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Singularly perturbed Cauchy problems of hyperbolic-parabolic type


Author: Rusu Galina
Degree:doctor of physics and mathematics
Speciality: 01.01.02 - Differential equations
Year:2011
Scientific adviser: Andrei Perjan
doctor habilitat, professor, Moldova State University
Institution: Moldova State University
Scientific council: DH 30-01.01.02-27.03.08
Moldova State University

Status

The thesis was presented on the 18 February, 2011
Approved by NCAA on the 31 March, 2011

Abstract

Adobe PDF document0.36 Mb / in romanian

Keywords

abstract differential equation, Cauchy problem, singular perturbations of hyperbolic-parabolic type, boundary layer, boundary layer function

Summary

Thesis structure: The thesis is written in Romanian and consists of an introduction, four chapters, conclusions, bibliography of 103 titles and 120 pages of basic text. In the thesis topic, 9 scientific papers are published.

Field of study of the thesis: The topic of thesis refers to the singular perturbation theory.

The aim of research: to study the behavior of solutions to the singularly perturbed Cauchy problem of hyperbolic-parabolic type for abstract second order differential equation with linear and lipschtzian operators in Hilbert spaces, relative to the corresponding solution to the unperturbed Cauchy problem; to establish the conditions when the solutions to the perturbed problem converge to the corresponding solutions to the unperturbed problem.

Scientific novelty and originality: a priory estimates, which are uniform relative to the small parameter, of solutions to the perturbed problem were established; the relation between solutions to the perturbed problem for abstract second order differential equations and the corresponding solutions to the abstract first order differential equations was established; convergence Theorems of solutions to the perturbed problem to the solution to the corresponding unperturbed problem were established; the problem of describing the behavior of solutions to the singularly perturbed problems of hyperbolic-parabolic type, relative to the small values of parameter, was solved.

The theoretical significance and applicative value of the thesis: The results of the thesis have a theoretical meaning and can be used to the qualitative study of various processes from Physics, Chemistry or other fields, especially to the qualitative study of wave processes and diffusion processes, whose evolution is nonuniform relative to some parameters.

The implementation of the scientific results: The results of the thesis can be used as content of some special courses for students and masters from specialities "Mathematics", "Applied Mathematics" and can serve as support for some master thesis.