Multiscale functions, scale dynamics, and applications to partial differential equations

Abstract

Abstract. -Modeling phenomena from experimental data, always begin with a choice of hypothesis on the observed dynamics such as determinism, randomness, derivability etc. Depending on these choices, different behaviors can be observed. The natural question associated to the modeling problem is the following : "With a finite set of data concerning a phenomenon, can we recover its underlying nature ? From this problem, we introduce in this paper the definition of multi-scale functions, scale calculus and scale dynamics based on the time-scale calculus (see [3]). These definitions will be illustrated on the multi-scale Okamoto's functions. IntroductionThis article deals with new mathematical tools to deal with scale phenomena and applications to partial differential equations. The framework that we have developed can be read from a mathematical point of view following each definitions and theorems. However, this framework can be seen as a synthesis of different tentative of one of the authors in order to deal with scales in geometry and analysis in the context of different physical problems (see [5,7,6,12,10]), in particular the scale relativity theory developed by L. Nottale [20,21,22], and more generally modelling problems. As a consequence, before coming to more mathematical considerations, we picture some important problems in modelling in Physics which are underlying our framework.Modelling a given phenomenon from experimental data using classical mathematical tools always assume, sometimes implicitly, a given framework hypothesis on the real nature of the phenomenon which can be also called the texture of reality. As an example, classical mechanics is developed using the classical differential calculus to write speed and acceleration MULTISCALE FUNCTIONS, SCALE DYNAMICS AND APPLICATIONS 3 of particle and implicitly assuming that the behaviour of these particles can be described using smooth curves on a given space. Depending on this framework, different behaviors will be predicted or not and will be confronted to reality. However, this assumption about the real nature of a phenomenon is in general not so easy to decide and in some sense depends on philosophical considerations (positivism, etc) which can not be proved. The classical debate between A. Einstein and N. Bohr about the nature of quantum physics is a famous example.The previous problem can be handled using a different approach, looking at the way mathematical models for a given phenomenon are constructed. Indeed, the framework question is in fact related to two different facts which are in general mixed in the literature. In order to put in evidence these points, we first remind very roughly the usual way to construct a model for a given phenomenon :-Acquiring experimental data.-Computations of relevant quantities (velocity, acceleration, etc).-Functional relation between these quantities (at a discrete level).-Asymptotic passage to a continuous model under a specific choice of hypothesis.-Comparison to reality using numerical simulations Putting ...

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