For a constructive analysis of the periodic boundary value problem for systems of non-linear non-autonomous ordinary differential equations, a numerical-analytic approach is developed, which allows one to both study the solvability and construct approximations to the solution. An interval halving technique, by using which one can weaken significantly the conditions required to guarantee the convergence, is introduced. The main assumption on the equation is that the non-linearity is locally Lipschitzian.

/pdf/constructive-analysis-of-periodic-solutions-with-interval-p32er8n4xk.pdf

Constructive analysis of periodic solutions with interval halving

We analyze the application of the numerical-analytic method proposed by A.M. Samoilenko in 1965 to multipoint boundary-value problems.

The theory of the numerical-analytic method: Achievements and new trends of development. VI

The article deals with approximate solutions of a nonlinear ordinary differential equation with homogeneous Dirichlet boundary conditions. We provide a scheme of numerical-analytic method based upon successive approximations constructed in analytic form. We give sufficient conditions for the solvability of the problem and prove the uniform convergence of the approximations to the parameterized limit function. We provide a justification of the polynomial version of the method with several illustrating examples. 2000 Mathematics Subject Classification: 34B15; 65L10.

/pdf/numerical-analytic-technique-for-investigation-of-solutions-d3xk2218ce.pdf

Numerical-analytic technique for investigation of solutions of some nonlinear equations with Dirichlet conditions

Численно-аналитический метод исследования периодическин систем обыкновенных дифференциальных уравнений. II

Nonconservative mechanical systems with one degree of freedom are considered. The goal is to provide the existence of steady-state oscillations with the prescribed properties. The system’s behavior is modeled by a second-order autonomous dynamical system with one variable parameter describing the amplifying coefficient of the control action. A numerical-analytic method to find the amplifying coefficient is proposed. Conditions of the orbital stability are obtained for the steady-state oscillations. An example of the application of the method is provided. The proposed approach can be applied to solve control problems and to find periodic solutions of second-order autonomous dynamical systems.

Method to Construct Periodic Solutions of Controlled Second-Order Dynamical Systems

Численно-аналитический метод исследования периодических систем обыкновенных дифференциальных уравнений. I

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Численно-аналитический метод исследования периодических систем обыкновенных дифференциальных уравнений. I