Nurlan Zh. Nauryzbayev

This paper investigates the exact orders of approximation of solutions to the Cauchy problem for the heat equation when the available information is inaccurate. The study identifies the optimal computational aggregates that achieve these approximation orders. The inaccuracy in the data arises from the Fourier coefficients of the initial condition, which belongs to the Nikol’skii–Besov class. The limiting error of the inaccurate information of computational aggregates constructed from Fourier coefficients with an arbitrary finite spectrum, is also determined. Furthermore, a set of computational aggregates is constructed, each having a limiting error not greater than the limiting error established in the C(N)D-2 problem. Numerical experiments have been carried out, confirming the theoretical results.

The notion of “randomness” in the mathematical theory of composites has typically been used abstractly within measure theory, making practical applications difficult. In contrast, engineering sciences often discuss randomness too loosely, lacking a theoretical foundation. This paper aims to bridge the gap between theory and applications, focusing on the effective properties of two-dimensional conducting composites with non-overlapping circular inclusions. It is shown that there is no universal minimum number of inclusions per cell in simulations of random composites. Even minor changes to Random Sequential Addition algorithms lead to different formulas for the effective constants. Application of the analytical representative volume element (aRVE) theory methodologically and practically addresses the diversity issue of random composites based on homogenization principles. In particular, it examines how the spatial arrangement of inclusions impacts the overall composite properties. The proposed method can be applied to a large number of inclusions and to symbolically given geometric and physical parameters relevant to optimal design problems. The method leverages structural sums and enables a more refined classification of different classes of composites, which was unattainable using previous approaches. The obtained results demonstrate a diversity of apparently similar composites. This paper outlines the investigation strategy and provides a detailed description of each step.

В данной статье изучается задача восстановления функций конечными суммами членов их тригонометрических рядов Фурье относительно вероятностных мер на функциональных классах, особенностью которых является невозможность нахождения "спектра больших коэффициентов", чем и объясняется рассмотрение произвольных "конечных сумм из членов рядов Фурье". Проблема вероятностного меровведения на классах с индивидуальными оценками на тригонометрические коэффициенты Фурье была решена на основе фундаментального характера свойства "Функция может быть задана двояко: либо как правило, либо как полный набор тригонометрических коэффициентов Фурье" из монографии В.М.Тихомирова, остальное было "делом техники". Переход к последовательностям коэффициентов Фурье с применением теоремы А.Н.Колмогорова о продолжении мер с конечных размерностей пространств на бесконечномерную, позволил ввести вероятностную меру на классах со взвешенными коэффициентами Фурье с доведением до окончательного, впервые построенного Стефаном Банахом в Приложении к книге Станислава Сакса "Теория интеграла". Здесь также предложены некоторые конструктивные детали процесса вероятностного меровведения.

The approximate calculation of integrals is a common part of research in various fields of mathematics, primarily in the theory of approximations and compu

We study approximate calculation of integrals of products of two functions by the method of tensor products of functionals. Under the assumption that one of them belongs to the Sobolev class with the...

We discuss two questions. First, we consider the existence of close to optimal quadrature formulas with a “bad” L2-discrepancy of their grids, and the second is the question of how much explicit...

In Ul’yanov classes, we compare computational aggregates constructed by the method of tensor products of functionals by means of trigonometric Fourier series.