Rigorous Numerical Computation of Polynomial Differential Equations Over Unbounded Domains

Abstract

In this abstract we present a rigorous numerical algorithm which solves initial-value problems (IVPs) defined with polynomial differential equations (i.e. IVPs of the type $y'=p(t,y)$, $y(t_0)=y_0$, where p is a vector of polynomials) for any value of t. The inputs of the algorithm are the data defining the initial-value problem, the time T at which we want to compute the solution of the IVP, and the maximum allowable error $\varepsilon >0$. Using these inputs, the algorithm will output a value $\tilde{y}_T$ such that $\Vert \tilde{y}_T-y(T)\Vert \le \varepsilon$ in time polynomial in T, $-\log \varepsilon$, and in several quantities related to the polynomial IVP.