Many of the systems studied by scientists, engineers, and applied mathematicians exhibit behavior that is characterized by long periods of slow change, punctuated by brief intervals of rapid change. Some commonly used names for these systems are "singularly perturbed systems", or "systems with multiple time scales".
Examples of singularly perturbed systems in fluid dynamics, chemistry, mechanics, biology, and other sciences can be found throughout the scientific literature. For example, in recent decades there has been an explosion of interest in the mathematical modeling of neurons. Models such as the Hodgkin-Huxley equations (and their simplifications) for the voltage across the cell membrane of a neuron are often studied in singularly perturbed regimes.
Because it is rare for the equations that model these systems to have explicit analytical solutions, they are often analyzed with either analytical approximations or computer simulations. There are powerful analytical tools that provide approximate solutions to singularly perturbed problems. Unfortunately, the analysis required to derive an approximate solution by these methods can be intractable. One can resort to computer simulations when analysis fails, but singularly perturbed systems are notoriously difficult to solve by computer simulation.
In this project, we will create a collection of computer programs that avoid the computational difficulties of solving the full singularly perturbed system by instead solving the slow and fast phases separately. The slow and fast solutions will be combined to create an approximate solution to the full system. The approximations generated will not be as accurate as those found by solving the original system, but they will be faster to compute. They will be accurate enough to answer many important questions, and they will also help researchers to determine where they should apply more sophisticated methods.
