Differential Equations and Various Notions of Derivative

Lukáš Malý received his Ph.D. in Mathematics at Linköping University in 2014. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position in Professor Nageswari Shanmugalingam’s research group at the University of Cincinnati, Ohio, USA.

Many phenomena occurring in nature can be described by means of differential equations. For example the temperature of a body can be described by the so-called heat equation, a partial differential equation with the position in the body and the time as variables.

In most cases, after imposing additional constraints, one can show that there exists a unique solution of a differential equation. However, explicit solutions are very difficult to find, and not known in most cases. On the other hand one can find an approximation of the solution through numerical computer calculations. The drawback of such an approximation is the difficulty in estimating how far from the exact solution the computed result is. The exact solution can be a complicated function which undergoes wild swings and diverges significantly from its approximation. It is therefore important to find a way to analyze properties of the solution.

The magnitude of the values the solution function assumes, and the velocity of its swings are some of the important properties one tries to assess in order to reach better understanding of the accuracy of the numerical approximation. There exist several methods to estimate such accuracy. Lukáš Malý’s project will research theoretical problems in the rich theory of such methods.