ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

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

On a Method of Solution for Singular Perturbed Boundary Problems


Margarita Besova

National Research University (MEU), Moscow
Post graduate student


In this paper a method based on the singular perturbations' holomorphic regularization was demonstrated on the example of the boundary value problem for a singularly perturbed second-order differential equation. This method follows the regularization method by S.A.Lomov and its main purpose is to construct so called pseudo-holomorphic solutions, i.e. solutions represented in the form of series in terms of powers of a small parameter convergent in the regular sense. Herewith a pseudo-holomorphic continuation algorithm was proposed to prove the global solvabiliy of the boundary value problem. The relevance of the problems studied in the article is dictated primarily by the necessity of further development in the analytical theory of singular perturbations, the fundamental principles of which are laid in the works of S.A. Lomov. In reference to the applications, the questions of solutions' smoothness with respect to the singularly incoming parameter also exist there. For example, in theoretical physics, the so-called Dyson argument is known, the essence of which is that the solution of a singularly perturbed problem cannot analytically depend on a small parameter (in astrophysics it depends on the gravitational constant). As proved in the paper, after a precise description of the singularities, the regular part of the solution will depend analytically on the parameter. It is also very important to study the Tikhonov systems and their use for constructing mathematical models in biology.



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