## Differentiating discontinuous functions and the modern generalization of differential equations

- Date: 03/29/2007

Cristian Rios (University of Calgary)

Calgary Place Tower (Shell)

Any professional interested in mathematical modeling of physical processes or industrial problems will no doubt find this lecture invaluable. The topic to be presented, the theory of generalized functions, is crucial for the treatment of many problems written in the language of differential equations. This language was developed from the beginnings of Calculus to express observed physical phenomena in a mathematical way.

In classical models, the physical world is modeled as a continuum, and the objects in study are thought as infinitely divisible and observable with arbitrarily good accuracy. In real life, physical phenomena are only observable to a maximum degree of precision dictated by the limitations of the instruments used or even by uncertainty principles inherent to the very nature of reality. Using the classical tools derived from Calculus, it is not only necessary to adopt this continuum model but often the quantities in study must satisfy regularity properties, they must show a certain degree of "smoothness". In many situations these assumptions are impractical and several important problems are not treatable using this classic approach to modeling.

Physicists, staring with the work of Dirac, again solved this shortcoming of the classical theory by introducing new objects (now called distribution or generalized functions) based in their physical intuition. This more modern approach opened the door to treat all sort of models where the smoothness assumptions are more relaxed, allowing for discontinuities and other types of singularities.

In this Lunchbox Lecture we will give a non-technical introduction to the theory of generalized functions and their application to mathematical modeling. We will show how the most basic Calculus tools provide the natural background for this more modern approach. A few examples will hopefully show that this framework is very intuitive perhaps more appropriate than the classical continuum approach.

PIMS is presenting a series of lectures at the Calgary Place Tower 1 in downtown Calgary. These lectures, given by experts from the PIMS Universities, will focus on mathematical techniques and applications relevant to the oil and gas industry and will demonstrate the utility and beauty of applied mathematics. The talks are aimed at a general audience. Attendance may qualify for APEGGA Professional Development Hours.