The author deals with the numerical solving the initial value problem \[ x'(t) = F(x(t)), \quad t\in[0,T]; \qquad x(0)=x_0, \tag{1} \] with \(F:\Omega\to \mathbb{R}^n\), \(\Omega\) being an open subset of \(\mathbb{R}^n\). To construct the approximations desired, the so-called optimal derivative procedure developed in previous works of the authors is used [Appl. Math. Comput. Sci. 5, No. 1, 33-48 (1995; Zbl 0823.34053) and Comput. Math. Appl. 31, No. 8, 69-84 (1996; Zbl 0855.65066)]. The original interval is partitioned into sufficiently many subdivisions. In every subinterval, say, \([t_k, t_{k+1}]\), the right-hand side of equation (1) is ‘centered’ around the approximate value of \(x(t_k)\) by introducing an appropriate change of variable. The resulting nonlinear problem is then replaced by a series of linear initial value problems, each of which is constructed through minimizing a certain functional. In the ‘limit’, the optimal, in the variational sense, approximation to the solution sought for is obtained.
A corresponding computational algorithm is described. On the assumption that \(F\) satisfies a one-sided Lipschitz condition, an error estimate is established.