Effective computability of solutions of ordinary differential equations: the thousand monkeys approach

• Main Author: Collins, Pieter
• Publication Date: 2008
• Other Authors: Graça, Daniel
• Language: eng
• Source: Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
• Download full: http://hdl.handle.net/10400.1/1038

In this note we consider the computability of the solution of the initial- value problem for ordinary di erential equations with continuous right- hand side. We present algorithms for the computation of the solution using the \thousand monkeys" approach, in which we generate all possi- ble solution tubes, and then check which are valid. In this way, we show that the solution of a di erential equation de ned by a locally Lipschitz function is computable even if the function is not e ectively locally Lips- chitz. We also recover a result of Ruohonen, in which it is shown that if the solution is unique, then it is computable, even if the right-hand side is not locally Lipschitz. We also prove that the maximal interval of existence for the solution must be e ectively enumerable open, and give an example of a computable locally Lipschitz function which is not e ectively locally Lipschitz.