Dzhumabaev D. S.
New general solutions of ordinary differential equations and the methods of solving the boundary-value problems
Ukr. Mat. Zh. - 2019. - 71, № 7. - pp. 884-905
New general solutions of ordinary differential equations are introduced and their properties are established. We develop new methods of solving the boundary-value problems based on the construction and solving of the systems of algebraic equations for arbitrary vectors of the general solutions. An approach to finding the initial approximation to the solution of a nonlinear boundary-value problem is proposed.
Ukr. Mat. Zh. - 2018. - 70, № 3. - pp. 356-365
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 sufficient conditions for the existence of an isolated solution to the boundary-value problem under consideration.
Ukr. Mat. Zh. - 2017. - 69, № 12. - pp. 1717-1722
On a finite interval, we consider a system of nonlinear ordinary differential equations with a singularity at the left endpoint of the interval. The definition of weighted limit solution is introduced and its attracting property is established.
Properties of the isolated solutions bounded on the entire axis for a system of nonlinear ordinary differential equations
Ukr. Mat. Zh. - 2016. - 68, № 8. - pp. 1132-1138
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.
Necessary and Sufficient Conditions for the Solvability of Linear Boundary-Value Problems for the Fredholm Integrodifferential Equations
Ukr. Mat. Zh. - 2014. - 66, № 8. - pp. 1074–1091
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 regular partitions is presented. It is shown that the set of regular partitions is nonempty. A criterion for the solvability of the analyzed problem is established and an algorithm for finding its solutions is constructed.
Ukr. Mat. Zh. - 2004. - 56, № 4. - pp. 562-572
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 conditions for the unique solvability of the problems under consideration.