scholarly journals AN ALGORITHM FOR SOLVING A CONTROL PROBLEM FOR A DIFFERENTIAL EQUATION WITH A PARAMETER

2018 ◽  
pp. 25-32
Author(s):  
Dzhumabaev D.S. ◽  
Bakirova E.A. ◽  
Kadirbayeva Zh.M.
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Control Problem ◽  
Ordinary Differential Equations ◽  
Boundary Value ◽  
Cauchy Problems ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

On a finite interval, a control problem for a linear ordinary differential equations with a parameter is considered. By partitioning the interval and introducing additional parameters, considered problem is reduced to the equivalent multipoint boundary value problem with parameters. To find the parameters introduced, the continuity conditions of the solution at the interior points of partition and boundary condition are used. For the fixed values of the parameters, the Cauchy problems for ordinary differential equations are solved. By substituting the Cauchy problem’s solutions into the boundary condition and the continuity conditions of the solution, a system of linear algebraic equations with respect to parameters is constructed. The solvability of this system ensures the existence of a solution to the original control problem. The system of linear algebraic equations is composed by the solutions of the matrix and vector Cauchy problems for ordinary differential equations on the subintervals. A numerical method for solving the origin control problem is offered based on the Runge-Kutta method of the 4-th order for solving the Cauchy problem for ordinary differential equations. Key words: boundary value problem with parameter, differential equation, solvability, algorithm.

scholarly journals A PROBLEM WITH PARAMETER FOR THE INTEGRO-DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 26 (1) ◽  
pp. 34-54
Author(s):  
Elmira A. Bakirova ◽  
Anar T. Assanova ◽  
Zhazira M. Kadirbayeva
Keyword(s):  
Differential Equation ◽  
General Solution ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Cauchy Problems ◽  
Algebraic Equations ◽  
Integro Differential Equation ◽  
Loaded Differential Equation ◽  
Linear Algebraic Equations

The article proposes a numerically approximate method for solving a boundary value problem for an integro-differential equation with a parameter and considers its convergence, stability, and accuracy. The integro-differential equation with a parameter is approximated by a loaded differential equation with a parameter. A new general solution to the loaded differential equation with a parameter is introduced and its properties are described. The solvability of the boundary value problem for the loaded differential equation with a parameter is reduced to the solvability of a system of linear algebraic equations with respect to arbitrary vectors of the introduced general solution. The coefficients and the right-hand sides of the system are compiled through solutions of the Cauchy problems for ordinary differential equations. Algorithms are proposed for solving the boundary value problem for the loaded differential equation with a parameter. The relationship between the qualitative properties of the initial and approximate problems is established, and estimates of the differences between their solutions are given.

scholarly journals ON A METHOD FOR SOLVING A LINEAR BOUNDARY VALUE PROBLEM FOR A SYSTEM OF INTEGRO-DIFFERENTIAL EQUATIONS CONTAINING THE PARAMETER

2021 ◽  
Vol 73 (1) ◽  
pp. 23-31
Author(s):  
N.B. Iskakova ◽  
◽  
G.S. Alihanova ◽  
А.K. Duisen ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Degenerate Kernel ◽  
Algebraic Equations ◽  
Integro Differential Equation ◽  
Linear Algebraic Equations ◽  
The Given

In the present work for a limited period, we consider the system of integro-differential equations of containing the parameter. The kernel of the integral term is assumed to be degenerate, and as additional conditions for finding the values of the parameter and the solution of the given integro-differential equation, the values of the solution at the initial and final points of the given segment are given. The boundary value problem under consideration is investigated by D.S. Dzhumabaev's parametrization method. Based on the parameterization method, additional parameters are introduced. For a fixed value of the desired parameter, the solvability of the special Cauchy problem for a system of integro-differential equations with a degenerate kernel is established. Using the fundamental matrix of the differential part of the integro-differential equation and assuming the solvability of the special Cauchy problem, the original boundary value problem is reduced to a system of linear algebraic equations with respect to the introduced additional parameters. The existence of a solution to this system ensures the solvability of the problem under study. An algorithm for finding the solution of the initial problem based on the construction and solutions of a system of linear algebraic equations is proposed.

scholarly journals On the Modified Boundary Value Problem of De La Vallée-Poussin for Nonlinear Ordinary Differential Equations

1994 ◽  
Vol 1 (4) ◽  
pp. 429-458
Author(s):  
G. Tskhovrebadze
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Nonlinear Ordinary Differential Equation ◽  
Nonlinear Ordinary Differential Equations ◽  
De La Vallée Poussin

Abstract The sufficient conditions of the existence, uniqueness, and correctness of the solution of the modified boundary value problem of de la Vallée-Poussin have been found for a nonlinear ordinary differential equation u (n) = f(t, u, u′, … , u (n–1)), where the function f has nonitegrable singularities with respect to the first argument.

scholarly journals ON THE ALGORITHM FOR SOLVING OF A LINEAR BOUNDARY VALUE PROBLEM FOR ORDINARY DIFFERENTIAL EQUATION WITH PARAMETER

2020 ◽  
Vol 70 (2) ◽  
pp. 71-76
Author(s):  
N.B. Iskakova ◽  
◽  
Zh. Kubanychbekkyzy ◽  
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Boundary Value ◽  
Linear Boundary ◽  
Existence Of A Solution ◽  
The Cauchy Problem ◽  
Linear Boundary Value Problem

A linear boundary value problem for a system of ordinary differential equations containing a parameter is considered on a bounded segment. For a fixed parameter value, the Cauchy problem for an ordinary differential equation is solved. Using the fundamental matrix of differential part and assuming uniqueness solvability of the Cauchy problem an origin boundary value problem is reduced to the system of linear algebraic equation with respect to unknown parameter. The existence of a solution to this system ensures the existence of a solution to the boundary value problem under study. The algorithm of finding of solution for initial problem is offered based on a construction and solving of the linear algebraic equation. The basic auxiliary problem of algorithm is: the Cauchy problem for ordinary differential equations. The numerical implementation of algorithm offered in the article uses the method of Runge-Kutta of fourth order to solve the Cauchy problem for ordinary differential equations.

scholarly journals On Discrete Solutions for Elliptic Pseudo-Differential Equations

2021 ◽  
Vol 103 (3) ◽  
pp. 117-123
Author(s):  
O.A. Tarasova ◽  
◽  
A.V. Vasilyev ◽  
V.B. Vasilyev ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Discrete Analogue ◽  
Continuous Solutions ◽  
Pseudo Differential Equation

We consider discrete analogue for simplest boundary value problem for elliptic pseudo-differential equation in a half-space with Dirichlet boundary condition in Sobolev–Slobodetskii spaces. Based on the theory of discrete boundary value problems for elliptic pseudo-differential equations we give a comparison between discrete and continuous solutions for certain model boundary value problem.

scholarly journals Nonlocal Inverse Problem for a Pseudohyperbolic- Pseudoelliptic Type Integro-Differential Equations

Axioms ◽  
2020 ◽  
Vol 9 (2) ◽  
pp. 45 ◽  
Author(s):  
Tursun K. Yuldashev
Keyword(s):  
Differential Equation ◽  
Inverse Problem ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Algebraic Equations ◽  
Spectral Parameters ◽  
Inverse Boundary ◽  
Inverse Boundary Value Problem ◽  
Infinite Set

The questions of solvability of a nonlocal inverse boundary value problem for a mixed pseudohyperbolic-pseudoelliptic integro-differential equation with spectral parameters are considered. Using the method of the Fourier series, a system of countable systems of ordinary integro-differential equations is obtained. To determine arbitrary integration constants, a system of algebraic equations is obtained. From this system regular and irregular values of the spectral parameters were calculated. The unique solvability of the inverse boundary value problem for regular values of spectral parameters is proved. For irregular values of spectral parameters is established a criterion of existence of an infinite set of solutions of the inverse boundary value problem. The results are formulated as a theorem.

scholarly journals AN ALGORITHM FOR SOLVING A BOUNDARY VALUE PROBLEM FOR ESSENTIALLY LOADED DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 2 (336) ◽  
pp. 6-14
Author(s):  
Zh. M. Kadirbayeva ◽  
E. A. Bakirova ◽  
A. Sh. Dauletbayeva ◽  
A. A. Kassymgali
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Numerical Implementation ◽  
Cauchy Problems ◽  
Linear Boundary ◽  
Algebraic Equations ◽  
Runge Kutta Method ◽  
The Cauchy Problem ◽  
Linear Boundary Value Problem

A linear boundary value problem for essentially loaded differential equations is considered. Using the properties of essentially loaded differential we reduce the considering problem to a two-point boundary value problem for loaded differential equations. This problem is investigated by parameterization method. We offer algorithm for solving to boundary value problem for the system of loaded differential equations. This algorithm includes of the numerical solving of the Cauchy problems for system of the ordinary differential equations and solving of the linear system of algebraic equations. For numerical solving of the Cauchy problem we apply the Runge–Kutta method of 4th order. The proposed numerical implementation is illustrated by example.

Boundary-Value Problems for Third-Order Lipschitz Ordinary Differential Equations

2014 ◽  
Vol 58 (1) ◽  
pp. 183-197 ◽  
Author(s):  
John R. Graef ◽  
Johnny Henderson ◽  
Rodrica Luca ◽  
Yu Tian
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Point Boundary ◽  
Third Order ◽  
Optimal Length ◽  
The Third

AbstractFor the third-order differential equationy′″ = ƒ(t, y, y′, y″), where, questions involving ‘uniqueness implies uniqueness’, ‘uniqueness implies existence’ and ‘optimal length subintervals of (a, b) on which solutions are unique’ are studied for a class of two-point boundary-value problems.

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