A Superexponentially Convergent Functional-Discrete Method for Solving   the Cauchy Problem for Systems of Ordinary Differential Equations
  In the paper a new numerical-analytical method for solving the Cauchy problem for systems of ordinary differential equations of special form is presented. The method is based on the idea of the FD-method for solving the operator equations of general form, which was proposed by V.L. Makarov. The sufficient conditions for the method converges with a superexponential convergence rate were obtained. We have generalized the known statement about the local properties of Adomian polynomials for scalar functions on the operator case. Using the numerical examples we make the comparison between the proposed method and the Adomian Decomposition Method. 
Makarov Volodymyr and Dragunov Denis
math.NA