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 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 The Laplace transform as an alternative general method for solving linear ordinary differential equations

2021 ◽  
Vol 1 (4) ◽  
pp. 309
Author(s):  
William Guo
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Laplace Transform ◽  
Ordinary Differential Equations ◽  
Case Studies ◽  
Initial Conditions ◽  
Linear Ordinary Differential Equations ◽  
The Laplace Transform ◽  
Linear Odes ◽  
General Method

<p style='text-indent:20px;'>The Laplace transform is a popular approach in solving ordinary differential equations (ODEs), particularly solving initial value problems (IVPs) of ODEs. Such stereotype may confuse students when they face a task of solving ODEs without explicit initial condition(s). In this paper, four case studies of solving ODEs by the Laplace transform are used to demonstrate that, firstly, how much influence of the stereotype of the Laplace transform was on student's perception of utilizing this method to solve ODEs under different initial conditions; secondly, how the generalization of the Laplace transform for solving linear ODEs with generic initial conditions can not only break down the stereotype but also broaden the applicability of the Laplace transform for solving constant-coefficient linear ODEs. These case studies also show that the Laplace transform is even more robust for obtaining the specific solutions directly from the general solution once the initial values are assigned later. This implies that the generic initial conditions in the general solution obtained by the Laplace transform could be used as a point of control for some dynamic systems.</p>

scholarly journals An implementation of the Chebyshev series method for the approximate analytical solution of second-order ordinary differential equations

2019 ◽  
pp. 97-103
Author(s):  
О.Б. Арушанян ◽  
С.Ф. Залеткин
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Canonical System ◽  
Second Order ◽  
Partial Sums ◽  
Problem Solution ◽  
Chebyshev Series ◽  
Canonical Systems ◽  
The Cauchy Problem ◽  
The Right

Описан один метод по применению рядов Чебышёва для интегрирования канонических систем обыкновенных дифференциальных уравнений второго порядка. Этот метод основан на аппроксимации решения задачи Коши, его первой и второй производных частичными суммами смещенных рядов Чебышёва. Коэффициенты рядов вычисляются итерационным способом с применением соотношений, связывающих коэффициенты Чебышёва решения задачи Коши, а также коэффициенты Чебышёва первой производной решения с коэффициентами Чебышёва правой части системы. Неотъемлемым элементом вычислительной схемы является использование формулы численного интегрирования Маркова для вычисления коэффициентов Чебышёва правой части системы. В статье не только сообщаются результаты, полученные численными расчетами, но и делается упор на высокоточном аналитическом представлении решения в виде частичной суммы ряда на промежутке интегрирования. A method used to apply the Chebyshev series for solving canonical systems of second order ordinary differential equations is described. This method is based on the approximation of the Cauchy problem solution and its first and second derivatives by partial sums of shifted Chebyshev series. The coefficients of these series are determined iteratively using the relations relating the Chebyshev coefficients of the solution and its first derivative with the Chebyshev coefficients found for the right-hand side of the canonical system by application of Markov's quadrature formula. The obtained numerical results are discussed and the high-precision analytical representations of the solution are proposed in the form of partial sums of Chebyshev series on a given integration segment.

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.

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 An Optimal Control Problem by a Hybrid System of Hyperbolic and Ordinary Differential Equations

Games ◽  
2021 ◽  
Vol 12 (1) ◽  
pp. 23
Author(s):  
Alexander Arguchintsev ◽  
Vasilisa Poplevko
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Ordinary Differential Equations ◽  
Hyperbolic Equations ◽  
Control Methods ◽  
The Right ◽  
The Cost ◽  
Optimal Control Methods

This paper deals with an optimal control problem for a linear system of first-order hyperbolic equations with a function on the right-hand side determined from controlled bilinear ordinary differential equations. These ordinary differential equations are linear with respect to state functions with controlled coefficients. Such problems arise in the simulation of some processes of chemical technology and population dynamics. Normally, general optimal control methods are used for these problems because of bilinear ordinary differential equations. In this paper, the problem is reduced to an optimal control problem for a system of ordinary differential equations. The reduction is based on non-classic exact increment formulas for the cost-functional. This treatment allows to use a number of efficient optimal control methods for the problem. An example illustrates the approach.

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