D. S. Dzhumabaev

• Professor
• Ministry of Education and Science of the Republic of Kazakhstan

The paper is concerned with a nonlocal problem for a partial Fredholm integro-differential equation (IDE) of hyperbolic type is investigated. This problem is reduced to a problem containing a family of boundary value problems for the ordinary Fredholm IDEs and some integral relationships. A novel concept of general solution to a family of the ordin...

New general solutions of ordinary differential equations are introduced and their properties are established. We develop new methods for the solution of boundary-value problems based on the construction and solution of the systems of algebraic equations for arbitrary vectors of the general solutions. An approach to finding the initial approximation...

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 th...

A nonlinear two-point boundary-value problem for an ordinary differential equation is studied by the method of parametrization. We construct systems of nonlinear algebraic equations that enable us to find the initial approximation to the solution to the posed problem. In terms of the properties of constructed systems, we establish necessary and suf...

On a finite interval, we consider a system of nonlinear ordinary differential equations with singularity at the left endpoint of the interval. The definition of weighted limit solution is introduced and its attracting property is established.

The article introduces a new general solution to a family of loaded ordinary differential equations and discusses its properties. It provides necessary and sufficient conditions for the well-posedness of a linear nonlocal boundary value problem for a system of loaded hyperbolic equations with mixed derivatives. Algorithms for solving the boundary v...

The article presents a new general solution to a loaded differential equation and describes its properties. Solving a linear boundary value problem for loaded differential equation is reduced to the solving a system of linear algebraic equations with respect to the arbitrary vectors of general solution introduced. The system's coefficients and righ...

This paper introduces the ΔN general solution to linear Fredholm integro-differential equations and sets its properties. The conditions for existence of classical general solution and solvability criteria for the equation are provided. Necessary and sufficient conditions for solvability of linear boundary value problems are established. Algorithms...

We establish the conditions of continuous dependence on the right-hand side for the “isolated” solutions of a system of nonlinear ordinary differential equations bounded on the entire axis.

We study a linear boundary-value problem for systems of Fredholm integrodifferential equations with degenerate kernels and present the definition of ν-regular partition of the interval. The coefficient necessary and sufficient conditions for the unique solvability of the considered problem are established.

Necessary and sufficient conditions for the well-posedness of linear boundary value problems for Fredholm integro-differential equations are established. Algorithms for finding their solutions are offered.

Abstract—We suggest a method for the study and solution of a linear boundary value problem for a Fredholm integro-differential equation with impulsive inputs at given times. The method is based on a partition of the interval and the introduction of auxiliary parameters as the values of the solution at the initial points of subintervals. For each pa...

We propose a method for the investigation and solution of linear boundary-value problems for the Fredholm integrodifferential equations based on the partition of the interval and introduction of additional parameters. Every partition of the interval is associated with a homogeneous Fredholm integral equation of the second kind. The definition of re...

A nonlinear two-point boundary-value problem for a system of ordinary differential equations is studied by splitting the interval and introducing additional parameters. We construct a system of equations with respect to the parameters, which enable us to determine the initial approximation to the solution of the boundary-value problem. We establish...

A nonlocal boundary value problem with integral condition for a system of hyperbolic equations is considered. Relationship with the family of boundary value problems for ordinary differential equations is established. Sufficient and necessary conditions of well-posedness of nonlocal boundary value problems with integral condition for a system of hy...

family of algorithms for solving a linear two-point boundary value problem is constructed in terms of the data of the integrodifferential equation and the boundary condition involved. The convergence conditions for the algorithms are established, and necessary and sufficient conditions for the well-posedness of the problem are found.

A method for solving the linear boundary value problem for an integro-differential equation is proposed that is based on interval partition and the introduction of additional parameters. Necessary and sufficient conditions for the solvability of the problem are obtained.

We consider a linear two-point boundary value problem for systems of integro-differential equations. By using the parametrization method and an approximation of the integro-differential equation by a loaded differential equation, we establish coefficient tests for the well-posedness of the considered problem and suggest an algorithm for finding the...

The paper considers the problem of finding a bounded solution of a one-parametric family of systems of ordinary differential equations. Using the parametrization method, the author proves necessary and sufficient conditions for the existence of a unique solution of the problem considered that is bounded on the whole axis in terms of a two-sided, in...

A sharper version of the local Hadamard theorem on the solvability of nonlinear equations is proved. Additional parameters are introduced, and a two-parameter family of algorithms for solving nonlinear two-point boundary value problems is proposed. Conditions for the convergence of these algorithms are given in terms of the initial data. Using the...

For a linear system of hyperbolic equations of the second order with two independent variables, we investigate the problem of the existence and uniqueness of a solution periodic in both variables and a solution periodic in one of the variables and bounded on a plane. By using the method of introduction of functional parameters, we obtain sufficient...

The aim of the paper is to consider a set of the hyperbolic linear (HL) equations, having the limited on the band solutions. In particular, the necessary and sufficient conditions are established for existence, uniqueness and continuous dependence on the vector-function of the classical solution of the set of these HL equations. The solution is fou...

Sufficient conditions for a well-posed solvability of the problem of finding solutions to systems of linear hyperbolic equations that are bounded in a strip are obtained in terms of bilaterally infinite functional matrices. Boundary value problems with data on characteristics in a finite domain that approximate the problem under consideration are c...

The conditions under which all the solutions of a non-linear ordinary differential equation contained in a certain sphere will coincide as t→∞ with a solution "in the limit as t→∞" are established. Lyapunov transformations of the linearized equations and solutions in the limit as t→∓∞ are used to construct regular two-point boundary-value problems...

Lyapunov transformations possessing certain properties are used to construct regular two-point boundary-value problems as approximations to the problem of determining a bounded solution in the general case. The concept of “limiting solutions as t→∞” is defined and the behaviour of solutions of linear ordinary differential equations as t→∞ is invest...

The method of parametrization is used to obtain necessary and sufficient conditons for the correct solvability of the problem of finding the solution bounded on the entire real axis. For systems whose matrices and right-hand sides are constant in the limit, two-point bounary-value problems which approximate the problem in question are constructed....

The parametrization method is used to study a two-point boundary-value problem. A mutual relationship is established between the unique solvability of the problem and invertibility of a matrix Qv(h) which depends on the boundary conditions and the matrix of the differential equation. On the basis of recurrence formulae for the inversion of Qv(h), n...