Uniform polynomial approximation to solutions of the Cauchy problem for O. D. E (Q2726128)

Several polynomial sequences that approximate uniformly the solution of some scalar initial value problems for ordinary differential equations (O.D.E.s) are proposed. Starting with Chaplygin's theorem in which the solution \( y(x)\) of an initial value problem for the equation \( y'(x) = f(x, y(x)) \) with \( \partial^2 f / \partial^2 y < 0\) is bounded by two sequences \((u_n)\) and \( (v_n)\) so that \( u_n < y < v_n \) which converge geometrically to \( y \), the authors propose other alternative sequences defined recursively by means of integral operators which converge uniformly to the solution. NEWLINENEWLINENEWLINEIn particular they consider sequences of Bernstein polynomials which, under suitable restrictions, converge uniformly to the solution of the initial value problem.