On the approximate solution of autonomous boundary-value problems by the Newton method

To the generalization of the Newton-Kantorovich theorem

“…The theorem just proved generalizes the corresponding results [2,3,4,6,7,8] to the case of matrix J k irreversibility and can be used in the theory of nonlinear Noetherian boundary-value problems [5,6,7,8], in the theory of stability of motion [10,11], in the theory of matrix boundary-value problems [12], and also in the theory of matrix linear differential-algebraic boundary value problem [13,14,15,16].…”

“…To construct an iteration scheme {z k }, that converges to the solution z ∈ R n , we use the Newton method [1,2,3]. Interest in the use of the Newton method is associated with its effective application in solving nonlinear equations, as well as in the theory of nonlinear oscillations [1,2,3,4], including in the theory of non-linear Noetherian boundary value problems [5,6,7,8].…”

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

“…( 1) obtained under condition (8) on the basis on the iterative scheme (10) does not lead to the appearance of secular terms. Unlike [16], to find the solution c."/ 2 R r of Eq. ( 4), we use a modification [13,14] of the traditional Newton-Kantorovich method [12].…”

“…Thus, the third approximation to the solution of the nonlinear Duffing equation ( 14) has the form: We compare the obtained zero-order and first three approximations to the periodic solution of the Duffing equation ( 14 The zero-order and first three approximations to the periodic solution of the Duffing equation ( 14) obtained by the Poincaré method give the following discrepancies: Hence, the obtained zero-order and first three approximations to the periodic solution of the Duffing equation ( 14) obtained by the iterative scheme (18) are more exact than the corresponding discrepancies of the first approximations obtained by the Poincaré method. Note that the investigated nonlinear periodic problem for the Duffing equation ( 14) is not weakly nonlinear unlike the best studied boundary-value problems for the ordinary differential equations [4,8,16]. In addition, unlike [20], in the construction of approximations to the solution of the periodic problem for the Duffing equation ( 14), the exact validity of the conditions of solvability guaranteeing the absence of secular terms is established in each step.…”