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) = y0, where f is a continuous function satisfying a Lipschitz condition. First, the function f(x,y) is approximated by a Bernstein polynomial in two variables, Bn(f; x,y), of an appropriate degree according to a prescribed accuracy. Second, by application of the Frobenius method to the initial value problem y′ = Bn(f; x,y), y(0) = y0, an exact series solution of the latter problem is constructed. Finally, the infinite series solution is truncated in order to obtain an explicit polynomial whose error with respect to the exact solution of the original problem is less than the prescribed accuracy, in an existence and uniqueness domain which is determined in terms of the initial data