arXiv2025
We consider the first order autonomous differential equation (ODE) ${\bf x}'={\bf f}({\bf x})$ where ${\bf f}: {\mathbb R}^n\to{\mathbb R}^n$ is locally Lipschitz. For ${\bf x}_0\in{\mathbb R}^n$ and $h>0$, the initial value problem (IVP) for $({\bf f},{\bf x}_0,h)$ is to determine if there is a unique solution, i.e., a function ${\bf x}:[0,h]\to{\mathbb R}^n$ that satisfies the ODE with ${\bf x}(0)={\bf x}_0$. Write ${\bf x} ={\tt IVP}_{\bf f}({\bf x}_0,h)$ for this unique solution. We pose a corresponding computational problem, called the End Enclosure Problem: given $({\bf f},B_0,h,\varepsilon_0)$ where $B_0\subseteq{\mathbb R}^n$ is a box and $\varepsilon_0>0$, to compute a pair of non-empty boxes $(\underline{B}_0,B_1)$ such that $\underline{B}_0\subseteq B_0$, width of $B_1$ is $<\varepsilon_0$, and for all ${\bf x}_0\in \underline{B}_0$, ${\bf x}={\tt IVP}_{\bf f}({\bf x}_0,h)$ exists and ${\bf x}(h)\in B_1$. We provide a complete validated algorithm for this problem. Under the assumption (promise) that for all ${\bf x}_0\in B_0$, ${\tt IVP}_{\bf f}({\bf x}_0,h)$ exists, we prove the halting of our algorithm. This is the first halting algorithm for IVP problems in such a general setting. We also introduce novel techniques for subroutines such as StepA and StepB, and a scaffold datastructure to support our End Enclosure algorithm. Among the techniques are new ways refine full- and end-enclosures based on a {\bf radical transform} combined with logarithm norms. Our preliminary implementation and experiments show considerable promise, and compare well with current validated algorithms.