Spline functions and the cauchy problems. II

Abstract

1. Introduction and description of the method The problem of approximating the solution of the non-linear second order differential equations was always of special interest. The problem was solved by K. D. SHARMA and R. G. GUPTA [4] by a one-step method depending upon the Lobatto four-point quadrature formula in which the function f is necessary to be sufficiently differentiable. The same problem has been also solved by GH. MICULA [2], [3] using spline functions, but under the restrictions that the first derivative is absent i.e. y" =f(x, y) and that fC C 2 at least. He has constructed a spline function of degree m~3 which approximates the solution of the Cauchy problem y'=f(x, y), y(xo)=Y0, y'(xo)=Yo and his convergence theorem is as follows. THEOREM 1.1 (Micula). Let s:[a, blur be the constructed spline function. If fCC m-1 where m>-3, then there is an h0>0 such that for all H<=ho and for all xE[a, b], there exists the constants 1<1, Ks, K~ such that Is(x)-y(x)l < K1H", ls'(x)-y(x)l < K2H m-a, Is'(x)-y'(x)] < KsH m-1 where y(x) is the exact solution. More details on this theorem may be found in [3]. In this paper, following the same method presented in [5], we are going to approximate the solution of the prolem avoiding the restrictions in the above theorem and proving theorems of high rate of convergence and of best approximation. For this purpose, consider the Cauchy problem in the non-linear ordinary differential equation