ПРО ДЕЯКІ ГРАНИЧНІ ВЛАСТИВОСТІ ІНТЕГРАЛІВ АБЕЛЯ–ПУАССОНА

• Дата: 2020
• Автори: Zhyhallo, T.V., Padalko, N.I.
• Формат: Стаття
• Мова: English
• Опубліковано: V.M. Glushkov Institute of Cybernetics of NAS of Ukraine 2020
• Теми: модуль неперервності; простори сумовних функцій; диференціальні рівняння в частинних похідних; крайова задача
• Онлайн доступ: https://jais.net.ua/index.php/files/article/view/496
• Назва журналу: Problems of Control and Informatics

When obtaining an analytical solution to many problems of applied mathematics using classical analytical methods, one encounters great mathematical difficulties. In most cases, these difficulties are caused by the huge amount of information (parameters) that is necessary for further mathematical processing. And here in some cases, it will probably be impossible not to use a numerical solution of the so-called boundary value problems of certain types of equations and systems of equations. Obviously, the type of equation or system of equations that we solve takes into account the peculiarities of formulation of this problem and, respectively, determines methods and properties of their solution. So, in the case of elliptic problem for a partial differential equation, its solution at some point of the considered domain is always influenced by the boundary conditions given on the entire boundary of the domain. In the foreign and national scientific literature on applied mathematics there is a number of results concerning the study of approximative properties of solutions of the classical Laplace equation inside the unit circle as well as on its boundary. As for similar studies in the upper coordinate half-plane, for the indicated above solutions of equations the successes were more moderate. That is why this work is devoted to the study of certain boundary properties of the Abel–Poisson integral, which in turn are solutions of the partial differential equations of elliptic type. The proved theorem on a concrete example (the Abel–Poisson integral) makes it possible to characterize the boundary properties of solutions of boundary value problems in flat domains (in the upper half-plane) in terms of the first order modulus of continuity of the spaces of functions that are summable on the entire numerical axis. The results obtained in this paper may be in demand in further studies of the modern applied mathematics.