Abstract
A concept of divided difference operators (DDO) is presented and combined with Newton-like methods for solving non-linear operator equations in high-dimensional spaces. The general idea of the concept lies in the approximation of local variations of non-linear operators with the aid of suitably fitted linear DDO's. The method is closely related to the conventional secant, regula falsi, and Steffensen's methods. The operator equations concerned are composed of ordinary first-order differential equations together with two-point boundary conditions. The method is tested in a numerical example originating in an optimal control problem for the classical Van der Pol system.
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