Constructive polynomial approximation with a priori error bounds for nonlinear initial value differential problems

In this paper, we consider an initial value problem y' = f(x, y), y(0) = y( 0), where f is a continuous function satisfying a Lipschitz condition. Firs t, the function f(x,y) is approximated by a Bernstein polynomial in two var iables, B-n(f; x,y), of an appropriate degree according to a prescribed acc uracy. Second, by application of the Frobenius method to the initial value problem y' = B-n(f; x, y), y(0) = y(0), an exact series solution of the lat ter problem is constructed. Finally, the infinite series solution is trunca ted in order to obtain an explicit polynomial whose error with respect to t he exact solution of the original problem is less than the prescribed accur acy, in an existence and uniqueness domain which is determined in terms of the initial data. (C) 1999 Elsevier Science Ltd. All rights reserved.