## Picard Iteration and Differential Equations: A New Approach?

- May 26, 1999

## Abstract

In general, the Picard iterates are difficult to obtain. However, if one poses the ordinary differential equation as an autonomous polynomial system with initial conditions at $t _ 0 = 0$ then the Picard iterates are straight forward to calculate. I discuss the types of ordinary differential equations that can be transformed to polynomial systems and the properties of the class of functions that solve these types of ordinary differential equations. We look at several important examples including hyperbolic partial differential equations without boundary conditions. The Picard method is robust on these types of ordinary differential equations. We look at computational considerations when using the method including parallelization.

*This work is joint with G. Edgar Parker.