SOLUTION OF DIFFERENTIAL EQUATIONS AND SYSTEMS OF DIFFERENTIAL EQUATIONS WITH THE USE OF MULTIPROCESSOR COMPUTERS

2018 ◽  
pp. 59-62 ◽  
Author(s):  
E. V. Glivenko ◽  
A. S. Fomochkina
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Computational Methods ◽  
Initial Conditions ◽  
Geometric Construction ◽  
Cauchy Problems ◽  
Systems Of Differential Equations ◽  
The Cauchy Problem

The paper proposes computational methods for solving differential equations and systems of two differential equations with initial conditions. At the beginning of the article the statement of the problem is given. Then we propose an essential complement to the classical methods for solving the Cauchy problem, which gives a new indication of the correctness of the estimation of the functionsolution. This feature is based on the geometric construction of the solution of several Cauchy problems that differ only in the initial conditions. The article describes an algorithm for constructing a solution on the investigated interval. After that, the possibility of generalizing a feature to systems of two differential equations is described, when the solution is not just a function but a function in space. The methods themselves are easily parallelized, therefore, they can effectively use multiprocessor computers.

scholarly journals The behavior of solutions of non-linear differential equations in Hilbert space. I

1988 ◽  
Vol 11 (1) ◽  
pp. 143-165 ◽  
Author(s):  
Vladimir Schuchman
Keyword(s):  
Hilbert Space ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Differential Equations ◽  
Linear Case ◽  
Cauchy Problems ◽  
The Cauchy Problem ◽  
Linear Differential ◽  
Degenerate Linear

This paper deals with the behavior of solutions of ordinary differential equations in a Hilbert Space. Under certain conditions, we obtain lower estimates or upper estimates (or both) for the norm of solutions of two kinds of equations. We also obtain results about the uniqueness and the quasi-uniqueness of the Cauchy problems of these equations. A method similar to that of Agmon-Nirenberg is used to study the uniqueness of the Cauchy problem for the non-degenerate linear case.

scholarly journals Visualization of Transition's Scenarios from Harmonic to Chaotic Flexible Nonlinear-elastic Nano Beam's Oscillations

2019 ◽  
Author(s):  
Вадим Крысько ◽  
Vadim Krys'ko ◽  
Ирина Папкова ◽  
Irina Papkova ◽  
Екатерина Крылова ◽  
...  
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Initial Conditions ◽  
Couple Stress Theory ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Physical Nonlinearity ◽  
Variable Load ◽  
Wide Range ◽  
The Cauchy Problem

In this study, a mathematical model of the nonlinear vibrations of a nano-beam under the action of a sign-variable load and an additive white noise was constructed and visualized. The beam is heterogeneous, isotropic, elastic. The physical nonlinearity of the nano-beam was taken into account. The dependence of stress intensity on deformations intensity for aluminum was taken into account. Geometric non-linearity according to Theodore von Karman’s theory was applied. The equations of motion, the boundary and initial conditions of the Hamilton-Ostrogradski principle with regard to the modified couple stress theory were obtained. The system of nonlinear partial differential equations to the Cauchy problem by the method of finite differences was reduced. The Cauchy problem by the finite-difference method in the time coordinate was solved. The Birger variable method was used. Data visualization is carried out from the standpoint of the qualitative theory of differential equations and nonlinear dynamics were carried out. Using a wide range of tools visualization allowed to established that the transition from ordered vibrations to chaos is carried out according to the scenario of Ruelle-Takens-Newhouse. With an increase of the size-dependent parameter, the zone of steady and regular vibrations increases. The transition from regular to chaotic vibrations is accompanied by a tough dynamic loss of stability. The proposed method is universal and can be extended to solve a wide class of various problems of mechanics of shells.

scholarly journals Interval Version of the Operational Calculation Method for Solving Ordinary Differential Equations

2021 ◽  
Vol 04 (10) ◽  
Author(s):  
Oybek Zhumaboyevich Khudayberdiyev ◽  
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Calculation Method ◽  
Initial Conditions ◽  
Operational Calculus ◽  
Real Solution ◽  
Proved Theorem ◽  
The Cauchy Problem

This article discusses the interval variant of solving ordinary differential equations with given initial conditions, i.e. the Cauchy problem, by the method of operational calculus. This is where the interval version of the operational calculus is motivated and built. As a result, on the basis of the proved theorem in this article, an analytic interval set of solutions is obtained that is guaranteed to contain a real solution to the problem.

scholarly journals DENSENESS OF SETS OF CAUCHY PROBLEMS WITHHOUT SOLUTIONS AND WITH NONUNIQUE SOLUTIONS IN THE SET OF ALL CAUCHY PROBLEMS

2020 ◽  
Vol 8 (2) ◽  
pp. 122-126
Author(s):  
V. Slyusarchuk
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Cauchy Problems ◽  
Number Of Solutions ◽  
Computational Mathematics ◽  
Infinite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
The Cauchy Problem ◽  
Solutions Of Equations

When finding solutions of differential equations it is necessary to take into account the theorems on innovation and unity of solutions of equations. In case of non-fulfillment of the conditions of these theorems, the methods of finding solutions of the studied equations used in computational mathematics may give erroneous results. It should also be borne in mind that the Cauchy problem for differential equations may have no solutions or have an infinite number of solutions. The author presents two statements obtained by the author about the denseness of sets of the Cauchy problem without solutions (in the case of infinite-dimensional Banach space) and with many solutions (in the case of an arbitrary Banach space) in the set of all Cauchy problems. Using two examples of the Cauchy problem for differential equations, the imperfection of some methods of computational mathematics for finding solutions of the studied equations is shown.

scholarly journals Guaranteed Estimation of Solutions to the Cauchy Problem When the Restrictions on Unknown Initial Data Are Not Posed

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3218
Author(s):  
Oleksandr Nakonechnyi ◽  
Yuri Podlipenko ◽  
Yury Shestopalov
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Conditions ◽  
Statistical Characteristics ◽  
Cauchy Problems ◽  
Linear Ordinary Differential Equations ◽  
Guaranteed Estimation ◽  
The Cauchy Problem ◽  
The Right ◽  
Bounded Sets

The paper deals with Cauchy problems for first-order systems of linear ordinary differential equations with unknown data. It is assumed that the right-hand sides of equations belong to certain bounded sets in the space of square-integrable vector-functions, and the information about the initial conditions is absent. From indirect noisy observations of solutions to the Cauchy problems on a finite system of points and intervals, the guaranteed mean square estimates of linear functionals on unknown solutions of the problems under consideration are obtained. Under an assumption that the statistical characteristics of noise in observations are not known exactly, it is proved that such estimates can be expressed in terms of solutions to well-defined boundary value problems for linear systems of impulsive ordinary differential equations.

scholarly journals On the solvability of the modified Cauchy problem for linear systems of generalized ordinary differential equations with singularities

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Malkhaz Ashordia ◽  
Inga Gabisonia ◽  
Mzia Talakhadze
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Systems ◽  
Unique Solvability ◽  
Sufficient Conditions ◽  
The Cauchy Problem ◽  
Generalized Ordinary Differential Equations

AbstractEffective sufficient conditions are given for the unique solvability of the Cauchy problem for linear systems of generalized ordinary differential equations with singularities.

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