Introduction to Diffusive Representation

Diffusive representation is an operator theory elaborated during the last years. It is devoted to time-nonlocal problems, allowing significant simplifications for analysis and numerical realization of integral time operators encountered in many physical situations. Namely, most of the shortcomings induced by time-nonlocal formulations are by-passed by use of suitable time-local state realizations deduced from diffusive representation and whose numerical approximations are straightforward, thanks to good properties of diffusion equations. In this paper, we introduce the notion of diffusive representation and its dual one: the diffusive symbol, and we briefly describe the associated mathematical framework and numerical techniques. The interest of this approach is highlighted by numerical examples.