Picard iteration algorithm combined with Gauss–Seidel technique for initial value problems

In this paper we introduce and implement a relatively new improvement for the well known Picard’s method, for studying linear and nonlinear systems of ordinary differential equations. In this method, the solution takes the form of a rapidly convergent series with easily computable components. A mathematica, computer algebra system (mathematica 5.1) is used to introduce a code which generates the series. Different illustrative examples have been considered as well as different comparison criterion between the proposed method and modern, reliable and conventional methods. The deeply mathematical roots of the method “fixed point theorem” and the correspondence with the theory of algebraic systems makes it is very interesting. Theoretical considerations are being discussed, and convergence of the method is addressed. Other extensions and applications for further work are mentioned.