The stability of Maxwell’s equations associated with an initial boundary value problem in quarter space

1991 ◽  
Vol 32 (2) ◽  
pp. 413-418
Author(s):  
J. H. Swart
Keyword(s):  
Boundary Value Problem ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Quarter Space ◽  
The Stability ◽  
Initial Boundary Value

scholarly journals Countable stability of a weak solution of a parabolic differential-difference system with distributed parameters on the graph

2020 ◽  
Vol 16 (4) ◽  
pp. 402-414
Author(s):  
Vyacheslav V. Provotorov ◽  
◽  
Sergey M. Sergeev ◽  
Van Nguyen Hoang ◽  
◽  
...  
Keyword(s):  
Weak Solution ◽  
Boundary Value Problem ◽  
Optimal Control Problems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Control Problems ◽  
Distributed Parameters ◽  
The Stability ◽  
Initial Boundary Value

The article proposes an analog of E. Rothe’s method (semi-discretization with respect to the time variable) for construction convergent different schemes when analyzing the countable stability of a weak solution of an initial boundary value problem of the parabolic type with distributed parameters on a graph in the class of summable functions. The proposed method leads to the study of the input initial boundary value problem to analyze the boundary value problem in a weak setting for elliptical type equations with distributed parameters on the graph. By virtue of the specifics of this method, the stability of a weak solution is understood in terms of the spectral criterion of stability (Neumann’s countable stability), which establishes the stability of the solution with respect to each harmonic of the generalized Fourier series of a weak solution or a segment of this series. Thus, there is another possibility indicated, in addition to the Faedo—Galerkin method, for constructing approaches to the desired solution of the initial boundary value problem, to analyze its stability and the way to prove the theorem of the existence of a weak solution to the input problem. The approach is applied to finding sufficient conditions for the countable stability of weak solutions to other initial boundary value problems with more general boundary conditions — in which elliptical equations are considered with the boundary conditions of the second or third type. Further analysis is possible to find the conditions under which Lyapunov stability is established. The approach can be used to analyze the optimal control problems, as well as the problems of stabilization and stability of differential systems with delay. Presented method of finite difference opens new ways for approximating the states of a parabolic system, analyzing their stability in the numerical implementation and algorithmization of optimal control problems.

RESEARCH OF THE IMPACT DEVICE MODEL OF THE ROD TYPE BY DIFFERENCE METHOD

2020 ◽  
pp. 73-80
Author(s):  
А.М. Слиденко ◽  
В.М. Слиденко
Keyword(s):  
Boundary Value Problem ◽  
Difference Method ◽  
Model Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Cross Sectional ◽  
Impact Device ◽  
Initial Boundary Value ◽  
The Impact

Приводится анализ механических колебаний элементов ударного устройства с помощью модели стержневого типа. Ударник и инструмент связаны упругими и диссипативными элементами, которые имитируют их взаимодействие. Аналогично моделируется взаимодействие инструмента с рабочей средой. Сформулирована начально-краевая задача для системы двух волновых уравнений с учетом переменных поперечных сечений стержней. Площади поперечных сечений определяются параметрическими формулами при сохранении объемов стержней. Параметрические формулы позволяют получать различного вида зависимости площади поперечного сечения стержня от его длины. Начальные условия отражают физическую картину взаимодействия инструмента с ударником и рабочей средой. Краевые условия описывают контактные взаимодействия ударника с инструментом и последнего с рабочей средой. В качестве модельной задачи рассматривается соударение ударника и инструмента через элемент большой жесткости. Начально-краевая задача исследуется разностным методом. Проводится сравнение решений задачи, полученных с помощью двухслойной и трехслойной разностных схем. Такие схемы реализованы в общей компьютерной программе в системе Mathcad. Показано, что при вычислениях распределения нормальных напряжений по длине стержня лучшими свойствами относительно устойчивости обладает двухслойная схема The article gives the analysis of mechanical vibrations of the impact device elements using the model of the rod type. The hammer and the tool are connected by elastic and dissipative elements that simulate their interaction. The interaction of the tool with the processing medium is simulated in a similar way. An initial boundary-value problem is formulated for a system of two wave equations taking into account the variable cross sections of the rods. Cross-sectional areas are determined by parametric formulas maintaining the volume of the rods. Parametric formulas allow one to obtain various dependence types of the cross-sectional area of the rod on its length. The initial and boundary conditions reflect the physical phenomenon of the tool interaction with the processing medium, and also describe the contact interactions of the hammer with the tool. The impacting of the hammer and the tool through an element of high rigidity is considered as a model problem. To control the limiting values, the solution of the model problem by the Fourier method is used. The initial-boundary-value problem is investigated by the difference method. A comparison of solutions obtained for the two-layer and three-layer difference schemes is given. Such schemes are realized in a common computer program in the Mathcad. It is shown that the two-layer scheme has the best properties in relation to stability while calculating the distribution of normal voltage along the length of the rod

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