteristics follows the existence of pseudoholomorphic in global sense solutions of weakly nonlinear singularly perturbed systems. It should be noted that the regularizing functions responsible for the description of the boundary layer are defined (as well as in the Lomov method) by the limit spectrum of the operator. In this method any preassigned accuracy of approximation is provided for a fixed value of the small parameter, not for tending it to zero, as it happens in classic asymptotic methods. This is very important in solving applied singularly perturbed problems arising in various fields of science. The holomorphic regularization method, along with the method of normal forms of V.F. Safonov, was specially designed to solve precisely nonlinear singularly perturbed equations and systems and to lay the foundation of the analytic theory of singular perturbations. In the future the described approach will be extended to other types of non-linear problems including equations in Banach space.

Keywords

• holomorphic regularization
• pseudoholomorphic solution

References:

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