Special Session 39: Polynomial Methods for Di↵erential Equations and Dynamical Systems

Since the advent of Taylor Series, polynomial methods have been used to solve di↵erential equations and learn the properties of di↵erential equations. There are many polynomial methods for solving di↵erential equations and understanding dynamical systems. For example, Taylor polynomials, Chebyshev Polynomials and Adomian Polynomials are used to generate approximate solutions. Automatic di↵erentiation can be used with power series and Cauchy products to generate solutions to di↵erential equations. Pade Methods are based on ratios of polynomial functions and products of polynomials and are used to understand singularities in di↵erential equations. Picard Iteration can be used with polynomial di↵erential equations to obtain error estimates and convergence rates. This session intends to bring together researchers in these areas to develop a community and to share ideas and develop new tools and theory for understanding di↵erential equations and dynamical systems and their relationship with polynomials.