5 results on '"Stability theory"'
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2. On the problem of recovering boundary conditions in the third boundary value problem for parabolic equation
- Author
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I.V. Boykov and V.A. Ryazantsev
- Subjects
inverse problems ,parabolic equations ,third boundary value problem ,stability theory ,continuous operator method ,Physics ,QC1-999 ,Mathematics ,QA1-939 - Abstract
Background. In recent decades the theory for solution of inverse and ill-posed problems has become one of the most important and fast-growing branch of modern mathematics. A relevancy of this theory is due to not only significant growth in the number of applications of inverse and ill-posed problems in different fields of physical and technical sciences but also rapid development of computer technology. It is known that most of inverse problems of mathematical physics belong to the class if ill-posed problems, and the most important property of these problems is their instability to small perturbations of the initial data of the problem. This property induces the need for the development of regularization methods of special type. The boundary value problems constitute one of the most significant classes of inverse problems. An inverse problem is termed boundary if it is required to recover the function that is the part of a boundary value. Such problems arise when direct measuring of heat field characteristics at a domain boundary is difficult or impossible at all. Constructing numerical methods for solution of these problems is very crucial due to a vast number of their physical and technical applications. Materials and methods. In this paper we propose a numerical method for simultaneous recovering of boundary value coefficients in the third boundary problem for a heat equation. At the core of the method there is continuous method for solution of operator equations in Banach spaces. The main idea of the method is composing and solving the auxiliary system of differential equations of special type relative to the unknown coefficients of the basic problem. This system is to be solved numerically with the help of some method for approximate solution of differential equations. Simultaneous recovering of the coefficients by means of the proposed method additionally requires knowledge of values of a solution of the basic parabolic equation at two different points. Results. A numerical method for solution of the problem of recovering boundary value coefficients in the third boundary value problem for one-dimensional heat equation is constructed. We show applicability of continuous operator method to solution of inverse boundary value problems for parabolic equations. Convergence of the method is proven with the help of the theory of stability of ordinary differential equations. Successful solution of the model example shows effectiveness of the proposed method. Conclusions. An efficient method for solution of the problem of recovering boundary value coefficients in the third boundary value problem for linear one-dimensional parabolic equation is described in the paper. The key advantages of the method are its simplicity, flexibility and also stability to perturbations of initial data of the problem. It is of significant theoretical and practical interest to extend the proposed method to a wider class of boundary problems and also to multidimensional and nonlinear parabolic equations.
- Published
- 2021
- Full Text
- View/download PDF
3. Sovremennye problemy teorii ustoichivosti i soderzhanie kursa matematiki v tekhnicheskom vuze
- Author
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Olga G. Antonovskaia and Antonina V. Besklubnaia
- Subjects
stability theory ,запас устойчивости ,фундаментальность образования ,ability to construct models of real phenomena ,умение строить модели реальных явлений ,robust stability ,робастная устойчивость ,fundamentals of education ,математические методы исследования ,professional activity of the trainee ,профессиональная деятельность обучаемого ,mathematical methods of research ,stability reserve ,теория устойчивости - Abstract
В работе рассматривается вопрос об изложении материала по теме «Запас устойчивости» раздела «Теория устойчивости» в курсах, посвященных динамике систем, дифференциальным уравнениям и т. д., для студентов технических специальностей. Вопросы нахождения запаса устойчивости реальной технической системы рассматриваются прежде всего с точки зрения доступности изложения материала и возможности их практического применения при решении прикладных задач на основе знаний, полученных в курсах математики студентами технических вузов., This paper addresses the problem of the "Stability reserve" issue in the "Stability theory" section in the system dynamics, differential equations and other courses for students of technical specialties. The problems of stability reserve determination are examined first of all from the point of view of accessibility of material account and possibility of its practical use when solving applied problems on the base of knowledge obtained by the students of technical high educational institutions in mathematical courses.
- Published
- 2021
4. Priamoi metod Liapunova i ego izlozhenie v sovremennom uchebnom protsesse
- Author
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Olga G. Antonovskaia and Antonina V. Besklubnaia
- Subjects
положительно определенная квадратичная форма ,stability theory ,Lyapunov's direct method ,прямой метод Ляпунова ,positive definite quadratic form ,фундаментальность образования ,ability to construct models of real phenomena ,умение строить модели реальных явлений ,Lyapunov matrix equation ,fundamentals of education ,математические методы исследования ,матричное уравнение Ляпунова ,professionally significant and practical goals of education ,professional activity of the trainee ,профессиональная деятельность обучаемого ,mathematical methods of research ,теория устойчивости ,профессионально-значимые и практические цели образования - Abstract
В работе рассматривается вопрос об изложении темы «Построение функций Ляпунова» раздела «Прямой метод Ляпунова» в курсах, посвященных динамике систем, дифференциальным уравнениям и т. д., для студентов математических и технических специальностей. Авторы подчеркивают, что метод функций Ляпунова рассматривается прежде всего, как метод, полезный при математическом исследовании динамики конкретных технических систем. Поскольку при практическом применении этого метода имеет смысл строить функции Ляпунова самого простого вида, авторами предлагается к изложению методика построения функций Ляпунова в виде положительно определенных квадратичных форм, обладающих свойствами, определенными особенностями исходной задачи., The article is devoted to the problem of the full treatment of the subject “Construction of Lyapunov functions” in the “Lyapunov's direct method” section in the courses on system dynamics, differential equations, etc., for students of mathematical and technical specialties. The authors of the article point out that first of all Lyapunov function method is considered the method useful for studying dynamics of concrete technical systems by mathematical methods. Since for practical use of this method it makes sense to construct Lyapunov functions in simplest form, the methodology for constructing Lyapunov functions in positive definite quadratic forms with given properties, certain features of original problems is proposed by the authors.
- Published
- 2020
5. THE STABILITY OF SOLUTION OF SYSTEMS WITH UNDEFINED PARAMETERS OPTIMAL DESIGN PROBLEM.
- Author
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Levin, V. I.
- Subjects
MATHEMATICAL optimization ,PARAMETER estimation ,STABILITY theory ,MATHEMATICAL functions ,ALGORITHMS - Abstract
Our article considers the problem of optimization of incompletely specified functions, namely, functions which parameters are given within range of possible values. It is shown that solution of this problem also requires solving problem of determining stability of optimum of such functions to variation of values of their parameters. A method for obtaining such optimum of incompletely defined functions is presented. Method uses determination of problem. It allows to split original non-deterministic problem into two optimization problem of deterministic functions, which are solved separately. After that solutions are combined into one which is a solution of original problem. The article also provides a method for determining the stability of the optimum found of incompletely defined functions by methods of interval mathematics. We formulate 5 theorems determining the conditions for existence of optimum of incompletely defined function and its resistance to changing the function parameters. Algorithms for verifying the stability function are given [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
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