On the approximate solution of an autonomous boundary-value problem by the Newton–Kantorovich method

Abstract

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.

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].…”

A finite point algorithm for soil water-salt movement equation

“…Equation (3.5) repre- Figure 11 shows the simulated results of the one-dimensional vertical (right) and horizontal (left) infiltration for the given soil. The solid red line is the solution of this method, the dotted line represents the solution of the Hydrus software [13,14].…”

“…The central idea of FEM and CFEM is that a problem domain can be split into small, non-overlapping elements so that simple interpolation functions can be used to approximate any field function within each element. In addition, the Newton-Kantorovich method can also better solve the problems associated with partial differential equations [11][12][13][14]. However, shape functions for heavily distorted elements will not produce acceptable numerical solutions.…”