On Approximate Solutions of Nonlinear Boundary-Value Problems by the Newton–Kantorovich Method

Abstract

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. ⇤.�/ π D 0:(3)

A Quasilinearization Approach for Identification Control Vectors in Fractional-Order Nonlinear Systems

“…The Newton-Kantorovich method for solving both classical and fractional nonlinear initialboundary values and initial value problems is employed in many papers, see e.g., [21][22][23][24]. In [25], the quasi-Newton's method, based on the Fréchet derivative, is developed for solving nonlinear fractional order differential equations.…”