Search

We investigate a weakly nonlinear boundary-value problem for a system of nonlinear ordinary differential equations with discontinuous right-hand side and periodic boundary conditions. A modified iterative scheme is proposed for the construction of approximate solutions of this problem. Unlike our previous results, this scheme is based on the refined expansion of the right-hand side of the analyzed differential equation and a new choice of the initial approximation. The proposed iterative procedure is applied to a mathematical model of nonisothermal chemical reaction with periodic switchings of the control functions. In the considered example, the convergence of our iterative scheme to the required solution is illustrated by the analysis of discrepancies in the boundary-value problem. We establish estimates for the range of a small parameter within which the operator used for the construction of the required solutions is contractive. The results of numerical simulations confirm the fact that the proposed computational scheme guarantees better accuracy than the method of simple iterations. [ABSTRACT FROM AUTHOR]

For the nonlinear periodic boundary-value problem posed for an ordinary differential equation in the critical and noncritical cases, we obtain constructive conditions of its solvability and propose a scheme for finding its solutions by using the Adomian decomposition method. [ABSTRACT FROM AUTHOR]

The investigation of differential-algebraic boundary-value problems was originated in the works by Weierstrass, Luzin, and Gantmakher. The works by Campbell, Boyarintsev, Chistyakov, Samoilenko, Perestyuk, Yakovets, Boichuk, Ilchmann, and Reis were devoted to the systematic study of differential-algebraic boundary-value problems. At the same time, the investigation of differential-algebraic boundary-value problems is closely connected with the analysis of linear boundary-value problems for ordinary differential equations originated in the works by Poincaré, Lyapunov, Krylov, Bogolyubov, Malkin, Myshkis, Grebenikov, Ryabov, Mitropolskii, Kiguradze, Samoilenko, Perestyuk, and Boichuk. The investigation of linear differential-algebraic boundary-value problems is connected with numerous applications of the corresponding mathematical models to the theory of nonlinear oscillations, mechanics, biology, radio-engineering, and the theory of stability of motion. Thus, the problem of generalization of the results obtained by Campbell, Samoilenko, and Boichuk to the case of nonlinear integrodifferential boundary-value problems unsolved with respect to the derivative is quite actual. In particular, this is true for the problem of finding necessary and sufficient conditions for the existence of solutions of nonlinear integrodifferential boundary-value problems unsolved with respect to the derivative. In the present paper, we establish conditions for the existence of solutions of the nonlinear integrodifferential boundary-value problem unsolved with respect to the derivative and present a constructive scheme for their finding. [ABSTRACT FROM AUTHOR]

We establish necessary and sufficient conditions of solvability of the nonlinear boundary-value problem in the critical case. By using the Newton–Kantorovich method, we propose a new iterative scheme for the construction of solutions of the weakly nonlinear boundary-value problem for a system of ordinary differential equations in the critical case. [ABSTRACT FROM AUTHOR]

We establish necessary and sufficient conditions for the solvability of the nonlinear boundary-value problem in the critical case and develop a scheme for the construction of solutions of this problem. By using the Newton–Kantorovich method, we propose a new iterative scheme for the determination of solutions to the weakly nonlinear boundary-value problem for a system of ordinary differential equations in the critical case. As examples of application of the constructed iterative scheme, we obtain approximations to the solutions of a periodic boundary-value problem for the Duffing and Liénard equations. To check the accuracy of the established approximations to the solutions of the periodic boundary-value problem for the Duffing and Liénard equations, we use discrepancies in the original equations. [ABSTRACT FROM AUTHOR]

We establish constructive conditions for the solvability of a nonlinear autonomous boundary-value problem in the presence of parametric resonance and develop a scheme for the construction of solutions of this problem. We propose a convergent iterative algorithm for finding approximate solutions of the nonlinear autonomous Noetherian boundary-value problem for a system of ordinary differential equations in the case of parametric resonance. As an example of application of the constructed iterative algorithm, we determine approximate solutions of the periodic boundary-value problem for the autonomous Duffing-type equation with parametric perturbation. [ABSTRACT FROM AUTHOR]

We establish necessary and sufficient conditions for the existence of solutions of a weakly nonlinear Noetherian boundary-value problem for a system of second-order ordinary differential equations in the critical case. [ABSTRACT FROM AUTHOR]

We establish the conditions of regularization of the linear boundary-value problem for a system of impulsive ordinary differential equations. We also construct a generalized Green's operator and determine the form of linear impulsive perturbation of the regularized linear boundary-value problem. [ABSTRACT FROM AUTHOR]

We establish necessary and sufficient conditions for the existence of solutions of a nonlinear autonomous Noetherian boundary-value problem for a system of second-order ordinary differential equations in a special critical case. The specific feature of the considered problem is the inapplicability of the traditional scheme developed in the works of Malkin, Samoilenko, Grebenikov, Ryabov, and Boichuk to the investigation of the critical boundary-value problem and its solution. For the construction of solutions of a nonlinear Noetherian boundary-value problem in a special critical case, we propose a scheme that combines the Newton method and the least-squares method. The efficiency of the proposed approach is illustrated by an example of a periodic problem for a Hill-type equation. [ABSTRACT FROM AUTHOR]

We establish necessary and sufficient conditions for the existence of solutions of an autonomous Noetherian boundary-value problem for a system of second-order ordinary differential equations in the critical case. For the construction of solutions of a nonlinear Noetherian boundary-value problem in the critical case, we propose an iterative scheme that combines the Newton-Kantorovich method and the leastsquares technique. The efficiency of the proposed method is demonstrated in the analysis of a periodic problem for a Liénard-type equation. [ABSTRACT FROM AUTHOR]