Let $(\alpha ,\beta )\subseteq \mathbb {R}$ denote the maximal interval of existence of solutions for the initial-value problem \[ $\left \{ \begin {array} [c]{l}\frac {dx}{dt}=f(t,x), x(t_{0})=x_{0}, \end {array} \right .$ \] where $E$ is an open subset of $\mathbb {R}^{m+1}$ , $f$ is continuous in $E$ and $(t_{0},x_{0})\in E$ . We show that, under the natural definition of computability from the point of view of applications, there exist initial-value problems with computable $f$ and $(t_{0},x_{0})$ whose maximal interval of existence $(\alpha ,\beta )$ is noncomputable. The fact that $f$ may be taken to be analytic shows that this is not a lack of the regularity phenomenon. Moreover, we get upper bounds for the “degree of noncomputability” by showing that $(\alpha ,\beta )$ is r.e. (recursively enumerable) open under very mild hypotheses. We also show that the problem of determining whether the maximal interval is bounded or unbounded is in general undecidable.

The theory of generalized functions is the foundation of the modern theory of partial differential equations (PDE). As computers are playing an ever-larger role in solving PDEs, it is important to know those operations involving generalized functions in analysis and PDE that can be computed on digital computers. In this article, we introduce natural concepts of computability on test functions and generalized functions, as well as computability on Schwartz test functions and tempered distributions. Type-2 Turing machines are used as the machine model [Weihrauch 2000]. It is shown here that differentiation and integration on distributions are computable operators, and various types of Fourier transforms and convolutions are also computable operators. As an application, it is shown that the solution operator of the distributional inhomogeneous three dimensional wave equation is computable.

The Lorenz attractor was introduced in 1963 by E. N. Lorenz as one of the first examples of strange attractors. However, Lorenz’ research was mainly based on (non-rigorous) numerical simulations, and, until recently, the proof of the existence of the Lorenz attractor remained elusive. To address that problem some authors introduced geometric Lorenz models and proved that geometric Lorenz models have a strange attractor. In 2002 it was shown that the original Lorenz model behaves like a geometric Lorenz model and thus has a strange attractor. In this paper we show that geometric Lorenz attractors are computable, as well as show their physical measures.