To Infinity and Beyond: Some ODE and PDE Case Studies

TL;DR

This paper explores methods to extend analysis and computation beyond infinity in differential equations using singular transformations, enabling solutions to remain well-behaved and computations to continue seamlessly across infinity.

Contribution

It introduces a systematic approach employing singular transformations to bypass infinity in ODE and PDE problems, including regularization via complexification and compactification techniques.

Findings

Infinity crossing can be instantaneous in ODEs

Finite-time infinity crossing in PDEs requires buffer zones

Transformations enable continued well-behaved solutions

Abstract

When mathematical/computational problems reach infinity, extending analysis and/or numerical computation beyond it becomes a notorious challenge. We suggest that, upon suitable singular transformations (that can in principle be computationally detected on the fly) it becomes possible to go beyond infinity to the other side, with the solution becoming again well behaved and the computations continuing normally. In our lumped, Ordinary Differential Equation (ODE) examples this infinity crossing can happen instantaneously; at the spatially distributed, Partial Differential Equation (PDE) level the crossing of infinity may even persist for finite time, necessitating the introduction of conceptual (and computational) buffer zones in which an appropriate singular transformation is continuously (locally) detected and performed. These observations (and associated tools) could set the stage for a systematic approach to bypassing infinity (and thus going beyond it) in a broader range of evolution equations; they also hold the promise of meaningfully and seamlessly performing the relevant computations. Along the path of our analysis, we present a regularization process via complexification and explore its impact on the dynamics; we also discuss a set of compactification transformations and their intuitive implications.